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Asymptotics of convergence to a wave travelling from a saddle to a node L. A. Kalyakin Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences
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    1. IntroductionThe notion of a travelling wave is used in the study of mathematical models for a broad range of applied problems. There is a number of monographs treating the problem of the use of travelling waves for various applications [1]–[4]. Based on the recent survey [5], we can say that this topic also remains relevant now. An extensive bibliography concerning various aspects of the theory of travelling waves was presented in [6] and [7]. These two large works by physicists contain interesting results and new ideas. They initiated the research presented in our paper. Original mathematical problems are commonly stated for differential or integro-differential equations of different types. For example, mathematical simulation in biology and chemistry (in particular, the theory of combustion) leads to semilinear parabolic equations associated with the names of Kolmogorov, Petrovskii, and Piskunov (the KPP equations) [8], [9]. In magnetodynamical problems travelling waves are used to describe the dynamics of the domain wall; the original equation is hyperbolic [10], [11]. The transition to a travelling wave leads to an ordinary differential equation, and this operation looks trivial. The problem simplified in this way is of little interest since we obtain only an isolated solution of the original partial differential equation. Problems of the relationship between a travelling wave and other solutions of the original equation turn out to be more interesting both for mathematics and applications. For instance, studying the stability of isolated solutions of this kind with respect to perturbations of different types uses deep mathematical results and has undoubted practical applications. Another area is related to the use of travelling waves to study the asymptotic behaviour of other, close solutions of original equations. The rigorous mathematical results on this topic obtained already in the first paper [8] remain a standard that has not been surpassed in most cases till the present time. For example, the fundamental problem of a logarithmic phase shift as the solution converges to a travelling wave has rigorously been solved only for the KPP equation [12]–[14]. 1.1. The original equationsThis paper considers the semilinear partial differential equation
$$
\begin{equation}
\begin{gathered} \, \delta\,\frac{\partial^2\varphi}{\partial t^2}- \frac{\partial^2\varphi}{\partial x^2}+\frac{\partial\varphi}{\partial t} +f(\varphi)=0, \\ x\in\mathbb{R},\quad t>0;\qquad \delta=\operatorname{const}\geqslant0, \nonumber \end{gathered}
\end{equation}
\tag{1.1}
$$
which can be hyperbolic or parabolic, depending on the value of the parameter $\delta\geqslant0$. A travelling wave is understood as a function of one variable (the phase $s$) of the form $x-S(t)$. A solution $\varphi=\Phi(s)$ in the form of a wave travelling at constant velocity $S'(t)\equiv V=\operatorname{const}>0$ is derived from an ordinary differential equation. In the hyperbolic case it is convenient to use the normalized variable thus, the travelling wave function $\Phi(s)$ satisfies the equation
$$
\begin{equation}
\frac{d^2\Phi}{ds^2}+\gamma\,\frac{d\Phi}{ds}-f(\Phi)=0.
\end{equation}
\tag{1.2}
$$
The coefficient $\gamma>0$ and the velocity $V$ are related by the equalities
$$
\begin{equation*}
\gamma=\frac{V}{\sqrt{1-\delta V^2}}\quad \Longleftrightarrow\quad V=\frac{\gamma}{\sqrt{1+\gamma^2\delta}}\,.
\end{equation*}
\notag
$$
The quantity $\gamma>0$ is assumed to be real and positive, which corresponds to ‘sublight’ wave velocities $0<V<1/\sqrt\delta$ . The purpose of this study is to find asymptotic formulae as $t\to\infty$ for solutions $\varphi(x,t)$ in which the leading term is described by $\Phi(s)$ with an appropriate phase $s$. 1.2. Equilibria and stabilityThe phase portraits of equation (1.2) for various positive constants $\gamma$ give comprehensive information on the structure of all possible waves travelling at constant velocities. In particular, fixed points correspond to equilibria $\Phi_{\rm st}=\operatorname{const}$, for which $f(\Phi_{\rm st})=0$. Constants of this kind are homogeneous solutions $\varphi(x,t)\equiv\Phi_{\rm st}$ of the original equation (1.1). These equilibrium states can be stable or unstable dynamically (in time) with respect to small perturbations of the solution. Stability in linear approximation is established by analyzing the corresponding linearized equation
$$
\begin{equation*}
\delta\,\frac{\partial^2\varphi}{\partial t^2}- \frac{\partial^2\varphi}{\partial x^2}+\frac{\partial\varphi}{\partial t} +f'(\Phi_{\rm st})\varphi=0,\qquad x\in\mathbb{R},\quad t>0.
\end{equation*}
\notag
$$
In the class of perturbations independent of $x$ we must study the ordinary differential equation
$$
\begin{equation*}
\delta\,\frac{d^2\varphi}{d t^2}+\frac{d\varphi}{dt}+ f'(\Phi_{\rm st})\varphi=0.
\end{equation*}
\notag
$$
The presence or absence of stability is governed by the structure of solutions $\varphi(t)=\exp(\lambda t)$ and depends on the sign of the real part of the characteristic roots
$$
\begin{equation*}
\begin{alignedat}{2} \lambda_{1,2}&=\frac{1}{2\delta} \bigl(-1\pm\sqrt{1-4\delta f'(\Phi_{\rm st})}\,\bigr)&\quad \text{for}\ \ \delta&\ne0, \\ \lambda&=-f'(\Phi_{\rm st})&\quad\text{for}\ \ \delta&=0. \end{alignedat}
\end{equation*}
\notag
$$
Negative real parts of both roots correspond to the exponential stability of the equilibrium; the case when at least one root has a positive real part corresponds to instability. The stability (instability) property is obviously determined by the sign of the derivative at the equilibrium point: $f'(\Phi_{\rm st})>0$ or $f'(\Phi_{\rm st})<0$. This conclusion remains valid for the wider class of perturbations that can be represented as Fourier series or integrals. In this case the equation linearized on solutions
$$
\begin{equation*}
\varphi(x,t)=\exp(\lambda t+i\kappa x),\qquad \kappa=\operatorname{const}\in\mathbb{R},
\end{equation*}
\notag
$$
leads to the dispersion relation
$$
\begin{equation*}
\delta\lambda^2+\lambda+\kappa^2+f'(\Phi_{\rm st})=0.
\end{equation*}
\notag
$$
The instability is the case when $f'(\Phi_{\rm st})<0$ on harmonics for not too large $\kappa^2(<-f'(\Phi_{\rm st}))$. 1.3. Waves connecting equilibriaWaves describing dynamical transitions from one equilibrium to another are of interest for applications. As we can shift and normalize the variable $\varphi$, we can assume that a pair of equilibria coincides with $\varphi\equiv0$ and $\varphi\equiv1$. Then a wave travelling from one equilibrium to the other is distinguished by the boundary conditions
$$
\begin{equation}
\Phi(s)\to 0\quad \text{as}\ \ s\to-\infty\quad\text{and}\quad \Phi(s)\to 1\quad \text{as}\ \ s\to+\infty.
\end{equation}
\tag{1.3}
$$
Equation (1.2) with conditions (1.3) is a nonlinear analogue of the spectral problem: we need to find the velocity $V$ and the corresponding solution $\Phi_v(s)$. The function $\Phi_v(s)$ is defined up to a constant phase shift $s\ \Rightarrow\ s+\operatorname{const}$. Waves describing the replacement of an unstable equilibrium by a stable one are of particular interest. In this case the spectrum of velocities $V$ turns out to be continuous.1 Just this situation, fixed by the conditions $f'(0)>0$ and $f'(1)<0$, is considered in this paper. The value of $f'(\varphi)$ at one equilibrium can be made equal to a prescribed quantity by normalizing the independent variables $x$ and $t$ in the original equation (1.1). For less cumbersome formulae, we assume in what follows that the derivative at the unstable equilibrium is equal to $-1$: $f'(1)=-1$. We must take into account that for the travelling wave equation (1.2) the stability properties of the same equilibria are different. For the equation linearized in a neighbourhood of $\Phi=0$, the characteristic roots
$$
\begin{equation*}
\lambda^0_{\pm}=-\frac{\gamma}{2}\pm\frac{1}{2}\sqrt{\gamma^2+4f'(0)}\,,\qquad f'(0)>0,\quad \gamma=\frac{V}{\sqrt{1-\delta V^2}}\,,
\end{equation*}
\notag
$$
correspond to the unstable fixed point $(0,0)$ of saddle type. For the equilibrium $\Phi=1$ the characteristic roots
$$
\begin{equation*}
\lambda_{\pm}=-\frac{\gamma}{2}\pm\frac{1}{2}\sqrt{\gamma^2-4} \quad \text{for} \ \ f'(1)=-1
\end{equation*}
\notag
$$
correspond to the fixed point $(1,0)$ of node or focus type (depending on the value of $\gamma$). This equilibrium is stable for equation (1.2) for any $\gamma>0$. On the phase plane with coordinates $(\Phi,\dot\Phi)$ a solution $\Phi_v(s)$ satisfying (1.3) corresponds to a trajectory connecting a saddle and a node (a focus). A solution of this type is said to be a wave on a saddle-node (saddle-focus) trajectory. The presence of such a trajectory is a robust property preserved under perturbations of the equation ([15], p. 80). Therefore, there is a continuous spectrum of velocities $V$ for which the boundary value problem (1.2), (1.3) is solvable.2 The solution of the ordinary differential equation (1.2) stabilizes to equilibria in an exponential way. The exponents of exponentials are specified by the characteristic roots of the equation linearized near an equilibrium. Oscillations are possible only in the case of a focus, when $\gamma^2-4<0$. Hence for $\gamma^2-4>0$ we have the asymptotic behaviour
$$
\begin{equation*}
\Phi_v(s)=\begin{cases} \exp(\lambda^0_+ s)\bigl[\operatorname{const}+ \mathcal{O}\bigl(\exp(\lambda^0_+ s)\bigr)\bigr],& s\to-\infty, \\ 1+\exp(\lambda_+s) \bigl[\beta+\mathcal{O}\bigl(\exp(\lambda_+s)\bigr)\bigr],& s\to+\infty, \end{cases}\qquad \beta=\operatorname{const},
\end{equation*}
\notag
$$
with the greatest exponent $\lambda_+<0$ in the general case. The leading terms in the formula correspond to solutions of the equations linearized near equilibria; corrections or remainders are specified by the nonlinearities. The least exponent $\lambda_-<\lambda_+$ appears only for a non-generic exceptional solution [16], Chap. I, § 1. Example 1. In the simplest case the KPP equation [8], [9] has the form (1.1), where $f(\varphi)=\varphi(1-\varphi)$. The corresponding travelling wave equation for $\varphi=\Phi(x-Vt)$ can be written as (1.2) for $\gamma=V>0$ and $f(\Phi)=\Phi(1-\Phi)$. The fixed points are the saddle $\Phi_0=0$ and the stable node $\Phi_1=1$ for $\gamma\geqslant2$. Example 2. Before the above normalization of the variables $(\varphi,x,t)$, the magnetodynamics equation [10] has the form (1.1), where $f(\varphi)=\sin\varphi\cos\varphi+w\sin\varphi$, $w=\operatorname{const}>0$. The corresponding travelling wave equation for $\varphi=\Phi(s)$, where $s=(x-V t)/\sqrt{1-\delta V^2}$, can be written as (1.2) for $\gamma=V/\sqrt{1-\delta V^2}$ and $f(\Phi)=\sin\Phi\cos\Phi+w\sin\Phi$. The fixed points are the saddle $\Phi_0=0$ and the stable node $\Phi_1=\pi$ (for $\gamma\geqslant2\sqrt{w-1}$ , $w>1$). 2. Known resultsFor the original partial differential equation (1.1) the existence result for travelling waves looks trivial. A more interesting and complicated question is the existence of solutions $\varphi(x,t)$ approaching the profile of one or another travelling wave at long times, asymptotically as $t\to\infty$. It is precisely this problem that was solved for a parabolic equation in [8]. In this case fundamental restrictions on the admissible velocity and the structure of a travelling wave were established. These restrictions are related to the stability of travelling wave solutions with respect to perturbations of the initial data and have most consistently been analyzed in [6] and [17]. The stability is identified with the absence of an exponential growth in $t$ in the first correction, which is derived from an equation obtained by linearizing (1.1) on the solution
$$
\begin{equation*}
\varphi=\Phi_v\biggl(\frac{x-Vt}{\sqrt{1-\delta V^2}}\biggr).
\end{equation*}
\notag
$$
The situation reduces to the study of the spectrum of the operator of the travelling wave equation (1.2) that is linearized on $\Phi_v(s)$ [17], [18]. A necessary stability condition is that $\Phi_v(s)$ is monotone. It follows that a fixed point corresponding to the equilibrium $\Phi=1$ cannot be a focus in the case of a stable wave. Therefore, we arrive at the necessary restriction on the velocity
$$
\begin{equation*}
\frac{V^2}{1-\delta V^2}=\gamma^2\geqslant4\quad (\text{for } f'(1)=-1).
\end{equation*}
\notag
$$
The problem of the identification of the wave to which the solution converges in dependence on the initial data is known as the choice problem. Results based on stability considerations were presented in [6]. The case of a wave of threshold (critical) velocity $V=V_*$ is particularly distinguished, namely, the case when If the corresponding function $\Phi_{*}(s)\stackrel{\rm def}{=}\Phi_{v_*}(s)$ is monotone, then the solution $\varphi(x,t)$ converges as $t\to\infty$ to the profile of precisely this wave when the initial data stabilize sufficiently rapidly. In other words, the leading term in the asymptotic formula as $t\to\infty$ is described by $\Phi_{*}(s)$. However, the time dependence of the phase $s$ is nonlinear, and the velocity has power asymptotics $S'(t)=V_*+\mathcal{O}(t^{-1})$. A similar result for the KPP equation was rigorously proved in [12]. There are no results of the same degree of rigour for other equations. Convergence to a wave with another velocity $V>V_*$ takes place when the initial data stabilize more slowly. In this case the velocity has exponential asymptotics: $S'(t)=V+\mathcal{O}(e^{-\kappa t})$, $\kappa>0$ [6].We explain how to solve the choice problem by taking the example of equation (1.2). The exponent $\lambda<0$, characterizing the stabilization rate of the travelling wave at the leading front, more precisely,
$$
\begin{equation*}
\Phi(s)=1+\exp(\lambda s)\bigl[\beta+\mathcal{O} \bigl(\exp(\lambda s)\bigr)\bigr],\qquad s\to+\infty, \quad \beta=\operatorname{const},
\end{equation*}
\notag
$$
determines the coefficient $\gamma=V\sqrt{1+4\delta}$ in (1.2) (or, which is the same, the wave velocity $V$). An intermediate step here is the characteristic equation for the linearization of (1.2) near $\Phi=1$: However, there are two roots $\lambda_-<-1<\lambda_+<0$ for each $\gamma>2$. They correspond to different scenarios for the solution $\Phi(s)$ of (1.2) with boundary conditions (1.3). For fixed $\gamma$ the solution stabilizes to $1$ either slowly, with exponent $\lambda=\lambda_+>-1$, or rapidly, with exponent $\lambda=\lambda_-<-1$. The realization of one scenario or another depends on the global properties of the nonlinear function $f(\varphi)$, $0<\varphi<1$. This is clear from the phase portrait of (1.2). The trajectory outgoing from the saddle $(0,0)$ approaches the node $(1,0)$ tangentially to one of the characteristic directions on the phase plane: $\dot\varphi=(\varphi-1)\lambda_\pm$; the stabilization rate is $\lambda_+$ or $\lambda_-$ respectively. The direction corresponding to $\lambda_-$ is exceptional: only two trajectories are tangent to it ([16], Chap. I, § 1, or [15], p. 80). It can occur that one of these trajectories goes out of the saddle (a separatrix). However, it is clear that this picture is generically destroyed by a small perturbation of the right-hand side of the equation. A trajectory from the saddle almost always approaches the node tangentially to the common direction corresponding to $\lambda_+$ (see Fig. 1). Thus, the travelling wave problem (1.2), (1.3) with the additional condition of rapid stabilization at the leading front is unstable with respect to perturbations of the equation. For the original equation (1.1) this instability implicitly indicates that rapid stabilization does not guarantee convergence to a fast wave. Initial data with rapid stabilization, for example of step-like form (as in [8]), lead to a wave with minimum velocity $V_*$, when $\gamma=2$ (provided that this wave is monotone). Of course, a similar assertion in a precise formulation must be proved, as it was done in [8] and [12]. The existence and stability of a fast (pushed) wave were discussed in [6]. In our paper the Cauchy problem is not under consideration, and the above reasoning just provides a basis for choosing the (pulled) wave $\Phi_*(s)$ with critical velocity $V=V_*$ as the leading term in the asymptotic solution. For $V=V_*$ the fixed point $(1,0)$ is a degenerate node with the multiple characteristic root3 For the corresponding function $\Phi_{*}(s)\equiv\Phi_{v_*}(s)$ the asymptotic formula as $s\to+\infty$ contains generically a specific factor linear in $s$ with coefficient $\alpha\ne0$ generically:
$$
\begin{equation}
\Phi_*(s)=\begin{cases} \exp(\lambda^0_+\, s)\bigl[\operatorname{const}+ \mathcal{O}\bigl(\exp(\lambda^0_+\,s)\bigr)\bigr],& s\to-\infty,\vphantom{\biggl\}} \\ 1+\exp(-s)\bigl[\alpha s+\beta+\mathcal{O}\bigl(s^2\exp(-s)\bigr)\bigr],& s\to+\infty, \end{cases}\quad \alpha,\beta=\operatorname{const}.
\end{equation}
\tag{2.1}
$$
In any case, one of the coefficients $\alpha$ and $\beta$ is non-zero. The presentation that follows concerns the construction of asymptotic expressions for solutions of (1.1) converging to the wave profile $\Phi_*(s)$. A solution in the form of a wave travelling at constant velocity $V$ is clearly isolated for equation (1.1). The following issues are usually discussed among the mathematical problems. 1. Identifying the class of initial data for which a solution $\varphi(x,t)$ of (1.1) converges as $t\to\infty$ to a solution $\Phi_v(s)$ of (1.2) for a prescribed parameter $V$. The phase of such a wave $\xi=x-S(t)$ does not coincide with $x-Vt$ in general. Relevant rigorous mathematical results were obtained in [12] for the KPP equation. 2. Calculating the asymptotics as $t\to\infty$ for the solution $\varphi(x,t)$ and, in particular, for the phase $S(t)$ (or wave velocity $dS(t)/dt$). This was a subject of many works; an extensive bibliography can be found in [6]. The fundamental problem of the presence of the logarithm in the asymptotic formula for the phase in the case $V=V_*$ was actually solved in [12] and, by another method, in [13] and [14]. The results in [12] and [14] were derived by analyzing the exact solution, only for the KPP equation again. Such an approach looks quite complicated; the further corrections in the asymptotic formula for the solution were not discussed.3. Constructing a formal asymptotic solution as an asymptotic series in inverse powers of the variable $t$ (and possibly in $\log t$) with no relation to any initial data seems to be simpler. Substituting a segment of such a series into the original equation yields a residual of the corresponding order of smallness as $1/t\to0$. Formal approaches of this type are widely used in various problems when the asymptotic behaviour is considered. For the KPP equation this approach was consistently implemented in [6], where a segment of the asymptotic solution in half-integer negative powers $t^{-n/2}$ was constructed. The leading term of this asymptotic expansion corresponds to the monotone solution $\Phi=\Phi_*(s)$ of (1.2) (for $\delta=0$ and $V=V_*$). In this case the time dependence of the phase $s=x-S(t)$ is nonlinear. If $s=0$ is identified with the coordinate of the centre of the wave, then the curve $x=S(t)$ on the plane $(x,t)$ is the trajectory of the centre. For the velocity of the centre of the wave the asymptotic formula was produced in the form In the case when $\delta=0$ the constants $c_0=-3/2$ and $c_1=3\sqrt\pi/2$ were calculated and some arguments for their universality, namely, independence of the initial data were given [6], [7]. Similar results for other equations were also presented in these papers. In particular, in the hyperbolic case of (1.1) (for $\delta>0$) the coefficients in the asymptotic formula (2.2) depend on the parameter $\delta$.4. Justification of the formal construction, in combination with the proof of an existence theorem for the exact solution and an estimate for the remainder in the asymptotic formula remains an open problem. Relevant results are known only for special cases of the KPP equation [19]. However, even without such a substantiation, some problems with formal solution remain. 3. Problem statement and approaches to its solution3.1. The formulation of questionsThe results presented in [6] and [7] turn out to be limited, as indicated in [20] and [21]. Asymptotic constructions for the velocity must include not only negative powers $t^{-k/2}$ but also powers of $\log t$. A fundamental question is whether there are logarithms in the universal part of the asymptotic formula, independent of the particuar solution. Since a formal asymptotic solution is constructed with no relation to initial data, the construction must contain arbitrary quantities that are responsible for the choice of a specific solution. In addition, it is necessary to specify the dependence of the solution on the parameter $\delta$, which governs the type of the original equation. We solve these problems under certain restrictions. 3.2. A clarification of the original restrictionsThe function $f(\varphi)$ is considered as infinitely differentiable on the interval $\varphi\in[0,1]$. We assume that $f(0)=f(1)=0$, the derivative satisfies the inequalities $f'(0)>0$ and $f'(1)<0$, and $f(\varphi)$ has no other zeros on the interval $0<\varphi<1$. The scale of the variables $x$ and $t$ is chosen so that $f'(1)=-1$. Therefore, the critical velocity $V_*$ is found from the relation that is, The whole construction is based on the solution $\Phi_*(s)$ of problem (1.2), (1.3) for the critical velocity $V=V_*$. The parameters $\alpha$ and $\beta$ specify the asymptotic behaviour (2.1) of this function as $s\to+\infty$. The generic case4 when $\alpha\ne0$ is under consideration.An essential condition is the monotonicity of the function $\Phi_*(s)$. Monotonicity depends on the global behaviour of $f(\varphi)$ in the interval $0<\varphi<1$ between the equilibria. In each particular case this property can be verified by analyzing numerically the phase portrait of equation (1.2) (see Fig. 2), as demonstrated in [22], [18], and [23]. The situation when the wave $\Phi_*(s)$ is not monotone is discussed in the conclusions (§ 9). 3.3. The methods usedThe matching method used below is usually applied to problems with a small parameter characterized by certain specific features. This specificity manifests itself through differences in the structure of the asymptotic formulae (with respect to the small parameter) in different domains of the independent variables. Fixing the structure of the asymptotic formula (an ansatz) often imposes restrictions on the suitability domain of the resulting constructions. The problem of the extension of the suitability domain in various situations can be solved by various methods. There is a broad class of problems with so-called bisingular perturbations [24], which can be solved only by the matching method and for which no alternative approaches are known. This method is based on the idea of representing the asymptotic behaviour of a solution in different forms in different domains of the independent variables. The boundaries of these domains are ‘blurred’ and depend on the small parameter. Because of this, distinct domains overlap. The requirement that different asymptotic solutions coincide asymptotically (match) in overlapping areas (in the intermediate layer) makes it possible to determine uniquely the coefficients of all asymptotic expansions. These ideas of the asymptotic extension of the solution are used in our paper. However, we must take into account that the problem under consideration does not contain any small parameter.5 Asymptotic solutions are constructed in our work in terms of the independent variable as $t\to\infty$, as series in the negative powers $t^{-k/2}$ and $\log t$. The coefficients of the asymptotic expansions depend on the spatial variable $x$. Suitability domains in $x$ for different expansions are described in terms of the variable $t$. Matching in the intermediate layer (overlapping area) is used to determine uniquely the coefficients. Such an approach was implemented in [6] and resulted in determining the constant $c_1$ in the asymptotic formula for the velocity. Using the matching method consistently makes it possible to construct full asymptotic expansions and, in particular, determine the coefficient of the logarithm in the velocity asymptotics. Relations from which the constants in asymptotic formulae are derived are solvability conditions for a relevant linear equation in a certain class of solutions. These ideas are like those used in soliton perturbation theory for the problem of determining the phase shift at long times [25]. A distant analogy with the averaging method [26] can be seen here. We indicate additionally the efficiency of numerical methods. They are used here to calculate integrals of non-elementary functions. These calculations play a key role in proving that critically important coefficients in asymptotic formulae are distinct from zero. 4. Formulation of the result4.1. The main resultThe following refinement of formula (2.2) for the velocity asymptotics as $t\to\infty$ is the main result of our study:
$$
\begin{equation}
\frac{dS}{dt}=\frac{1}{\sqrt{1+4\delta}}\bigl[2+t^{-1}\bigl(c_0+c_1t^{-1/2}+ t^{-1}(c_{2,1}\log t+c_2)+\mathcal{O}(t^{-3.2}\log t)\bigr)\bigr].
\end{equation}
\tag{4.1}
$$
The coefficient in the first correction is a half-integer number $c_0\leqslant-3/2$. The other coefficients depend on its choice. The coefficient $c_0$ is not uniquely determined in formal constructions. The value $c_0=-3/2$ is chosen on the basis of coincidence with the known rigorously proved result for the KPP equation [12], [14]. In this particular case the next coefficient $c_1=3\sqrt\pi\,\sqrt{1+4\delta}/2$ coincides with the one indicated in [6]. The expression for the coefficient of the logarithm $c_{2,1}= c_{2,1}(\delta)$ as a function of the parameter $\delta$ looks more complicated. However, it is important that this coefficient is not zero and universal, just like $c_0$ and $c_1$ (Theorems 3 and 4). This completes extracting the universal part of the asymptotic expansion of a solution converging to a travelling wave. The further construction contains arbitrary parameters in the form of constants $c_{2n}$ in the terms with even indices, beginning with $c_2t^{-1}$. The constants $c_{2n}$ depend on the initial data and cannot be determined uniquely in the framework of the formal asymptotics proposed. The constructions below are of interest not only because they lead to (4.1) but also because they demonstrate the matching method, which is not often used in problems of asymptotic behaviour with respect to the independent variable. 4.2. The original ansatzWhen an asymptotic formula is constructed, it is convenient to use a normalized travelling wave variable (instead of $x$), by making in (1.1) the substitution
$$
\begin{equation*}
\varphi(x,t)=\phi(s,t),\quad s=\frac{x-V_*t}{\sqrt{1-\delta V_*^2}}-\sigma(t)\quad\Longleftrightarrow\quad s=x\sqrt{1+4\delta}-2t-\sigma(t).
\end{equation*}
\notag
$$
The new independent variable $s$ differs by an additional term $\sigma(t)$ from the variable used in the reduction to equation (1.2) of a travelling wave with constant velocity. Therefore, now we obtain the partial differential equation
$$
\begin{equation}
\begin{aligned} \, &\frac{\partial^2\phi}{\partial s^2}+(2+\nu)\frac{\partial\phi}{\partial s}- \frac{\partial\phi}{\partial t}-f(\phi)-\delta(4\nu+\nu^2) \frac{\partial^2\phi}{\partial s^2}+ 2\delta(2+\nu)\frac{\partial^2\phi}{\partial s\,\partial t}\nonumber\\ &\qquad=\delta\biggl[\frac{\partial^2\phi}{\partial t^2}- \nu'(t)\frac{\partial\phi}{\partial s}\biggr], \end{aligned}
\end{equation}
\tag{4.2}
$$
which contains two unknown functions, $\phi(s,t)$ and $\sigma'(t)=\nu(t)$. The solution $\phi(s,t)$ we have constructed depends on the choice of $\nu(t)$. In the theory of averaging and in perturbations of solitons a similar technique is used to construct asymptotic solutions suitable for long times (see, for example, [26] and [25]). In our work the function $\nu(t)$ is chosen in asymptotic constructions below to make them suitable far away at the leading wave front, where $s\gg\sqrt t$ . The formula for the wave velocity (in view of the normalization) assumes the form Since we deal with the construction of an asymptotic solution, only the asymptotic behaviour as $t\to\infty$ is considered for $\nu(t)$ in the form
$$
\begin{equation}
\nu=t^{-1}\biggl[c_0+c_1t^{-1/2}+t^{-1}(c_{2,1}\log t+c_2)+ t^{-3/2}(c_{3,1}\log t+c_3)+\sum_{n,k} c_{n,k}t^{-n/2}\log^k t\biggr].
\end{equation}
\tag{4.4}
$$
The exponents in the general term of the series are specified by the indices
$$
\begin{equation*}
n=2,3,\dots\quad\text{and}\quad k=0,1,\dots,\biggl\lfloor \frac{n}{2}\biggr\rfloor.
\end{equation*}
\notag
$$
We omit the index $k=0$ for the coefficients of the zeroth power of the logarithm and simply write $c_n$. Explicit expressions will be given for the first coefficients. As usual in asymptotic studies, higher corrections are only of academic interest. Infinite series are used here only to show the general structure of asymptotic expressions. The main goal of the constructions below is to indicate an efficient method for calculating the constants $c_{n,k}$ in the velocity asymptotics. For this purpose it is necessary to construct an asymptotic solution as $t\to\infty$ that is suitable up to distant values $s\gg\sqrt t$ , that is, far away at the leading front. No other method for finding $c_{n,k}$ is known. The asymptotic expansion as $t\to\infty$ obtained at the first step turns out to be unsuitable for $s\approx\sqrt t$ , and the constants $c_{n,k}$ remain undetermined. At the next step we construct a formal asymptotic solution which is suitable on a larger interval $\sqrt t\leqslant s\ll t$ and find the $c_{n,k}$. Asymptotic expressions as $t\to\infty$ that differ on different intervals of values of $s$ are constructed using the matching method [24]. The terminology associated with this method is inner and outer expansions. The justification of asymptotic expressions is not discussed. 5. Inner expansionIn the domain where the wave changes rapidly (the inner layer), the asymptotic solution is constructed as a segment of a series of the form
$$
\begin{equation}
\begin{aligned} \, \phi(s,t)&=\Phi_*(s)+t^{-1}\bigl\{\Phi_0(s)+t^{-1/2}\Phi_1(s)+ t^{-1}\bigl[\log t\,\Phi_{2,1}(s)+\Phi_2(s)\bigr] \nonumber \\ &\qquad+t^{-3/2}\bigl[\log t\,\Phi_{3,1}(s)+\Phi_3(s)\bigr]+ \cdots\bigr\},\qquad t\to\infty, \end{aligned}
\end{equation}
\tag{5.1}
$$
with coefficients depending on $s$. The occurrence of fractional powers and logarithms in the expansion is due only to the ansatz (4.4) for the velocity $\nu(t)$, the constants $c_{n,k}$ in which cannot be determined at this stage. Thus, the construction (5.1) looks tentative. The index $k =0$ is dropped throughout. 5.1. The recurrence systemAs the leading term we take the solution $\Phi_*(s)$ of the ordinary differential equation (1.2) in which the parameter $\gamma$ is $2$, or, which is the same, $V=V_*$. The linearized operator is
$$
\begin{equation*}
\mathcal{L}=\frac{d^2}{d s^2}+2\,\frac{d}{ds}-f'(\Phi_*(s)),\qquad s\in\mathbb{R}.
\end{equation*}
\notag
$$
Corrections in (5.1) are obtained using the recurrence system of inhomogeneous linear equations which we derive from (4.2) by isolating terms with like powers $t^{-n/2}\log ^kt$. At each step the right-hand sides are determined by using the previous approximations. For one of the frequently encountered expressions, it is convenient to introduce the notation
$$
\begin{equation*}
F_*(s)\overset{\rm def}=4\delta\Phi_*^{\prime\prime}(s)-\Phi_*'(s).
\end{equation*}
\notag
$$
At the first steps we have
$$
\begin{equation*}
F_0(s)=c_0F_*(s),\quad F_1(s)= c_1F_*(s),\quad F_{2,1}(s)=c_{2,1}F_*(s),\quad F_{3,1}(s)=c_{3,1}F_*(s),
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, F_2(s)&=c_2F_*(s)+c_0\bigl[4\delta\Phi_0^{\prime\prime}(s)+ \Phi_0'(s)\bigr]-\Phi_0(s) \\ &\qquad+\frac{1}{2}f^{\prime\prime}(\Phi_*(s))\Phi_0^2(s)+ 4\delta\,\Phi_0^{\prime}(s)+\delta c_0\Phi_*^{\prime}(s). \end{aligned}
\end{equation*}
\notag
$$
The equations are supplemented with the homogeneous boundary conditions 5.2. The solvability of the recurrence system of problemsWe let $\mathcal{P}$ denote the class of smooth functions $\Phi(s)$, $s\in\mathbb{R}$, whose asymptotics at infinities has the form of quasi-polynomials:
$$
\begin{equation*}
\Phi(s)=\begin{cases} Q(s)\exp(\lambda_+^0 s)\bigl[1+\mathcal{O} \bigl(s^q\exp(\lambda_+^0s)\bigr)\bigr],& s\to-\infty; \\ P(s)\exp(-s)\bigl[1+\mathcal{O}\bigl(s^p\exp(-s)\bigr)\bigr], & s\to+\infty. \end{cases}
\end{equation*}
\notag
$$
Here $Q(s)$ and $P(s)$ denote polynomials; the finite exponents $0\leqslant p,q<\infty$ in the remainders are of no importance, and their values are not determined. Since the coefficient $\gamma$ in the travelling wave equation for the velocity $V_*$ under consideration is 2, the characteristic roots for the saddle point are given by the formulae
$$
\begin{equation*}
\lambda_\pm^0=-1\pm\sqrt{1+f'(0)},\quad\text{where}\ \ f'(0)>0.
\end{equation*}
\notag
$$
Lemma 1. For any set of constants $c_{n,k}$ the recurrence system of problems (5.2), (5.3) for $\Phi_{n,k}(s)$ is solvable in the class $\mathcal{P}$. Proof. A fundamental system of solutions of the homogeneous equation $\mathcal{L}\Phi=0$ can be chosen in different ways; for example, we can fix the asymptotic beaviour as $s\to+\infty$. If we focus on formulae using the original wave $\Phi_*(s)$, then it is convenient to take a pair of solutions in the form
$$
\begin{equation*}
\varphi_+(s)=\Phi_*'(s),\qquad \varphi_-(s)=\varphi_+(s)\int_\infty^s \frac{\exp(-2\zeta)}{\varphi_+^2(\zeta)}\,d\zeta
\end{equation*}
\notag
$$
with Wronskian Below we establish the properties of these functions. According to (2.1), the solution $\varphi_+(s)$ vanishes exponentially at both infinities, namely,
$$
\begin{equation}
\varphi_+(s)=\begin{cases} \exp(\lambda_+^0s)\bigl[\operatorname{const}+\mathcal{O} \bigl(\exp(\lambda_+^0s)\bigr)\bigr],& s\to-\infty,\ \ \lambda_+>0; \\ \exp(-s)\bigl[-(\alpha s+\beta_1)+ \mathcal{O}\bigl(s^2\exp(-s)\bigr)\bigr],& s\to+\infty,\ \ \beta_1=\beta-\alpha. \end{cases}
\end{equation}
\tag{5.4}
$$
The second (linearly independent) solution $\varphi_-(s)$ has asymptotics growing as $s\to-\infty$, with exponent $\lambda_-^0<0$, the second characteristic root for the saddle point $(0,0)$. We can conclude from the explicit formula for $\varphi_-(s)$ that the integral converges for $\alpha\ne0$, since by the monotonicity of the wave $\varphi_+(\zeta)\ne0$ and the integrand is small at infinity:
$$
\begin{equation*}
\frac{\exp(-2\zeta)}{\varphi_+^2(\zeta)}\approx (\alpha\zeta+\beta_1)^{-2},\qquad \zeta\to+\infty.
\end{equation*}
\notag
$$
At minus infinity the integrand has the asymptotic behaviour
$$
\begin{equation*}
\frac{\exp(-2\zeta)}{\varphi_+^2(\zeta)}= \exp\bigl((\lambda_-^0-\lambda_+^0)\zeta\bigr) \bigl[\operatorname{const}+\mathcal{O}\bigl(\exp(\lambda_+^0\zeta)\bigr)\bigr],\qquad \zeta\to-\infty.
\end{equation*}
\notag
$$
Therefore, after integration we obtain exponentially growing asymptotics of the form
$$
\begin{equation*}
\varphi_-(s)={\exp(\lambda_-^0s)} \bigl[\operatorname{const}+\mathcal{O}\bigl(\exp(\lambda_+^0s)\bigr)\bigr],\qquad s\to-\infty,\quad \lambda_-<0,
\end{equation*}
\notag
$$
for $\varphi_-(s)$ at this infinity. It corresponds to the second (linearly independent) solution on the trajectory to the saddle point $(0,0)$.
At plus infinity the integrand has the asymptotic behaviour
$$
\begin{equation*}
\frac{\exp(-2\zeta)}{\varphi_+^2(\zeta)}=(\alpha\zeta+\beta_1)^{-2}+ \mathcal{O}\bigl(\exp(-\zeta)\bigr),\qquad \zeta\to+\infty.
\end{equation*}
\notag
$$
Upon integration, we obtain an asymptotic expression for $\varphi_-(s)$ as $s\to+\infty$, which resembles the expression for $\varphi_+(s)$, with an explicitly specified coefficient, namely,
$$
\begin{equation*}
\varphi_-(s)=\frac{1}{\alpha} \exp(-s)\bigl[1+\mathcal{O}\bigl(\exp(-s)\bigr)\bigr],\qquad s\to+\infty.
\end{equation*}
\notag
$$
Thus, the second basis solution is as follows:
$$
\begin{equation}
\varphi_-(s)=\begin{cases} \exp(\lambda_-^0s)\bigl[\operatorname{const}+ \mathcal{O}\bigl(\exp(\lambda_+^0s)\bigr)\bigr],& s\to-\infty,\ \ \lambda_-^0<0; \\ \exp(-s)\biggl[\dfrac{1}{\alpha}+ \mathcal{O}\bigl(s^2\exp(-s)\bigr)\biggr],& s\to+\infty. \end{cases}
\end{equation}
\tag{5.5}
$$
If $\alpha=0$, then the asymptotic formulae for $\varphi_+(s)$ and $\varphi_-(s)$ as $s\to+\infty$ interchange. We must take the integral in the formula for $\varphi_-(s)$ over the finite interval $\zeta\in[0,s]$. In this case we obtain an asymptotic formula with a growing factor for the second basis solution:
$$
\begin{equation*}
\varphi_-(s)=\frac{s}{\beta} \exp(-s)\bigl[1+\mathcal{O}\bigl(\exp(-s)\bigr)\bigr],\qquad s\to+\infty.
\end{equation*}
\notag
$$
A particular solution of the inhomogeneous equation $\mathcal{L}\Phi=F(s)$ is taken in the form
$$
\begin{equation}
\Phi(s)=-\varphi_+(s)\int_0^s\varphi_-(\zeta)F(\zeta)\exp(2\zeta)\,d\zeta+ \varphi_-(s)\int_{-\infty}^s\varphi_+(\zeta)F(\zeta)\exp(2\zeta)\,d\zeta.
\end{equation}
\tag{5.6}
$$
In the case when $F(s)$ corresponds to an exponentially small remainder, namely,
$$
\begin{equation*}
F(s)=\begin{cases} \mathcal{O}\bigl(s^q\exp(2\lambda_+^0\,s)\bigr),& s\to-\infty; \\ \mathcal{O}\bigl(s^p\exp(-2s)\bigr),& s\to+\infty, \end{cases}
\end{equation*}
\notag
$$
the integrands in (5.6) are exponentially small at infinity. Therefore, the integrals are exponentially close to constants, and the solution asymptotics is formed of the basis solutions $\varphi_\pm(s)$. Taking account of the asymptotic behaviour of these functions, we arrive at the required asymptotics for the solution with polynomials $Q_0=\operatorname{const}$ and $P_1(s)$ of degrees zero and one, respectively.
In the general case, when $F(s)\in\mathcal{P}$, the integrands are exponentially close to polynomials. Upon integration, in view of multiplication by basis solutions, the degree of the polynomials involved in the asymptotic expansions increases by two as $s\to+\infty$ and by one as $s\to-\infty$. The class $\mathcal{P}$ is invariant under differentiation and multiplication of functions. This makes it possible to complete the proof of the lemma by referring to induction on $n$. $\Box$ Supplement. Note that the first corrections are expressed in terms of the same function It is straightforward to derive from the explicit representation
$$
\begin{equation*}
F_*(\zeta)=4\delta\varphi_+^{\prime}(\zeta)-\varphi_+(\zeta)
\end{equation*}
\notag
$$
that the asymptotic behaviour at infinity has the form
$$
\begin{equation*}
\widetilde\Phi(s)=\exp(-s)\bigl[\widetilde P_3(s)+ \mathcal{O}\bigl(s^4\exp(-s)\bigr)\bigr],\qquad s\to+\infty,
\end{equation*}
\notag
$$
where The coefficients of higher powers are calculated here using the leading terms of the asymptotic expressions for integrands in (5.7). The simplest formulae are obtained in the case when $\delta=0$, namely,
$$
\begin{equation*}
A_3=\frac{\alpha}{6}\quad\text{and}\quad A_2=\frac{\beta_1}{2}\quad (\beta_1=\beta-\alpha).
\end{equation*}
\notag
$$
The lower-order coefficients are calculated in terms of integrals of the remainders as follows:
$$
\begin{equation*}
A_1\big|_{\delta=0}=-\alpha\int_0^\infty\biggl[\varphi_+(\zeta) \varphi_-(\zeta)\exp(2\zeta)+ \frac{\alpha\zeta+\beta_1}{\alpha}\biggr]\,d\zeta
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, A_0\big|_{\delta=0}&=-\beta_1\int_0^\infty\biggl[\varphi_+(\zeta) \varphi_-(\zeta)\exp(2\zeta)+\frac{\alpha\zeta+\beta_1}{\alpha}\biggr]\,d\zeta \\ &\qquad-\frac{1}{\alpha}\int_{-\infty}^\infty\bigl[(\varphi_+(\zeta))^2 \exp(2\zeta)-(\alpha\zeta+\beta_1)^2\bigr]\,d\zeta. \end{aligned}
\end{equation*}
\notag
$$
5.3. The non-uniqueness of an asymptotic solutionThe above asymptotic solution (5.1) depends on the constants $c_{n}$, $c_{n,k}$, which have not been determined yet. It is possible to supplement at each step solutions of (5.2) and (5.3) with a solution of the homogeneous problem $D\,\Phi_*'(s)$ that decreases exponentially at infinities, with an arbitrary coefficient $D=\operatorname{const}$ or, which is the same, to change the lower limit in the first integral in (5.6). We do not do this because an equivalent freedom in the asymptotic solution is associated with the velocity constants $c_n$ and $c_{n,k}$. In addition, we mention an ambiguity in the construction (5.1), which is a consequence of the two-scale structure of the asymptotic solution and is not related to the choice of the constants $c_{n,k}$. The coefficients of (5.1) involve the variable
$$
\begin{equation*}
s=x\sqrt{1+4\delta}-2t-\sigma(t),\qquad \sigma'(t)=\nu(t),
\end{equation*}
\notag
$$
for which an asymptotic expression as $t\to\infty$ is constructed using the expansion (4.4) for $\nu(t)$. We can extract a non-decreasing part
$$
\begin{equation*}
s_0=x\sqrt{1+4\delta}-2t-c_0\log t+\operatorname{const}
\end{equation*}
\notag
$$
from this asymptotic expression by isolating it from the decreasing terms:
$$
\begin{equation*}
s=s_0+2c_1t^{-1/2}+c_{2,1}t^{-1}\log t+(c_2+c_{2,1})t^{-1}+ c_{3,1}t^{-3/2}\log t+\cdots\,.
\end{equation*}
\notag
$$
After that, the coefficients $\Phi_{n,k}(s)$ in the series (5.1) are expanded in series as $t\to\infty$. Hence we obtain another series with another phase variable $s_0$ for the asymptotic solution. Unlike (5.1), the first summands in this new series contain terms with slower decrease, namely,
$$
\begin{equation}
\begin{aligned} \, \phi(s,t)&=\Phi_*(s_0)+\Phi_*'(s_0)\bigl[2c_1t^{-1/2}+c_{2,1}t^{-1}\log t+ (c_2+c_{2,1})t^{-1}+\cdots\bigr] \nonumber \\ &\qquad+t^{-1}\bigl\{\Phi_0(s_0)+\Phi_0'(s_0)\bigl[2c_1t^{-1/2} +c_{2,1}t^{-1}\log t+(c_2+c_{2,1})t^{-1}+\cdots\bigr] \nonumber \\ &\qquad+t^{-1/2}\Phi_1(s_0)+t^{-1}\log t\,\Phi_{2,1}(s_0)+ t^{-1}\Phi_2(s_0)+\cdots\bigr\}+\cdots, \quad t\to\infty. \end{aligned}
\end{equation}
\tag{5.9}
$$
We can supplement the isolated part $s_0$ of the phase with several decreasing terms of $s$. As a result, we arrive at many distinct asymptotic solutions of the form (5.9); all of them coincide asymptotically as $t\to\infty$, since they are re-expansions of the same original series. There is nothing new in these re-expansions. A similar situation often occurs when two-scale expansions with fast and slow variables are used. In our case the phase $s$ is a fast variable. The remainder of the solution varies slowly and can be expanded in inverse powers of $t$. An expansion in the form (5.1) with phase
$$
\begin{equation*}
s=x\sqrt{1+4\delta}-2t-\sigma(t),\qquad \sigma'(t)=\nu(t),
\end{equation*}
\notag
$$
is most general, and we will use it. In addition, we can see by comparing with (5.9) that the first two terms in the decreasing part of the phase do not mix with coefficients of the expansion (5.1) and, as will be shown, are universal, that is, independent of the initial data. It is natural to relate them to slow changes in the wave velocity.
6. Intermediate layerIt is straightforward to see that the degrees of the polynomials $P(s)$ in the asymptotics for $\Phi_{n,k}(s)$ as $s\to+\infty$ increase by two at each step $n$ in comparison to the asymptotics of the right-hand side $F_{n,k}(s)$. For this reason, the series (5.1) is not asymptotic uniformly with respect to $s$. The property of asymptoticity fails for $s\geqslant\mathcal{O}(\sqrt t\,)$, since the subsequent correction turns out to be not less than the previous one in the order of $1/t$. In perturbation methods a similar situation is described by the notion of ‘secular terms’ [26]. In our case, secularities are identified as the factors $(s/\sqrt t\,)^{n}$. The construction of the asymptotic solution must be changed far away at the leading front, for $s\approx\sqrt t$. This is done by using the matching method [24]. To implement it, it is necessary to find a more suitable representation for the asymptotic expression (5.1) in the domain $s\gg1$, that is, when we leave the inner layer. In the domain $s\gg1$ the asymptotic solution can be simplified by using the asymptotic behaviour of the coefficients $\Phi_{n,k}(s)$ as $s\to+\infty$. These functions vanish exponentially as $s\to+\infty$. There are polynomials multiplying exponentials whose degree grows with $n$. At each step such an asymptotic behaviour can be derived from the integrals (5.6), as done for the first corrections. As a result, we arrive at the relation
$$
\begin{equation*}
\begin{aligned} \, [\phi(s,t)-1]e^s&=[\alpha s+\beta]+t^{-1}\widetilde P_3(s) \bigl[c_0+c_1t^{-1/2}+c_{2,1}t^{-1}\log t+\cdots\bigr] \\ &\qquad+t^{-2}\bigl[c_2\widetilde P_3(s)+P_5(s)\bigr]+\cdots\,. \end{aligned}
\end{equation*}
\notag
$$
The structure of the higher corrections to the asymptotics (for example, the polynomial $P_5(s)$) can be clarified using a simpler (formal) method, by distinguishing a decreasing exponential in the original equation. To do this we change the unknown function and make a slight normalization of time as follows6:
$$
\begin{equation*}
[\phi(s,t)-1]e^{s}=\psi(s,\tau),\qquad t=(1+4\delta)\tau.
\end{equation*}
\notag
$$
Since and
$$
\begin{equation*}
\partial_s\phi=[\partial_s\psi-\psi]e^{-s},\quad \partial^2_s\phi=[\partial^2_s\psi-2\partial_s\psi+\psi] e^{-s}
\end{equation*}
\notag
$$
in this case, the original equation (4.2) for the new function $\psi(s,\tau)$ reduces to the form
$$
\begin{equation}
\begin{aligned} \, &\frac{\partial^2\psi}{\partial s^2}- \frac{\partial\psi}{\partial \tau}- \nu(1+4\delta)\psi+\nu(1+8\delta)\frac{\partial\psi}{\partial s}+ \frac{4\delta}{1+4\delta}\,\frac{\partial^2\psi}{\partial s\,\partial\tau} \nonumber \\ &\qquad=\delta\biggl\{\nu(4+\nu)\partial^2_s\psi-\nu^2[2\partial_s\psi-\psi]- 2\nu\,\frac{1}{1+4\delta}\,\partial_\tau[\partial_s\psi-\psi] \nonumber \\ &\qquad\qquad\qquad\qquad+\frac{1}{(1+4\delta)^2}\,\partial^2_\tau\psi- \nu'[\partial_s\psi-\psi]\biggr\}+\mathcal{O}(e^{-s}\psi^2). \end{aligned}
\end{equation}
\tag{6.1}
$$
The nonlinear terms turn out to be exponentially small and do not participate in the further constructions in the domain $s\gg1$. Actually, beginning with this step, the problem (at the leading wave front) is linearized. The asymptotic solution is constructed as a series in powers and logarithms of $\tau$:
$$
\begin{equation}
\psi(s,\tau)=\Psi_*(s)+\tau^{-1}\bigl[\Psi_0(s)+\tau^{-1/2}\Psi_1(s) +\tau^{-1}\bigl(\log\tau\,\Psi_{2,1}(s)+\Psi_2(s)\bigr)+\cdots\bigr].
\end{equation}
\tag{6.2}
$$
The aim of this section is to clarify the structure of the coefficients as $s\to+\infty$ in order to distinguish the secular terms like $s/\sqrt t$ . The coefficients of the expansion can be found from the recurrence system of trivial equations
$$
\begin{equation}
\frac{d^2\Psi_*}{ds^2}=0,\quad \frac{d^2\Psi_{n,k}}{ds^2}=G_{n,k}(s),\qquad n=0,1,2,\dots,\quad 0\leqslant k\leqslant\biggl\lfloor\frac{n}{2}\biggr\rfloor;
\end{equation}
\tag{6.3}
$$
they certainly turn out to be polynomials. The integration constants determining the linear part with respect to $s$ are chosen from matching with $\Phi_{n,k}(s)$. Hence using this approach we need to know the asymptotic behaviour of the integrals (5.6) again; no other methods are known for the calculation of the linear parts of the polynomials $\Psi_{n,k}(s)$. Lemma 2. For any set of constants $c_{n,k}$ the recurrence system of problems (6.3) with matching conditions for the $\Psi_{n,k}(s)$ is uniquely solvable in the class of polynomials. The degree of a polynomial is determined by the integer part of $n/2$, and in the case when $\alpha\ne0$. If $c_0<0$, then for $k=0$ all constants $A_n$ are distinct from zero. Proof. The first assertion is obvious because of the structure of the recurrence system (6.3).
At the initial step the equation is homogeneous; its solution is a first-degree polynomial determined by matching with $\Phi_*(s)$: This yields the $s$-linear function
$$
\begin{equation*}
G_*=(1+4\delta)\Psi_*-(1+8\delta)\,\frac{d\Psi_*}{ds}= (1+4\delta)(\alpha s+\beta_1)-4\alpha\delta,\qquad \beta_1=\beta-\alpha,
\end{equation*}
\notag
$$
which specifies the right-hand sides of the equations for the first corrections:
$$
\begin{equation*}
G_{0}=c_{0}G_*,\quad G_{1}=c_{1}G_*,\quad G_{2,1}=c_{2,1}G_*,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
G_2=c_2G_*+c_0\biggl[\Psi_0-\frac{d\Psi_0}{ds}\biggr]-\Psi_0.
\end{equation*}
\notag
$$
The first corrections have the form
$$
\begin{equation*}
\Psi_0=c_0\widetilde P_3(s),\quad \Psi_1=c_1\widetilde P_3(s),\quad\text{and}\quad \Psi_{2,1}=c_{2,1}\widetilde P_3(s),
\end{equation*}
\notag
$$
where $\widetilde P_3(s)$ is a polynomial of third degree; we encountered it in the inner layer (see (5.8)):
$$
\begin{equation*}
\begin{gathered} \, \widetilde P_3(s)=A_0+A_1 s+A_2s^2+A_3s^3, \\ A_3=(1+4\delta)\,\frac{\alpha}{3!}\,,\qquad A_2=\frac{1}{2}(\beta_1-8\alpha\delta). \end{gathered}
\end{equation*}
\notag
$$
Now the coefficients of higher powers are found by integration. The $s$-linear part is obtained from matching. As we can see, the coefficient of the highest power is non-zero here. Verifying this property in higher corrections is the key aspect in the proof of the lemma.
It follows from the structure of equation (6.1) that the right-hand sides $G_n(s)$ of the recurrence system, $n\geqslant2$, are formed of the terms multiplying $t^{n/2}$ and are calculated in terms of the previous approximations $t^{m/2}\Psi_{m,k}$ with $m\leqslant n-2$. The highest power $s^{2\lfloor (n-2)/2\rfloor+3}$ in $G_n(s)$ is determined by two terms in the original equation (6.1),
$$
\begin{equation*}
\frac{\partial\psi}{\partial\tau}+\nu(1+4\delta)\psi.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
G_n(s)=\biggl(-\frac{n-2}{2}+c_0\biggr) A_{n-2}s^{2\lfloor(n-2)/2\rfloor+3}[1+\mathcal{O}(s^{-1})].
\end{equation*}
\notag
$$
After integration, going over to $\Psi_n(s)$ we obtain a polynomial whose degree is higher by two. If $c_0<0$, then the coefficient $A_n$ of the highest power is not equal to zero.
The structure of the polynomials $\Psi_{n,k}(s)$ multiplying $t^n\log^kt$ is established similarly. $\Box$ Remark. In the case $\alpha=0$ the original polynomial has degree zero: $\Psi_*(s)=\beta$. Thus, the degree of all polynomials $\Psi_{n}(s)$ in solutions of the recurrence system decreases by one and is equal to $2\lfloor n/2\rfloor+2$. The fact that the degrees increase by two in going over from $n-2$ to $n$ remains valid. The increase of the degrees of $\Psi_{n}(s)$ indicates that the sequence $\tau^{-n/2}\Psi_{n}(s)$, $n=0,1,2,\dots$, involved in (6.2) is not asymptotic as $\tau\to\infty$ uniformly in $s$. For distant values of $s\approx\sqrt\tau$ subsequent terms of the series are not corrections with respect to the previous ones. Terms like $\tau^{-n/2}s^{2\lfloor n/2\rfloor+3}$ correspond to secular terms of perturbation theory. They result in restrictions on the suitability domain of the asymptotic solution (6.2). These restrictions, following from the requirement that the series must be asymptotic, are described by $s\ll\sqrt\tau$ . In addition, we must take into account that there is also the restriction $s\gg1$ on the variable $s$. So, when using the series (6.2) as an asymptotic solution, we must identify the intermediate layer $1\ll s\ll\sqrt\tau$ . This relation should be put in a more concrete form. Since we deal with power asymptotics, the intermediate layer is described by the inequalities7
$$
\begin{equation*}
\tau^{\delta_1/2}\leqslant s\leqslant\tau^{(1-\delta_2)/2}
\end{equation*}
\notag
$$
for some (arbitrary) positive constants $\delta_1$ and $\delta_2$, $\delta_1+\delta_2<1$. Note that the exponential smallness with respect to $s$ in the intermediate layer implies the exponential smallness with respect to $t$, since
$$
\begin{equation*}
\exp\bigl(-2\tau^{(1-\delta_2)/2}\bigr)\leqslant\exp(-s).
\end{equation*}
\notag
$$
This justifies ignoring the remainder with nonlinear terms in (6.1) in the construction of the expansion (6.2) in powers of $\tau$. Like in perturbation theory, to construct an asymptotic solution for distant $s\approx \sqrt\tau$ we need to change the variable $s$ in an appropriate way. As usual, such a change depends on the structure of the secular terms, which have the form $s/\sqrt\tau$ in our case. The choice of a new variable determines the structure of the standard equation at the next step. One appropriate change of the variable of the form $z=s^2/(4\tau)$ was indicated in [6]. Making this change and moving out $\tau^{1/2}$ as a common factor, we can write the resulting expansion in the intermediate layer as follows:
$$
\begin{equation}
\begin{aligned} \, \phi(s,t)&=1+\exp(-s)\tau^{1/2}\bigl\{\bigl[\alpha\sqrt{4z}+c_0A_3 (4z)^{3/2}+\mathcal{O}(z^{5/2})\bigr] \nonumber \\ &\quad+\tau^{-1/2}\bigl[\beta+c_0A_2\,4z+c_1A_3(4z)^{3/2}+ \mathcal{O}(z^{2})\bigr] \nonumber \\ &\quad+\tau^{-1}\log\tau\,\bigl[c_{2,1}A_3(4z)^{3/2}+\mathcal{O}(z^{2})\bigr]+ \tau^{-3/2}\log \tau\,\bigl[c_{3,1}A_3(4z)^{3/2}+\mathcal{O}(z^{2})\bigr] \nonumber \\ &\quad+\tau^{-1}\bigl[c_0A_1\sqrt{4z}+c_1A_2\,4z+c_2A_3 (4z)^{3/2}+\mathcal{O}(z^{2})\bigr]+\cdots\bigr\}. \end{aligned}
\end{equation}
\tag{6.4}
$$
Here
$$
\begin{equation*}
z=\frac{s^2}{4\tau}\quad\text{and}\quad \tau=\frac{t}{1+4\delta}\,.
\end{equation*}
\notag
$$
Note that now the coefficient of each expression $\tau^{-n/2}\log^k\tau$ is an infinite series in $z^{m/2}$, provided that we take account of all terms in the original relation (6.2). If we take a finite segment of the series (6.2), then we obtain finite sums in $z^{m/2}$ in (6.4). We restrict ourselves to indicating the leading terms of the asymptotic expressions as $z\to0$ for coefficients in (6.4). This turns out to suffice for a well-defined construction of an asymptotic solution at the next step using the matching method. In terms of the variable $z$ the intermediate layer is located in the set In a more concrete form it is described by the inequalities
$$
\begin{equation*}
\tau^{-1+\delta_1}\leqslant z=\frac{s^2}{4\tau}\leqslant\tau^{-\delta_2}.
\end{equation*}
\notag
$$
We formulate the result of this section in terms of $t$. Theorem 1. For any set of constants $c_n$ and $c_{n,k}$, an asymptotic solution in the form (5.1) is suitable in the domain $-t\ll s\ll \sqrt t$ . In the intermediate layer $1\ll s\ll \sqrt t$ the expansions (5.1) and (6.4) coincide asymptotically. 7. Outer expansionAn asymptotic solution that is suitable for large $s\approx\sqrt t$ is constructed using the ‘slow’ variable $z$. To simplify the construction, it is convenient to change the function by detaching the decreasing exponentials:
$$
\begin{equation*}
\phi(s,t)=1+2\alpha\,\exp(-s)\,\tau^{1/2}\exp(-z)\,u(z,\tau),\qquad z=\frac{s^2}{4\tau}\,,\quad t=(1+4\delta)\tau.
\end{equation*}
\notag
$$
Since the nonlinearities are exponentially small far away at the leading front, for $s\gg1$, they are not taken into account in the asymptotic power expansion. Hence only the linear part of equation (6.1) participates in the constructions, and the change of variables can be written as
$$
\begin{equation*}
\psi(s,\tau)=2\alpha\,\tau^{1/2}e^{-z}u(z,\tau), \qquad z=\frac{s^2}{4\tau}\,.
\end{equation*}
\notag
$$
The factor $\alpha\ne0$ is derived from the original wave asymptotics (2.1) at the leading front. It is convenient to detach it to avoiding multiple repetitions in the formulae that follow. In the case when $\alpha=0$ we must move out the remaining parameter $\beta$ as a coefficient and drop the increasing factor $\tau^{1/2}$. The exponential $\exp(-s^2/(4\tau))$ is a significant part of the fundamental solution of the heat equation. For $\delta=0$ such an ansatz looks natural for equation (6.1), whose main part is the heat operator, whereas the remaining terms with coefficient $\nu\approx \tau^{-1}$ are small. For $\delta\ne0$ these arguments do not look sound because of the terms containing the second-order derivatives $\partial^2_{s,\tau}\psi$ and $\partial^2_{\tau}\psi$, which enter (6.1) without small coefficients. That these terms are small in comparison to the leading terms of the asymptotic expression as $\tau\to\infty$ can be seen by comparing the expressions for the first derivatives of the new variable
$$
\begin{equation*}
\frac{\partial z}{\partial s}=\tau^{-1/2}\sqrt z\,,\qquad \frac{\partial^2 z}{\partial s^2}=\frac{1}{2}\,\tau^{-1}, \quad\text{and}\quad \frac{\partial z}{\partial\tau}=-\tau^{-1}z
\end{equation*}
\notag
$$
and the expressions for higher derivatives
$$
\begin{equation*}
\frac{\partial^2 z}{\partial s\,\partial \tau}= \frac{1}{2}\,\tau^{-3/2}\biggl(\sqrt z-\frac{1}{\sqrt z}\biggr)\quad\text{and}\quad \frac{\partial^2 z}{\partial\tau^2}=2\tau^{-2}z.
\end{equation*}
\notag
$$
In view of these relations, after the above change the expressions for derivatives reduce to the form The leading terms in the equation are the ones with the derivatives
$$
\begin{equation*}
\partial^2_s\psi=\tau^{-1/2}\sqrt z\,\partial_z \bigl[\sqrt z\,\partial_z(u\,e^{-z})\bigr]
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\partial_\tau\psi=\tau^{-1/2}\biggl[\biggl(\frac{1}{2}u+ \tau\partial_\tau u\biggr)e^{-z}-z\partial_z(u\,e^{-z})\biggr].
\end{equation*}
\notag
$$
The other derivatives contain smaller factors in the order of the small quantity $1/\tau\to0$:
$$
\begin{equation*}
\begin{aligned} \, \partial^2_{\tau,s}\psi&=\tau^{-1}\sqrt z\,\partial_z\biggl[\biggl(\frac{1}{2}u+\tau\partial_\tau u\biggr)e^{-z}-z\partial_z(u\,e^{-z})\biggr] \\ &=\sqrt z\,\partial_z(\partial_\tau ue^{-z})- \tau^{-1}z\,\partial_z\bigl[\sqrt z\,\partial_z(u\,e^{-z})\bigr] \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \partial^2_{\tau}\psi&=\tau^{-3/2}\biggl\{\biggl(-\frac{1}{4}\,u+ \tau\partial_\tau u+\tau^2\partial^2_\tau u\biggr)e^{-z}+ z\,\partial_z[z\partial_z(u\,e^{-z})] \\ &\qquad-2z\partial_z\biggl[\biggl(\frac{1}{2}\,u+\tau\partial_\tau u\biggr)e^{-z}\biggr]\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
We write the equation for the new function $u(z,\tau)$ in the form
$$
\begin{equation}
z\,\frac{\partial^2u}{\partial z^2}+\biggl(\frac{1}{2}-z\biggr)\, \frac{\partial u}{\partial z}-\bigl[1+\nu\tau(1+4\delta)\bigr]u- \tau\,\frac{\partial u}{\partial\tau}=\tau^{-1/2}h_1+\tau^{-1}\delta\,h_2.
\end{equation}
\tag{7.1}
$$
Here the left-hand side combines the terms with factors that are not small as $\tau\to\infty$; in particular, The right-hand side combines the terms with factors of other orders of smallness, namely,
$$
\begin{equation*}
\begin{aligned} \, h_1={}&{-}\tau\,\nu(1+8\delta)e^{z}\sqrt z\,\partial_z(u\,e^{-z})\\ &-\frac{4\delta}{1+4\delta}\,e^{z}\sqrt z\,\partial_z\biggl[\biggl(\frac{1}{2}u +\tau\partial_\tau u\biggr)e^{-z}-z\partial_z(u\,e^{-z})\biggr] \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, h_2={}&\tau\,\nu(4+\nu)e^{z}\sqrt z\,\partial_z\bigl[\sqrt z\,\partial_z (u\,e^{-z})\bigr]-\tau^2\,\nu^2\bigl[\tau^{-1/2}\,2e^{z}\sqrt z\, \partial_z(u\,e^{-z})-u\bigr] \\ &-2\tau\nu\,\frac{1}{1+4\delta}\,e^{z}\biggl\{\tau^{-1/2}\sqrt z\, \partial_z\biggl[\biggl(\frac{1}{2}u+\tau\partial_\tau u\biggr)e^{-z} -z\partial_z(u\,e^{-z})\biggr] \\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad-\biggl[\biggl(\frac{1}{2}u+\tau\partial_\tau u\biggr)e^{-z}- z\partial_z(u\,e^{-z})\biggr]\biggr\} \\ &+\frac{1}{(1+4\delta)^2}\biggl\{\biggl(-\frac{1}{4}\,u+ \tau\partial_\tau u+\tau^2\partial^2_\tau u\biggr)+ e^{z}z\partial_z[z\partial_z(u\,e^{-z})] \\ &\qquad\qquad-2e^{z}z\partial_z\biggl[\biggl(\frac{1}{2}u+ \tau\partial_\tau u\biggr)e^{-z}\biggr]\biggr\}- \tau^2\,\nu'\bigl[\tau^{-1/2}e^{z}\sqrt z\, \partial_z(u\,e^{-z})-u\bigr]. \end{aligned}
\end{equation*}
\notag
$$
We need these detailed and cumbersome expressions justify the choice of the constants $c_{n,k}$ in asymptotic constructions. 7.1. The ansatz and recurrence systemThe expansion (6.4) in the intermediate layer suggests that the structure of the asymptotics as $t\to\infty$ has the form
$$
\begin{equation}
u(z,\tau)\,{=}\,U_{0}(z)+\tau^{-1/2}U_1(z)+\tau^{-1}\log\tau\,U_{2,1}(z) +\tau^{-1}U_{2}(z)+\tau^{-3/2}\log \tau\,U_{3,1}(z)+\dotsb.
\end{equation}
\tag{7.2}
$$
The asymptotic expansion (7.2) for (7.1) is constructed under an additional restriction: there must be matching with the inner expansion (6.4) in the intermediate layer as $z\to0$. This reduces to the condition that the asymptotics of the coefficients $U_{n,k}(z)$ as $z\to0$ must coincide with the coefficients of (6.4). Substituting the ansatz (7.2) into (7.1) and equating the coefficients of like powers $t^{-n/2}\log^kt$, we arrive at a recurrence system for the $U_{n,k}(z)$. In this case the original ansatz (4.4) is used for the velocity $\nu(t)$; in view of the time change $t=\tau(1+4\delta)$ we rewrite it in the form
$$
\begin{equation*}
\begin{aligned} \, \nu\tau(1+4\delta)&=c_0+c_1\tau^{-1/2}\,\frac{1}{\sqrt{1+4\delta}}+ c_{2,1}\frac{1}{1+4\delta}\,\tau^{-1}\log\tau \\ &\qquad+[c_{2}+c_{2,1}\log(1+4\delta)]\frac{1}{1+4\delta}\,\tau^{-1}+\cdots\,. \end{aligned}
\end{equation*}
\notag
$$
Now a key role is played by the Kummer operator
$$
\begin{equation*}
\mathcal{K}_{n}\stackrel{\rm def}{=} z\,\frac{d^2}{dz^2}+(b-z)\,\frac{d}{dz}-a_n,\qquad z>0,\quad n=0,1,2,\dots,
\end{equation*}
\notag
$$
where the coefficient $b$ is $1/2$ and the second coefficient $a_n$ depends on the constant $c_0$ and the step of approximation: $a_n=c_0+1-n/2$. At the first steps we obtain the equations
$$
\begin{equation}
\begin{aligned} \, \mathcal{K}_{n}U_{n}&\equiv z\,\frac{d^2U_{n}}{dz^2}+ \biggl(\frac{1}{2}-z\biggr)\frac{dU_{n}}{dz}- \biggl(c_0+1-\frac{n}{2}\biggr)U_{n}=H_n(z),\qquad n=0,1,2, \nonumber \\ \mathcal{K}_{2}U_{2,1}&\equiv z\,\frac{d^2U_{2,1}}{dz^2}+ \biggl(\frac{1}{2}-z\biggr)\,\frac{dU_{2,1}}{dz}-c_0U_{2,1}=H_{2,1}(z). \end{aligned}
\end{equation}
\tag{7.3}
$$
At the initial step the equation is homogeneous with $H_0(z)\equiv0$. At the subsequent steps the right-hand sides are as follows:
$$
\begin{equation}
\begin{aligned} \, H_1(z)&=\frac{c_1}{\sqrt{1+4\delta}}\,U_0- c_0\frac{1+8\delta}{1+4\delta}\,e^z\sqrt z\,\partial_z(U_0\,e^{-z})\\ &\qquad+ \frac{4\delta}{1+4\delta}\,e^z z\,\partial_z \bigl[\sqrt z\,\partial_z(U_0\,e^{-z})\bigr], \\ H_{2,1}&=c_{2,1}\,\frac{1}{{1+4\delta}}\,U_0, \\ H_2(z)&=\frac{c_{2}+c_{2,1}\log(1+4\delta)}{{1+4\delta}}\,U_0+U_{2,1}+ c_1\,\frac{1}{\sqrt{1+4\delta}}\,U_1 \\ &\qquad-c_1\,\frac{1+8\delta}{(1+4\delta)^{3/2}}\,\sqrt z\,e^z\partial_z (U_0\,e^{-z})-c_0\,\frac{1+8\delta}{1+4\delta}\,\sqrt z\, e^z\partial_z(U_1\,e^{-z}) \\ &\qquad-\frac{4\delta}{1+4\delta}\,\frac{1}{2}\,e^{z}\sqrt z\,\partial_z (U_1\,e^{-z})+\frac{4\delta}{1+4\delta}\,ze^z\partial_z \bigl[\sqrt z\,\partial_z(U_1\,e^{-z})\bigr] \\ &\qquad+\delta\biggl\{c_0\,\frac{4}{1+4\delta}\,e^{z}\sqrt z\,\partial_z\, \bigl[\sqrt z\,\partial_z(U_0\,e^{-z})\bigr]+ c_0^2\,\frac{1}{(1+4\delta)^2}\,U_0 \\ &\qquad\qquad+2c_0\,\frac{1}{(1+4\delta)^2}\biggl[\frac{1}{2}U_0- e^zz\partial_z(U_0e^{-z})\biggr] \\ &\qquad\qquad+\frac{1}{(1+4\delta)^2}\biggl[-\frac{1}{4}U_0- e^zz\partial_z(U_0\,e^{-z})+e^zz\partial_z \bigl[z\partial_z(U_0\,e^{-z})\bigr]\biggr] \\ &\qquad\qquad-c_0\,\frac{1}{(1+4\delta)^2}\,U_0\biggr\}. \end{aligned}
\end{equation}
\tag{7.4}
$$
It is important to note that the exponent of the logarithm in the general term $\tau^n\log^k\tau\,U_{n,k}(z)$ of the series (7.2) is non-negative and grows with $n$; however, this growth is bounded, namely, $k\leqslant n/2$. Therefore, the asymptotic expansion (7.2) is of power nature. The full procedure of the construction of an asymptotic solution in the form of a series (7.2) resembles the usual perturbation method for problems with a small parameter. However, there is a distinction: one feature of the recurrence system of equations (7.3) is the dependence of the operator $\mathcal{K}_{n}$ on $n$. This is because we seek an expansion in the independent variable and is due to the structure of the original equation (7.1), which contains the term $\tau\partial_\tau u$. Note that the Kummer operator in the equations for the coefficients $U_{n,k}$ depends on $n$ but is independent of $k$. Terms related to derivatives of $\log t$ occur on the right-hand sides of the subsequent corrections. For example, $U_{2,1}$ is involved in the formula for $H_2$. This term plays a crucial role in the construction we propose and makes it possible to avoid the problems associated with a pure power expansion [6] that were revealed in [20]. The differential equations (7.3) for the $U_{n,k}$ are supplemented with asymptotic conditions following from the matching condition. At the first steps they have the form
$$
\begin{equation}
\begin{gathered} \, U_0(z)=\sqrt z+\mathcal{O}(z),\quad U_1(z)=\frac{\beta}{2\alpha}+\mathcal{O}(z),\quad U_2(z)=\frac{c_0A_1}{\alpha}\sqrt z+\mathcal{O}(z), \\ U_{n,1}(z)=\mathcal{O}(z^{3/2}),\quad n=2,3;\qquad z\to0. \end{gathered}
\end{equation}
\tag{7.5}
$$
In the general case we have
$$
\begin{equation*}
U_{n,k}(z)=\beta_{n,k}+\alpha_{n,k}\sqrt z+\mathcal{O}(z),\qquad z\to0.
\end{equation*}
\notag
$$
The constants $\beta_{n,k}$ and $\alpha_{n,k}$ in the leading term of the asymptotic expression, for example, $c_0A_1/\alpha$, are found from integrals in the inner expansion, as explained in the previous section. The two leading terms in the asymptotic formulae (7.5) are sufficient for determining the coefficients $U_{n,k}(z)$, as we show below. 7.2. Solving the recurrence system of problems7.2.1. A fundamental system of solutionsFor the homogeneous equation a pair of solutions depends on $n$ and is written in terms of the well-known Kummer functions $M(a,b,z)$, which have representations by convergent power series [27]:
$$
\begin{equation}
\begin{gathered} \, \mathcal{M}_1(z;n)=M\biggl(a_{n},\frac{1}{2}\,,z\biggr)\quad\text{and}\quad \mathcal{M}_2(z;n)=\sqrt{z}\,M\biggl(a_{n}+\frac{1}{2}\,,\frac{3}{2}\,,z\biggr), \\ a_n=c_0+1-\frac{n}{2}\,. \nonumber \end{gathered}
\end{equation}
\tag{7.6}
$$
So, as $z\to0$, the functions $\mathcal{M}_1(z;n)$ and $\mathcal{M}_2(z;n)$ have expansions in integer and half-integer powers of $z$ of the form
$$
\begin{equation}
\begin{aligned} \, \mathcal{M}_1(z;n)&=1+2a_nz+\mathcal{O}(z^2), \\ \mathcal{M}_2(z;n)&=\sqrt z\,\biggl[1+\frac{2a_n+1}{3}\,z+ \mathcal{O}(z^2)\biggr], \end{aligned}\qquad z\to0.
\end{equation}
\tag{7.7}
$$
The asymptotic behaviour of the Kummer functions $M(a,b,z)$ as $z\to\infty$ depends essentially on the parameter $a$ and is characterized by exponential growth in the general case [27], namely,
$$
\begin{equation}
M(a,b,z)=\frac{\Gamma(b)}{\Gamma(a)}\,z^{a-b}\exp(z) \bigl[1+\mathcal{O}(z^{-1})\bigr], \qquad z\to\infty\quad (a\ne-m).
\end{equation}
\tag{7.8}
$$
If $a=-m$ is a non-positive integer, then the function $M(-m,b,z)=P_m(z)$ is a polynomial of degree $m$. In this case a zero factor occurs formally in the exponential asymptotics (7.8), which is due to poles of the gamma function [27]. For the pair of solutions under consideration the Wronskian $W$ does not depend on the parameter $a_n$ and has the form [27]
$$
\begin{equation*}
W[\mathcal{M}_1,\mathcal{M}_2]\equiv W(z)=\frac{1}{2}\,z^{-1/2}\exp(z).
\end{equation*}
\notag
$$
7.2.2. The solvability of the recurrence systemAll coefficients of the asymptotic solution (7.2) are expressed below in terms of the basis solutions $\mathcal{M}_1(z;n)$ and $\mathcal{M}_2(z;n)$, $n=0,1,2,\dots$, and integrals of them. To solve the inhomogeneous equation (7.3) with the Kummer operator $\mathcal{K}_{n}U=H(z)$, a particular solution of the form
$$
\begin{equation*}
\widetilde U(z)=\mathcal{M}_2(z;n)\int_0^z \frac{\mathcal{M}_1(\zeta;n)H(\zeta)}{\zeta W(\zeta)}\,d\zeta- \mathcal{M}_1(z;n)\int_0^z\frac{\mathcal{M}_2(\zeta;n) H(\zeta)}{\zeta W(\zeta)}\,d\zeta
\end{equation*}
\notag
$$
is used. It follows from the structure of the fundamental system of solutions $\mathcal{M}_1(z;n)$ and $\mathcal{M}_2(z;n)$ that the first two terms of the asymptotics in the boundary conditions (7.5) fully determine a solution of the second-order differential equations (7.3). Lemma 3. For any set of constants $c_n$ and $c_{n,k}$ the recurrence system of equations (7.3) with conditions (7.5) is uniquely solvable. Proof. At the initial step $n=0$ the general solution of the homogeneous equation contains two arbitrary constants, namely,
$$
\begin{equation*}
U_{0}(z)=\beta_0\mathcal{M}_1(z;0)+\alpha_0\mathcal{M}_2(z;0),\qquad \beta_0,\alpha_0=\operatorname{const}.
\end{equation*}
\notag
$$
In view of (7.8) this function has the asymptotic behaviour
$$
\begin{equation*}
U_{0}(z)=\beta_0+\alpha_0\sqrt z+\mathcal{O}(z),\qquad z\to0.
\end{equation*}
\notag
$$
The boundary conditions (7.5) fully determine the constants: The resulting solution $U_{0}(z)=\mathcal{M}_2(z;0)$ has an expansion in half-integer powers as $z\to0$.
At the next steps $n=1,2,\dots$ and $k=1,2,\dots,\lfloor n/2\rfloor$ the right-hand sides $H_{n,k}(z)$ are formed of the previous approximations by means of the operator $\sqrt z\,d/dz$. Hence the structure of an asymptotic expansion in integer and half-integer non-negative powers of $z$ is preserved in the transition from $U_{m,k}(z)$, $m<n$, to $H_{n,k}(z)$. To go over from $H_{n,k}(z)$ to $U_{n,k}(z)$ we use a particular solution of the inhomogeneous equation expressed in terms of integrals as follows:
$$
\begin{equation*}
\widetilde U_{n,k}(z)=\mathcal{M}_2(z;n)\int_0^z \frac{\mathcal{M}_1(\zeta;n)H_{n,k}(\zeta)}{\zeta W(\zeta)}\,d\zeta- \mathcal{M}_1(z;n)\int_0^z\frac{\mathcal{M}_2(\zeta;n) H_{n,k}(\zeta)}{\zeta W(\zeta)}\,d\zeta.
\end{equation*}
\notag
$$
The asymptotic behaviour at zero of the integrands is described by the relations
$$
\begin{equation*}
\frac{\mathcal{M}_1(\zeta;n)H_{n,k}(\zeta)}{\zeta W(\zeta)}= \frac{1}{\sqrt\zeta}\,\mathcal{O}(1)\quad\text{and}\quad \frac{\mathcal{M}_2(\zeta;n)H_{n,k}(\zeta)}{\zeta W(\zeta)}= \mathcal{O}(1),\qquad \zeta\to0.
\end{equation*}
\notag
$$
Therefore, the particular solution has the asymptotic behaviour and can be expanded in a series in integer and half-integer powers.
The required function $U_{n,k}(z)$ can also include a solution of the homogeneous equation, which is fully determined by the constants in the boundary condition, namely, for example, The solution $U_{n,k}(z)$ we have obtained can be expanded in integer and half-integer non-negative powers as $z\to0$. So the lemma is proved by induction. $\Box$The above lemma implies the following result. Theorem 2. For any set of constants $c_n$ and $c_{n,k}$ a segment of the series (7.2) is an asymptotic solution as $t\to\infty$ uniformly with respect to $z$ on any interval The asymptotic coincidence of the solutions follows from the uniqueness of coefficients of the in the intermediate layer [24]. The outer expansion (7.2) constructed is rewritten in the intermediate layer. In this case the asymptotics as $z\to0$ is used for the coefficients $U_{n,k}(z)\,e^{-z}$. As a result, after changing to the variables $s$ and $\tau$ we obtain an asymptotic series similar to (6.2), whose coefficients are polynomials. Since this series remains an asymptotic solution in the intermediate area, its coefficients satisfy the same recurrence system of equations (6.3). The matching condition we use ensures the equality of integration constants with the linear parts of the polynomials $\Psi_{n,k}(s)$ derived from the inner expansion. The polynomial derived from the outer expansion coincides with $\Psi_{n,k}(s)$ by the uniqueness of the solution of a second-order differential equation with two initial conditions (for $s=0$). The series (7.2) is asymptotic as $t\to\infty$ uniformly with respect to $z$ on each finite interval $0\leqslant z\leqslant Z<\infty$. As an appropriate asymptotic solution, this series is suitable beginning with the intermediate layer, namely, for $t^{-1}\ll z\leqslant Z$. We can clarify and extend the suitability domain of the asymptotic solution (7.2) by making it dependent on $t$, provided that in the asymptotics as $z\to\infty$ for the coefficients $U_{n,k}(z)$ we take the growth rate into account. These coefficients consist of terms with different behaviour at infinity. Some of these terms contain the growing exponential $\exp(z)$ in view of the structure of the basis solutions (7.6) and (7.8). Therefore, the comparison of different coefficients in power asymptotic expansions becomes inefficient. If we manage to exclude exponential growth, then the power growth $U_{n,k}(z)\approx z^{n/2}$ persists and the domain where the expansion is asymptotic is described by the usual relation $t^{-1}\ll z\ll t$ [6]. It must be emphasized that there are no formal grounds (like fighting against secular terms) for the exclusion of exponential growth, since the powers of the exponential function do not increase with the step $n$ of approximation. 8. Asymptotics of the wave velocityTo find the constants $c_{n,k}$ in the velocity asymptotics (4.1) we propose to use an additional condition on the coefficients of the outer expansion (7.2) in the form of the absence of exponential growth as $z\to\infty$. In addition to the above discussion, arguments in favour of this assumption were presented in [6]. On this way only the constants corresponding to the universal part of the asymptotic formula are uniquely determined. The constants with even indices $c_2,c_4,\dots$ remain arbitrary parameters of the solution. They correspond to the non-universal part of the asymptotic expansion, which depends on the initial data. Problems of the connection between the initial functions and these constants is not considered in our paper. 8.1. Calculating $c_0$At the initial step $n=0$, in view of the boundary conditions (7.5) the solution of the homogeneous equation is expressed as follows in terms of one of the Kummer functions:
$$
\begin{equation*}
U_{0}(z)=\mathcal{M}_2(z,0)\equiv \sqrt z\,M\biggl(c_0+\frac{3}{2}\,,\frac{3}{2}\,,z\biggr).
\end{equation*}
\notag
$$
An additional condition on $U_{0}(z)$ is the absence of an exponential growth as $z\to\infty$. This can occur only when $c_0+3/2=-m$ is a non-positive integer. In this case we have where $P_{m,2}(z)$ is a polynomial of degree $m$. In particular, for $c_0=-3/2$ we obtain a zero-degree polynomial:
$$
\begin{equation*}
U_{0}(z)=\sqrt{z}\quad \Longleftrightarrow\quad c_0=-\frac{3}{2}\,.
\end{equation*}
\notag
$$
As $c_0$ we can take any negative half-integer, namely,
$$
\begin{equation*}
c_0=-\frac{3}{2}\,,-\frac{5}{2}\,,-\frac{7}{2}\,,\dots\,.
\end{equation*}
\notag
$$
The further construction is independent of the choice of $c_0$. We fix $c_0= -3/2$ only to find the numerical values of the velocity constants $c_1$ and $c_{2,1}$ in corrections. The condition of the absence of exponential growth at infinity in the higher corrections $U_{n,k}(z)$ leads to equations for $c_{n,k}$ of the form The constant coefficients $\Delta_{n}$ and $J_{n,k}$ are expressed by integrals. Although the equations look trivial, they are unsolvable for even $n$ since their coefficients vanish: while $J_{2}\ne0$. This is a consequence of the orthogonality of the basis solutions, which was established in [20]. The proof of this property is briefly reproduced below.8.2. Orthogonality propertiesIn the series of Kummer’s equations considered for a half-integer constant $c_0\leqslant-3/2$ the parameter $a_n=c_0+1-n/2$ ($n=0,1,2,\dots$) can be an integer or a half-integer. Therefore, the fundamental system of solutions (7.6) has the following property. For each $n$ one of the functions $\mathcal{M}_{1}(z,n)$ and $\mathcal{M}_{2}(z,n)$ is expressed in terms of a polynomial of degree $m$, namely,
$$
\begin{equation*}
\begin{alignedat}{3} \mathcal{M}_1(z;n)&=P_{m,1}(z),&\qquad -m&=a_n,&&\quad \text{if $n$ is odd}, \\ \mathcal{M}_2(z;n)&=\sqrt z\,P_{m,2}(z),&\qquad -m&=a_n+\frac{1}{2}\,,&&\quad \text{if $n$ is even}. \end{alignedat}
\end{equation*}
\notag
$$
For a pair of smooth functions $u(z)$, $v(z)$ we define their inner product by
$$
\begin{equation*}
\langle u(z),v(z)\rangle=\int_0^\infty\frac{u(z)\,v(z)}{z\,W(z)}\,dz,\qquad W(z)=\frac{1}{2}\,z^{-1/2}\exp(z),
\end{equation*}
\notag
$$
provided that the integral converges. Lemma 4. For distinct numbers $n=i,j$, $i\ne j$, the fundamental system of solutions $\mathcal{M}_{1}(z,n)$, $\mathcal{M}_{2}(z,n)$ of Kummer’s equation has the following property: and Proof. The orthogonality property (8.1) follows from the orthogonality of Hermite polynomials. In the case when the Kummer functions $M(a,b,z)$ are polynomials, they correspond to the Hermite polynomials $\operatorname{He}(y)$, where the variables are related by $z=y^2/2$ ([27], Chap. 13.6). In particular, the basis solutions chosen are expressed in terms of Hermitec polynomials of degrees $2m$ and $2m+1$, namely,
$$
\begin{equation*}
\begin{alignedat}{3} \mathcal{M}_1(z;n)&=\operatorname{const}\cdot\operatorname{He}_{2m}(y),&\qquad -m&=a_n,&&\quad \text{if $n$ is odd}, \\ \mathcal{M}_2(z;n)&=\operatorname{const}\cdot \operatorname{He}_{2m+1}(y),&\qquad -m&=a_n+\frac{1}{2}\,,&&\quad \text{if $n$ is even}. \end{alignedat}
\end{equation*}
\notag
$$
In (8.1) we have integrals of even functions of $y$ with exponential weight over the half-axis. They obviously reduce to integrals over the whole axis and vanish because of the orthogonality of Hermite polynomials.
In (8.2) we have integrals of functions odd in $y$ over the half-axis. In this case the integral is calculated using Kummer’s equation. For the homogeneous equation
$$
\begin{equation*}
\mathcal{K}_{i}u\equiv z\,\frac{d^2u}{dz^2}+ \biggl(\frac{1}{2}-z\biggr)\frac{du}{dz}-a_iu=0,\qquad z>0,
\end{equation*}
\notag
$$
we consider solutions $u(z;i)$. They are specified by conditions at zero of the first type or of the second type The equation for $u(z;i)$ is multiplied by $u(z;j)/(z W(z))$. Upon integration, we arrive at the relation
$$
\begin{equation*}
a_i\langle u (z;i),u (z;j)\rangle=\int_0^\infty \biggl[u^{\prime\prime}(z;i)\,u (z;j)+\biggl(\frac{1}{2z}-1\biggr) u'(z;i)\,u(z;j)\biggr]\frac{1}{W(z)}\,dz.
\end{equation*}
\notag
$$
The integral of the term containing a second derivative is taken by parts. In view of the relation for the Wronskian and by virtue of a similar relation for $u(z;j)$ we have the equality
$$
\begin{equation*}
(a_i-a_{j})\langle u (z;i),u (z;j)\rangle= \frac{u'(z;i)u(z;j)-u(z;i)u'(z;j)}{W(z)}\bigg|_{z=0}^\infty.
\end{equation*}
\notag
$$
Note that the expression in the numerator does not coincide with the Wronskian $W(z)$ of the fundamental system of solutions, since $u(z;i)$ and $u(z;j)$ correspond to different equations.
Furthermore, we must take account of the expression for the Wronskian, which yields and of the asymptotic behaviour of solutions at the endpoints of the interval $z>0$. In the case when $u(z;i)$ and $u(z,j)$ are of the same type, the expression tends to zero as $z\to0$. In the case when $u(z;i)$ and $u(z;j)$ are of different types, the same expression tends to $+1$ or $-1$ as $z\to0$, depending on which function is of the second type.For solutions of polynomial type under consideration the expression (8.3) yields zero in the limit at infinity. Finally, in view of the relation $a_i-a_j=(j-i)/2$ we obtain the required equality (8.2). On the way we have also verified (8.1). $\Box$ 8.3. Calculating $c_1$At the step $n=1$ we have an inhomogeneous Kummer equation with parameter $a_1=c_0+1/2$ for the correction $U_1(z)$. The quantity $a_1=-m$, where $m\geqslant1$, turns out to be a negative integer. The corresponding fundamental system of solutions has the following properties:
$$
\begin{equation*}
\mathcal{M}_1(z;1)=M\biggl(c_0+\frac{1}{2}\,,\frac{1}{2}\,,z\biggr)\equiv P_{m,1}(z)=1-2mz+\mathcal{O}(z^{2})
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{M}_2(z;1)=\sqrt z\,M\biggl(1+ c_0,\frac{3}{2}\,,z\biggr)= \frac{\Gamma(1/2)}{\Gamma(1+c_0)}\,z^{c_0}\exp(z)[1+\mathcal{O}(z^{-1})]
\end{equation*}
\notag
$$
as $z\to\infty$. A particular solution of the inhomogeneous equation is written in terms of integrals as follows:
$$
\begin{equation}
\begin{aligned} \, \widetilde U_1(z)&=2\mathcal{M}_2(z;1)\int_0^z \frac{\mathcal{M}_1(\zeta;1)H_1(\zeta)}{2\zeta W(\zeta)}\,d\zeta \nonumber \\ &\qquad-2\mathcal{M}_1(z;1)\int_0^z \frac{\mathcal{M}_2(\zeta;1)H_1(\zeta)}{2\zeta W(\zeta)}\,d\zeta. \end{aligned}
\end{equation}
\tag{8.4}
$$
Here the coefficient 2 in front of the integral is related to the Wronskian:
$$
\begin{equation}
\begin{aligned} \, \frac{H_1(\zeta)}{2\zeta W(\zeta)}&=\frac{c_1}{\sqrt{1+4\delta}}\, \frac{U_0e^{-\zeta}}{\sqrt \zeta} \nonumber \\ &\qquad-\frac{1}{1+4\delta}\bigl\{c_0{(1+8\delta)} \partial_\zeta(U_0\,e^{-\zeta})-4\delta\sqrt\zeta\,\partial_z \bigl[\sqrt\zeta\,\partial_\zeta(U_0\,e^{-\zeta})\bigr]\bigr\}. \end{aligned}
\end{equation}
\tag{8.5}
$$
Since $H_{1}(\zeta)=\mathcal{O}(1)$ as $\zeta\to0$, the particular solution (8.4) has the asymptotic behaviour The required function $U_1(z)$ can also include a solution of the homogeneous equation. It follows from a comparison with the boundary condition (7.5) that only the one of the two solutions that turns out to be a polynomial is suitable in our case: The resulting function $U_1(z)$ can exponentially grow at infinity because of the factor $\mathcal{M}_2(z,1)$ involved in the expression for $\widetilde U_1(z)$. The condition of the absence of exponential growth in the solution $U_1(z)$ leads to vanishing of the integral over the half-axis that multiplies $\mathcal{M}_2(z,1)$, which can be stated as the peculiar orthogonality condition In view of the formula for $H_1$ and the expression for $U_0(z)=\mathcal{M}_2(z;0)$, this gives an equation for the constant $c_1$:
$$
\begin{equation*}
\begin{aligned} \, &\frac{c_1}{\sqrt{1+4\delta}} \langle \mathcal{M}_1(z;1),\mathcal{M}_2(z;0)\rangle \\ &\qquad=\frac{1}{1+4\delta}\bigl\langle e^z\mathcal{M}_1(z;1), \bigr[c_0(1+8\delta)\sqrt z\,\partial_z\bigl(\mathcal{M}_2(z;0)\,e^{-z}\bigr) \\ &\qquad\qquad-4\delta z\,\partial_z\bigl[\sqrt z\,\partial_z \bigl(\mathcal{M}_2(z;0)\,e^{-z}\bigr)\bigr]\bigr]\bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
The coefficient on the left-hand side is $-2$ owing to Lemma 4. We rewrite the right-hand side in terms of integrals in an explicit form, using the formula for the inner product. Cancelling $-2$ we arrive at the relation
$$
\begin{equation*}
\begin{aligned} \, \frac{c_1}{\sqrt{1+4\delta}}&=-\frac{1}{1+4\delta}\int_0^\infty \mathcal{M}_1(z;1)\bigl[\bigl(c_0(1+8\delta)-2\delta\bigr) \partial_z(\mathcal{M}_2(z;0)\,e^{-z}) \\ &\qquad-4\delta z\,\partial_z^2 \bigl(\mathcal{M}_2(z;0)\,e^{-z}\bigr)\bigr]\,dz. \end{aligned}
\end{equation*}
\notag
$$
If $c_0$ is fixed, then the basis solutions are specified and the integral is calculated in terms of the gamma function. For the convenience of calculations we can split the integral into two and reduce the equation to the form
$$
\begin{equation*}
\frac{c_1}{\sqrt{1+4\delta}}=c_0A_0+\frac{\delta}{1+4\delta}\,A_1.
\end{equation*}
\notag
$$
We express the coefficients $A_0$ and $A_1$ in terms of integrals as follows:
$$
\begin{equation*}
A_0=-\int_0^\infty\mathcal{M}_1(z;1)\,\partial_z(\mathcal{M}_2(z;0)\, e^{-z})\,dz=\int_0^\infty \partial_z\mathcal{M}_1(z;1)\,\mathcal{M}_2(z;0)\, e^{-z}\,dz
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
A_1 =-\int_0^\infty\bigl[(4c_0-2)\mathcal{M}_1(z;1)\, \partial_z(\mathcal{M}_2(z;0)\,e^{-z})-4z\,\partial_z^2(\mathcal{M}_2(z;0)\, e^{-z})\bigr]\,dz.
\end{equation*}
\notag
$$
Taking the last integral by parts, we reduce it to
$$
\begin{equation*}
A_1=\int_0^\infty\bigl[(4c_0+6)\,\partial_z\mathcal{M}_1(z;1)\, \mathcal{M}_2(z;0)\,e^{-z}+4z\,\partial_z^2\mathcal{M}_1(z;1)\, \mathcal{M}_2(z;0)\,e^{-z}\bigr]\,dz.
\end{equation*}
\notag
$$
If $c_0=-3/2$, then the basis solutions are explicitly defined:
$$
\begin{equation*}
\mathcal{M}_2(z;0)=\sqrt z\quad\text{and}\quad \mathcal{M}_1(z;1)=1-2z
\end{equation*}
\notag
$$
and the integrals can be calculated:
$$
\begin{equation*}
A_0=-2\int_0^\infty\sqrt z\,e^{-z}\,dz=-\sqrt\pi,\,\qquad A_1=0.
\end{equation*}
\notag
$$
As a result, we infer the relation It follows, in particular, that the second correction in the velocity asymptotics (4.3) is independent of the parameter $\delta$ [6], namely,
$$
\begin{equation*}
\frac{dS}{dt}=\frac{1}{\sqrt{1+4\delta}}\biggl[2-\frac{3}{2}t^{-1}\biggr]+ t^{-3/2}\,\frac{3}{2}\,\sqrt\pi+\mathcal{O}(t^{-2}\log t).
\end{equation*}
\notag
$$
As for the next correction of order $\mathcal{O}(t^{-2}\log t)$ in the universal part of the asymptotics, we know that it is non-trivial for $\delta=0$ [21]. Its dependence on $\delta$ is to be clarified below. If $c_0$ is chosen in another way, then the basis solutions and the corresponding integrals change. For example, if $c_0=-5/2$, then
$$
\begin{equation*}
\mathcal{M}_2(z;0) =\sqrt zM\biggl(-1,\frac{3}{2}\,,z\biggr)= \sqrt z\,\biggl(1-\frac{2}{3}z\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{M}_1(z;1) =M\biggl(-2,\frac{1}{2}\,,z\biggr)=1-4z+\frac{4}{3}z^2.
\end{equation*}
\notag
$$
In this case one of the integrals remains equal to zero, namely,
$$
\begin{equation*}
A_0=-4\int_0^\infty\sqrt z\,\biggl(1-\frac{2}{3}z\biggr)^2\,e^{-z}\,dz= -\frac{4}{3}\sqrt\pi\quad\text{and}\quad A_1=0.
\end{equation*}
\notag
$$
The formula for the velocity for $c_0=-5/2$ assumes the form
$$
\begin{equation*}
\frac{dS}{dt}= \frac{1}{\sqrt{1+4\delta}}\,\biggl[2-\frac{5}{2}t^{-1}\biggr]+ t^{-3/2}\,\frac{4}{3}\,\sqrt\pi+\mathcal{O}(t^{-2}\log t).
\end{equation*}
\notag
$$
8.4. Calculating the correction with logarithmFor $U_{2,1}(z)$ we have the inhomogeneous equation (7.3) with a half-integer parameter $a_2=c_0$. The quantity where $m\geqslant1$, turns out to be a negative integer. Therefore, the corresponding system of fundamental solutions has the folowing properties:
$$
\begin{equation*}
\mathcal{M}_1(z;2)=M\biggl(c_0,\frac{1}{2}\,,z\biggr)= \frac{\Gamma(1/2)}{\Gamma(c_0)}\,z^{c_0-1/2}\exp(z)[1+\mathcal{O}(z^{-1})]
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathcal{M}_2(z;2) =\sqrt z\,M\biggl(c_0+\frac{1}{2}\,,\frac{3}{2}\,,z\biggr) \equiv\sqrt z\,P_m(z)=\sqrt z\,\biggl[1-\frac{2}{3}mz+ \mathcal{O}(z^{2})\biggr]
\end{equation*}
\notag
$$
as $z\to\infty$. A solution of the inhomogeneous equation is expressed in terms of integrals in the form
$$
\begin{equation}
\begin{aligned} \, \widetilde U_{2,1}(z)&=\mathcal{M}_2(z;2)\int_0^z \frac{\mathcal{M}_1(\zeta;2)H_{2,1}(\zeta)}{\zeta W(\zeta)}\,d\zeta \nonumber \\ &\qquad-\mathcal{M}_1(z;2)\int_0^z\frac{\mathcal{M}_2(\zeta;2) H_{2,1}(\zeta)}{\zeta W(\zeta)}\, d\zeta. \end{aligned}
\end{equation}
\tag{8.6}
$$
Since
$$
\begin{equation*}
H_{2,1}(\zeta)=c_{2,1}U_{0}(\zeta)=c_{2,1}\mathcal{M}_2(\zeta;0)= \mathcal{O}\bigl(\sqrt\zeta\,\bigr),\qquad \zeta\to0,
\end{equation*}
\notag
$$
the particular solution (8.6) is characterized by the asymptotic behaviour It follows from a comparison with boundary condition (7.5) that none of the basis solutions suits in this case. Hence $U_{2,1}$ coincides with the particular solution (8.6), which we express as where the remaining part is represented in terms of integrals of the basis solutions as
$$
\begin{equation}
\begin{aligned} \, \widehat U_{2,1}(z)&= \mathcal{M}_2(z;2)\int_0^z \frac{\mathcal{M}_1(\zeta;2)\mathcal{M}_2(\zeta;0)}{\zeta W(\zeta)}\,d\zeta \nonumber \\ &\qquad-\mathcal{M}_1(z;2)\int_0^z \frac{\mathcal{M}_2(\zeta;2)\mathcal{M}_2(\zeta;0)}{\zeta W(\zeta)}\, d\zeta. \end{aligned}
\end{equation}
\tag{8.7}
$$
Here the function $\mathcal{M}_1(z;2)$ in front of the second integral grows exponentially at infinity. However, the solution $U_{2,1}(z)$ does not have exponential growth, since the integral over the infinite interval is zero because of the orthogonality $\langle\mathcal{M}_2(\zeta;2),\mathcal{M}_2(\zeta;0)\rangle= 0$. The constant $c_{2,1}$ remains undetermined until the next step. 8.5. Calculating $c_{2,1}$At the next step the terms forming the coefficient of $t^{-1}$ yield the inhomogeneous equation (7.3) for $U_{2}(z)$ with the same parameter $a_2=c_0$ and the same basis solutions. A particular solution is expressed in terms of integrals as
$$
\begin{equation*}
\widetilde U_2(z)=\mathcal{M}_2(z;2)\int_0^z \frac{\mathcal{M}_1(\zeta;2)H_2(\zeta)}{\zeta W(\zeta)}\,d\zeta- \mathcal{M}_1(z;2)\int_0^z \frac{\mathcal{M}_2(\zeta;2)H_2(\zeta)}{\zeta W(\zeta)}\,d\zeta.
\end{equation*}
\notag
$$
The asymptotic behaviour of the integrands at zero is described by the relations
$$
\begin{equation*}
\frac{\mathcal{M}_1(\zeta;2)H_2(\zeta)}{\zeta W(\zeta)}= \frac{1}{\sqrt\zeta}\,\mathcal{O}(1)\quad\text{and}\quad \frac{\mathcal{M}_2(\zeta;2)H_2(\zeta)}{\zeta W(\zeta)}=\mathcal{O}(1),\qquad \zeta\to0.
\end{equation*}
\notag
$$
Thus, the particular solution has the asymptotic behaviour The required function $U_2(z)$ can also contain a solution of the homogeneous equation. It follows from a comparison with the boundary condition (7.5) that only the one of the two solutions that is expressed in terms of a polynomial, namely, $\mathcal{M}_2(z;2)=\sqrt z\,P_m(z)$, can be used in this case; hence we have
$$
\begin{equation*}
U_2(z)=\widetilde U_2(z)+\frac{c_0A_1}{\alpha}\,\mathcal{M}_2(z;2).
\end{equation*}
\notag
$$
The function $U_2(z)$ constructed in this way can grow exponentially at infinity because of the factor $\mathcal{M}_1(z,2)$ involved in the expression for the particular solution $\widetilde U_2(z)$. The condition of the absence of exponential growth in $U_2(z)$ leads to the orthogonality condition In view of formula (7.4) for the right-hand side $H_2(z)$, the expression for the leading term $U_0=\mathcal{M}_2(\zeta;0)$, and the equality $U_{2,1}(z)=c_{2,1}\widehat U_{2,1}(z)$, this condition yields an equation for the constant $c_{2,1}$. Of course, the resulting relation does not contain terms proportional to $U_0(z)=\mathcal{M}_2(\zeta;0)$ in view of the orthogonality $\langle \mathcal{M}_2(z;2),\mathcal{M}_2(\zeta;0)\rangle=0$; in particular, there is no term with $c_2$. The equation with coefficients determined by inner products has the trivial form:
$$
\begin{equation*}
c_{2,1}\langle\mathcal{M}_2(z;2),\widehat U_{2,1}(z)\rangle+ \langle \mathcal{M}_2(z;2),\widehat H_2(z)\rangle=0.
\end{equation*}
\notag
$$
Here the coefficients are specified in terms of the above approximations $U_0,U_1$, and $U_{2,1}$, taking the relation
$$
\begin{equation*}
\begin{aligned} \, \widehat H_2(z)&=c_1\frac{1}{\sqrt{1+4\delta}}\,U_1 \\ &\qquad-c_1\frac{1+8\delta}{(1+4\delta)^{3/2}} \sqrt z\,e^z\partial_z(U_0\,e^{-z})- c_0\frac{1+8\delta}{1+4\delta}\sqrt z\,e^z\partial_z(U_1\,e^{-z}) \\ &\qquad-\frac{4\delta}{1+4\delta}\,\frac{1}{2}\,e^{z} \sqrt z\,\partial_z(U_1\,e^{-z})+ \frac{4\delta}{1+4\delta}\,ze^z\partial_z[\sqrt z\,\partial_z(U_1\,e^{-z})] \\ &\qquad+\delta\biggl\{c_0\frac{4}{1+4\delta}\,e^{z}\sqrt z\,\partial_z \bigl[\sqrt z\,\partial_z(U_0\,e^{-z})\bigr]+ 2c_0\,\frac{1}{(1+4\delta)^2}[-e^zz\partial_z(U_0e^{-z})] \\ &\qquad+\frac{1}{{(1+4\delta)^2}}\bigl[-e^zz\partial_z(U_0\,e^{-z})+ e^zz\partial_z[z\partial_z(U_0\,e^{-z})]\bigr]\biggr\} \end{aligned}
\end{equation*}
\notag
$$
into account. To simplify formulae we introduce the notation
$$
\begin{equation}
\Delta=\langle \mathcal{M}_2(z;2),\widehat U_{2,1}(z)\rangle\quad\text{and}\quad J(\delta)=\langle \mathcal{M}_2(z;2),\widehat H_2(z)\rangle.
\end{equation}
\tag{8.8}
$$
Then the equation reads The fundamental questions as to whether the constant $c_{2,1}$ can be found from (8.9) and whether it can be distinct from zero are answered by analyzing the integrals in the inner products (8.8). We perform such an analysis below for $c_{0}=-3/2$, when the Kummer functions are simplest. In this case
$$
\begin{equation*}
\frac{c_1}{\sqrt{1+4\delta}}=-c_{0}\sqrt\pi\quad\text{and}\quad U_0=\sqrt z
\end{equation*}
\notag
$$
and the basis solutions are
$$
\begin{equation*}
\begin{gathered} \, \mathcal{M}_2(z;0)=\sqrt z\,,\qquad \mathcal{M}_1(z;1)=1-2z,\qquad \mathcal{M}_2(z;1)=\sqrt z\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,,z\biggr), \\ \mathcal{M}_1(z;2)=M\biggl(-\frac{3}{2}\,,\frac{1}{2}\,,z\biggr),\quad\text{and}\quad \mathcal{M}_2(z;2)=\sqrt z\,\biggl(1-\frac{2}{3}z\biggr). \end{gathered}
\end{equation*}
\notag
$$
Proof. In view of the formula for the Wronskian, the expression (8.7) can be written as
$$
\begin{equation}
\widehat U_{2,1}(z)\bigl|_{c_0=-3/2}= 2\int_0^z[\mathcal{M}_2(z;2)\mathcal{M}_1(\zeta;2)- \mathcal{M}_1(z;2)\mathcal{M}_2(\zeta;2)]\exp(-\zeta)\, d\zeta.
\end{equation}
\tag{8.10}
$$
Taking (8.10) into account we obtain an expression for $\Delta$ as a repeated integral:
$$
\begin{equation*}
\begin{aligned} \, \Delta&=4\int_0^\infty\,\int_0^z\biggl(1-\frac{2}{3}z\biggr)\exp(-z-\zeta) \\ &\qquad\times\biggl[\sqrt z\,\biggl(1-\frac{2}{3}z\biggr) M\biggl(-\frac{3}{2}\,,\frac{1}{2}\,,\zeta\biggr)- M\biggl(-\frac{3}{2}\,,\frac{1}{2}\,,z\biggr) \sqrt\zeta\,\biggl(1-\frac{2}{3}\zeta\biggr)\biggr]\,d\zeta\,dz. \end{aligned}
\end{equation*}
\notag
$$
We see special Kummer functions, not reducible to elementary functions, under the integral sign. Numerical calculations using well-known software packages yield the non-zero value Since this value is far beyond the limits of calculation error, we have $\Delta\ne0$. The lemma is proved. Thus, equation (8.9) is solvable with respect to the constant $c_{2,1}=J(\delta)/\Delta$. The coefficient $J(\delta)$ defined in (8.8) can be written in terms of integrals of the basis solutions. Unlike $\Delta$, it depends on the parameter $\delta$ via long cumbersome expressions derived from the formula for the function $\widehat H_2(z)$. In the case under consideration, when $c_{0}=-3/2$, the representation of $\widehat H_2(z)$ is slightly simplified to
$$
\begin{equation}
\begin{aligned} \, \widehat H_2(z)&=-c_0\sqrt\pi\,U_1+c_0\sqrt\pi\, \frac{1+8\delta}{1+4\delta}\sqrt z\,e^z\partial_z\bigl(\sqrt z\,e^{-z}\bigr) \nonumber \\ &\qquad-c_0\,\frac{1+8\delta}{1+4\delta}\sqrt z\,e^z\partial_z(U_1\,e^{-z})+ \frac{4\delta}{1+4\delta}\,z^{3/2}e^z\partial_z^2(U_1\,e^{-z}) \nonumber \\ &\qquad+\delta\biggl\{c_0\,\frac{4}{1+4\delta}\,e^{z}\sqrt z\,\partial_z \bigl[\sqrt z\,\partial_z(\sqrt z\,e^{-z})\bigr] \nonumber \\ &\qquad+2c_0\,\frac{1}{(1+4\delta)^2} \bigl[-e^zz\,\partial_z(\sqrt z\,e^{-z})\bigr] \nonumber \\ &\qquad+\frac{1}{(1+4\delta)^2}\, e^zz^2\partial_z^2\bigl(\sqrt z\,e^{-z}\bigr)\biggr\}. \end{aligned}
\end{equation}
\tag{8.11}
$$
The situation is simplest for $\delta=0$, when the original equation is the KPP equation. 8.5.1. Calculating $c_{2,1}$ in the case $\delta=0$ (the KPP equation)Theorem 3. If $c_0=-3/2$, then the orthogonality condition in the case $\delta=0$ fully determines the non-zero constant $c_{2,1}\approx-3.55$. Proof. In our case, when $c_{0}=-3/2$ and $\delta=0$, we have
$$
\begin{equation*}
\widehat H_2(z)=c_0\bigl[\sqrt\pi\,\sqrt z\,e^z\partial_z \bigl(\sqrt z\,e^{-z}\bigr)-\sqrt\pi\,U_1- \sqrt z\,e^z\partial_z(U_1\,e^{-z})\bigr].
\end{equation*}
\notag
$$
This relation involves the first correction part of which is expressed as the integral
$$
\begin{equation*}
\widetilde U_1(z)=\int_0^z\bigl[\mathcal{M}_2(z;1)\mathcal{M}_1(\zeta;1)- \mathcal{M}_1(z;1)\mathcal{M}_2(\zeta;1)\bigr] \frac{H_1(\zeta)}{\zeta W(\zeta)}\,d\zeta,
\end{equation*}
\notag
$$
containing the function
$$
\begin{equation*}
\frac{H_1(\zeta)}{\zeta W(\zeta)}=2\biggl[c_1\,\frac{U_0e^{-\zeta}}{\sqrt\zeta} -c_0\partial_\zeta(U_0\,e^{-\zeta})\biggr]= -2c_0\biggl[\sqrt\pi+\frac{1}{2\sqrt\zeta}-\sqrt\zeta\,\biggr]e^{-\zeta}.
\end{equation*}
\notag
$$
Therefore, the expression for the required coefficient $J(0)$ includes a double repeated integral as one of the terms. The full expression for $J(0)$ can be written as three terms which are calculated as the inner products of certain known functions as follows:
$$
\begin{equation*}
\begin{aligned} \, I_0&=\bigl\langle\mathcal{M}_2(z;2),\sqrt\pi\,\sqrt z\,e^z\partial_z \bigl(\sqrt z\,e^{-z}\bigr)\bigr\rangle, \\ I&=-\bigl\langle\mathcal{M}_2(z;2),\sqrt\pi\,\widetilde U_1+ \sqrt z\,e^z\partial_z(\widetilde U_1\,e^{-z})\bigr\rangle, \\ B&=-\bigl\langle\mathcal{M}_2(z;2),\bigl[\sqrt\pi\,\mathcal{M}_1(z;1)+ \sqrt z\,e^z\partial_z(\mathcal{M}_1(z;1)\,e^{-z})\bigr]\bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
The quantities $B$ and $I_0$, which are expressed in terms of single integrals, can be calculated. One of these turns out to be zero, namely,
$$
\begin{equation*}
\begin{aligned} \, B&=-2\sqrt\pi-\bigl\langle\mathcal{M}_2(z;2),\sqrt z\,e^z\partial_z (\mathcal{M}_1(z;1)\,e^{-z})\bigr\rangle \\ &\qquad-2\sqrt\pi+2\int_0^\infty\sqrt z\, \biggl(1-\frac{2}{3}z\biggr)[3-2z]e^{-z}\,dz \\ &=-2\sqrt\pi+4\int_0^\infty \sqrt z\,e^{-z}\,dz=0. \end{aligned}
\end{equation*}
\notag
$$
The second one reduces to the following integral of elementary functions:
Taking the integral for $I$ by parts, we reduce it to the form
$$
\begin{equation*}
\begin{aligned} \, I&=-2\int_0^\infty\biggl[\biggl(1-\frac{2}{3}z\biggr)\sqrt\pi\,\widetilde U_1 e^{-z}+\sqrt z\,\biggl(1-\frac{2}{3}z\biggr) \partial_z(\widetilde U_1\,e^{-z})\biggr]\,dz \\ &=2\int_0^\infty\biggl[\biggl(1-\frac{2}{3}z\biggr)\sqrt\pi- \biggl(\frac{1}{2\sqrt z}-\sqrt z\biggr)\biggr]\widetilde U_1\,e^{-z}\,dz. \end{aligned}
\end{equation*}
\notag
$$
Substituting in the expression for the particular solution $\widetilde U_1$, we obtain the repeated integral
$$
\begin{equation*}
\begin{aligned} \, I&=4c_0\int_0^\infty\,\int_0^z\biggl[\sqrt\pi\,\biggl(1-\frac{2}{3}z\biggr)- \biggl(\frac{1}{2\sqrt z}-\sqrt z\,\biggr)\biggr]\exp(-z-\zeta) \\ &\qquad\times\biggl[\sqrt\pi+\frac{1}{2\sqrt\zeta}-\sqrt\zeta\,\biggr] \biggl[(1-2\zeta)\sqrt z\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,,z\biggr) \\ &\qquad-\sqrt\zeta\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,,\zeta\biggr) (1-2z)\biggr]\,d\zeta\,dz. \end{aligned}
\end{equation*}
\notag
$$
To calculate it we use numerical methods. Approximate computations using the Mathematica software package yield Since the approximate value is far beyond the limits of calculation error, the integral $I$ is not zero. Equation (8.9) yields an approximate value of the required constant: Below we discuss the dependence of $c_{2,1}$ on the parameter $\delta>0$. 8.5.2. Calculating $c_{2,1}$ in the case $\delta>0$ (a hyperbolic equation)Theorem 4. If $c_0=-3/2$, then for $\delta>0$ the orthogonality condition fully determines the constant $c_{2,1}=J(\delta)/\Delta$. The dependence on the parameter $\delta$ is described by the relation with the non-zero coefficient $J_2\approx0.88$. Proof. Since the coefficient $\Delta$ in (8.9) is non-zero and independent of $\delta$, the equation is solvable with respect to $c_{2,1}$ for all $\delta>0$; thus, $c_{2,1}={J(\delta)}/{\Delta}$. The $\delta$-dependence of $J(\delta)$ manifests itself in the formula8
$$
\begin{equation*}
J(\delta)=\langle\mathcal{M}_2(z;2),\widehat H_2(z)\rangle= 2\int_0^\infty\mathcal{M}_2(z;2)\frac{\widehat H_2(z)}{2zW(z)}\,dz
\end{equation*}
\notag
$$
through rational expressions of $1+4\delta$ in the function (8.11), which lead to the relation
$$
\begin{equation*}
\begin{aligned} \, \frac{\widehat H_2(z)}{2zW(z)}&=-c_0\sqrt\pi\,\frac{1}{\sqrt z}\,U_1e^{-z} +c_0\sqrt\pi\,\frac{1+8\delta}{1+4\delta}\,\partial_z \bigl(\sqrt z\,e^{-z}\bigr) \\ &\qquad-c_0\,\frac{1+8\delta}{1+4\delta}\, \partial_z(U_1\,e^{-z})+\frac{4\delta}{1+4\delta}\,z\, \partial_z^2(U_1\,e^{-z}) \\ &\qquad+\delta\biggl\{c_0\frac{4}{1+4\delta}\,\partial_z \bigl[\sqrt z\,\partial_z(\sqrt z\,e^{-z})\bigr]+ 2c_0\,\frac{1}{(1+4\delta)^2}\bigl[-\sqrt z\,\partial_z \bigl(\sqrt z\,e^{-z}\bigr)\bigr] \\ &\qquad+\frac{1}{(1+4\delta)^2}\,z^{3/2}\partial_z^2(\sqrt z\,e^{-z})\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
From this we can see the required structure with respect to $\delta$ in the form (8.12), where the coefficients $J_0$, $J_1$, and $J_2$ are expressed by integrals.
When calculating coefficients in (8.12) we must take account of the fact that the first correction $U_1(z)=\widetilde U_1(z)+\beta\mathcal{M}_1(z;1)$ also involves rational dependence on $1+4\delta$ via the part expressed in terms of integrals in (8.4):
$$
\begin{equation*}
\widetilde U_1(z)=2\int_0^z\bigl[\mathcal{M}_2(z;1)\mathcal{M}_1(\zeta;1)- \mathcal{M}_1(z;1)\mathcal{M}_2(\zeta;1)\bigr]\, \frac{H_1(\zeta)}{2\zeta W(\zeta)}\,d\zeta.
\end{equation*}
\notag
$$
The fraction in the integrand is specified by (8.5) and in our case, when $c_0=-3/2$ and $U_0=\sqrt \zeta$, it is represented in the form
$$
\begin{equation}
\begin{aligned} \, \frac{H_1(\zeta)}{2\zeta W(\zeta)}&=\frac{3}{2}\sqrt\pi\,U_0e^{-\zeta} \nonumber \\ &\qquad+\frac{1}{1+4\delta}\biggl\{\frac{3}{2}{(1+8\delta)}\partial_\zeta (U_0\,e^{-\zeta})+{4\delta}\sqrt\zeta\,\partial_\zeta \bigl[\sqrt\zeta\,\partial_\zeta(U_0\,e^{-\zeta})\bigr]\biggr\} \nonumber \\ &=\frac{3}{2}\sqrt\pi\,\sqrt\zeta\,e^{-\zeta}+3\partial_\zeta \bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+\sqrt\zeta\,\partial_\zeta\, \bigl[\sqrt\zeta\,\partial_\zeta(\zeta\,e^{-\zeta})\bigr] \nonumber \\ &\qquad-\frac{1}{1+4\delta}\biggl[\frac{3}{2}\,\partial_\zeta \bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+\sqrt\zeta\,\partial_\zeta \bigl[\sqrt\zeta\,\partial_\zeta \bigl(\sqrt\zeta\,e^{-\zeta}\bigr)\bigr]\biggr]. \end{aligned}
\end{equation}
\tag{8.13}
$$
This relation can be written in a more suitable form as
$$
\begin{equation*}
\begin{aligned} \, \frac{H_1(\zeta)}{2\zeta W(\zeta)}&=\frac{3}{2}\sqrt\pi\,\sqrt\zeta\, e^{-\zeta}+\frac{5}{2}\,\partial_\zeta\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+ \zeta\,\partial_\zeta^2\bigl(\sqrt\zeta\,e^{-\zeta}\bigr) \\ &\qquad-\frac{1}{1+4\delta}\bigl[2\partial_\zeta \bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+ \zeta\,\partial_\zeta^2\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)\bigr]. \end{aligned}
\end{equation*}
\notag
$$
This makes it possible to identify as follows the dependence on $1+4\delta$ in the first correction: Here,
$$
\begin{equation*}
\begin{aligned} \, U_1^1(z)&=-\int_0^z\bigl[\mathcal{M}_2(z;1)\mathcal{M}_1(\zeta;1)- \mathcal{M}_1(z;1)\mathcal{M}_2(\zeta;1)\bigr] \\ &\qquad\times\bigl[2\partial_\zeta\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+ \zeta\,\partial_\zeta^2\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)\bigr]\,d\zeta. \end{aligned}
\end{equation*}
\notag
$$
Thus, the expression for $J(\delta)$ reduces to the form (8.12). The coefficients $J_0$, $J_1$, and $J_2$ do not depend on $\delta$ and are expressed in terms of integrals, single and double alike. To show that $J(\delta)$ depends essentially on $\delta$ we need to verify that one of the coefficients $J_1$ and $J_2$ is not zero. The shortest expression is the one for $J_2$, which we split into two terms, by isolating the part depending on $U_0=\sqrt z$ and calculated as a single integral, namely,
$$
\begin{equation*}
J_2^0=-\frac{1}{4}\int_0^\infty \mathcal{M}_2(z;2) \bigl[3\sqrt z\,\partial_z(U_0e^{-z})+ z^{3/2}\partial_z^2(U_0\,e^{-z})\bigr]\,dz.
\end{equation*}
\notag
$$
The second part is defined in terms of the first correction as
$$
\begin{equation*}
J_2^1=-\int_0^\infty \mathcal{M}_2(z;2)\biggl[\frac{3}{2}\,\partial_z( U_1^1\,e^{-z})+z\partial_z^2(U_1^1\,e^{-z})\biggr]\,dz
\end{equation*}
\notag
$$
and reduces to a double integral.
In the case $c_0=-3/2$ under consideration we have
$$
\begin{equation*}
\mathcal{M}_2(z;2)=\sqrt z\,\biggl(1-\frac{2}{3}z\biggr).
\end{equation*}
\notag
$$
Therefore, the first integral (of elementary functions) is calculated in terms of the gamma function as follows:
$$
\begin{equation*}
\begin{aligned} \, J_2^0&=-\frac{1}{4}\int_0^\infty\biggl[z\biggl(1-\frac{2}{3}z\biggr)\, 3\,\partial_z\bigl(\sqrt z\,e^{-z}\bigr)+z^{2}\biggl(1-\frac{2}{3}z\biggr) \partial_z^2\bigl(\sqrt z\,e^{-z}\bigr)\biggr]\,dz \\ &=\frac{1}{4}\int_0^\infty\sqrt z\,e^{-z}\,dz=\frac{\sqrt\pi}{8}\,. \end{aligned}
\end{equation*}
\notag
$$
The outer integral in the second term $J_2^1$ is taken by parts, which gives
$$
\begin{equation*}
\begin{aligned} \, J_2^1&=-\int_0^\infty\mathcal{M}_2(z;2)\bigl[2\partial_z(U_1^1\,e^{-z})+ z \partial_z^2(U_1^1\,e^{-z})\bigr]\,dz \\ &=-\int_0^\infty\biggl[\sqrt z\,\biggl(1-\frac{2}{3}z\biggr)\frac{3}{2}\, \partial_z(U_1^1\,e^{-z})+z^{3/2}\biggl(1-\frac{2}{3}z\biggr) \partial_z^2(U_1^1\,e^{-z})\biggr]\,dz \\ &=\int_0^\infty \sqrt z\,U_1^1\,e^{-z}\,dz. \end{aligned}
\end{equation*}
\notag
$$
As a result, we obtain the double integral
$$
\begin{equation*}
\begin{aligned} \, J_2^1&=-\int_0^\infty\sqrt z\,e^{-z}\int_0^z \bigl[\mathcal{M}_2(z;1)\mathcal{M}_1(\zeta;1)- \mathcal{M}_1(z;1)\mathcal{M}_2(\zeta;1)\bigr] \\ &\qquad\qquad\qquad\qquad\;\times\bigl[2\partial_\zeta\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)+ \zeta\partial_\zeta^2\bigl(\sqrt\zeta\,e^{-\zeta}\bigr)\bigr]\,d\zeta\,dz. \end{aligned}
\end{equation*}
\notag
$$
Under the integral sign we have Kummers functions, which can be elementary and special alike, namely,
$$
\begin{equation*}
\mathcal{M}_1(z;1)=1-2z\quad\text{and}\quad \mathcal{M}_2(z;1)=\sqrt z\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,,z\biggr).
\end{equation*}
\notag
$$
The integral reduces to a form convenient for numerical calculations, namely,
$$
\begin{equation*}
\begin{aligned} \, J_2^1&=-\int_0^\infty\,\int_0^z\sqrt z\,\biggl[\frac{3}{4}\, \frac{1}{\sqrt\zeta}-3\sqrt\zeta+\zeta^{3/2}\biggr]e^{-z-\zeta} \\ &\qquad\qquad\;\times\biggl[\sqrt z\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,,z\biggr) (1-2\zeta)-(1-2z)\sqrt\zeta\,M\biggl(-\frac{1}{2}\,,\frac{3}{2}\,, \zeta\biggr)\biggr]\,d\zeta\,dz. \end{aligned}
\end{equation*}
\notag
$$
Approximate computations using the Mathematica software package yield Since the approximate value is far beyond the limits of calculation error, the coefficient $J_2$ is not zero. Theorem 4 is proved.Corollary. Since $J(\delta)$ is continuous with respect to $\delta$ and not zero at $\delta=0$, $J(\delta)\ne 0$ for $\delta$ in an interval $0<\delta<\delta_0$. In addition, the structure of the expression (8.12) for $J(\delta)$, which is rational in $\delta$, shows that $J(\delta)$ depends on $\delta$ essentially and can vanish for at most two values of $\delta$. The calculating of the coefficient $J_1$ leads to similar integrals, which we will have to calculate numerically. No fundamentally new results for the wave velocity are expected along this way. 9. ConclusionsThe asymptotic solution (5.1) (as $t\to\infty$) has been constructed for the semilinear equation (1.1), which converges in the leading term to a travelling monotonic wave $\Phi_*(s)$, $s=x\sqrt{1+4\delta}-2t-\sigma(t)$. Such a wave describes the dynamics of the replacement of an unstable equilibrium by a stable one. The function $\Phi_*(s)$ satisfies the ordinary differential equation (1.2) with conditions (1.3) of stabilization to equilibria for the critical value $V=V_*$ of the parameter (that is, for $\gamma=2$). The quantity $V_*$, as a value of the bifurcation parameter $V$, corresponds to the stability bound for a wave travelling at constant velocity $V$. Asymptotically, the wave velocity is determined by (4.3) and (4.4). When the original variables are appropriately normalized, the asymptotic behaviour of the velocity is described by (4.1). The first terms of this asymptotic expression depend on the constants $c_0$, $c_1$, and $c_{2,1}$, which are uniquely determined when $c_0$ is fixed. Their uniqueness indicates that these constants are independent of the choice of a solution (for instance, of picking a solution by specifying initial data). There is still freedom in the choice of $c_0\leqslant -3/2$ among the negative half-integers. In [6] arguments for taking $c_0=-3/2$ were presented. Well-based rigorous arguments for the case of the KPP equation are the essence of [12]. An analysis of subsequent corrections in the asymptotic solution (7.2) aimed at ruling out exponential growth in higher orders of the asymptotic expansion results in the calculation of the constants $c_{3,1}$, $c_3$, $c_{4,2}$, $c_{4,1},\dots$ . The constants with even indices $c_2,c_4,\dots$ remain arbitrary parameters in any order. The fact that the constants with even indices $c_2,c_4,\dots$ can be arbitrary corresponds to the non-uniqueness of the asymptotic construction in higher terms. The choice of these constants must depend on the initial data in the Cauchy problem and must therefore specify the solution. This relationship remains an open problem for equations (1.1) under consideration, as the substantiation of asymptotic expansions with estimates for the remainder also is. Note that a connection between constants in asymptotics and the initial data is known for a number of equations integrable by the inverse scattering method (see [29] and [28]). However, for the KPP equation and other equations treated in [6] the approach from [29] and [28] is not suitable, since these dissipative systems do not have the property of integrability of this type. All constructions have been performed using the solution $\Phi_*(s)$ of the travelling wave equation (1.2) for the particular velocity $V=V_*$ (that is, for $\gamma=2$). For completeness, the situation with waves that have other velocities is worth highlighting. When we choose a monotone solution $\Phi_v(s)$ of (1.2) with another value $V>V_*$ (that is, for $\gamma>2$) as the leading term in the asymptotic formula, the construction simplifies significantly. An asymptotic solution for $\phi(s,t)$ is constructed as a series in negative integer powers:
$$
\begin{equation}
\phi(s,t)=\Phi_v(s)+\sum_{n\geqslant1}t^{-n}\Phi_n(s).
\end{equation}
\tag{9.1}
$$
The wave velocity remains constant, namely $s=(x-Vt)/\sqrt{1-\delta V^2}$ . As concerns the suitability interval $|s|\ll t$ of the asymptotic representation, there is no need to divide it into an inner and an outer layer. The freedom in the construction of the series lies in the terms $t^{-n}D_n\Phi_v'(s)$ with indefinite constants $D_n$. All distinctions from the case $\gamma=2$ are explained by the fact that for $\gamma>2$ the fixed point $(1,0)$ is a non-degenerate node. The structure (5.4), (5.5) of the fundamental system of solutions for the equation linearized near this point changes in this case. The stabilization rate of the coefficients $\Phi_n(s)\approx\exp(\lambda_+s)$ at the leading front $s\to+\infty$ is governed by the greater characteristic root $-1<\lambda_+<0$. The leading term $\Phi_v(s)$ must be of the same order. The above constructions are also possible for the leading term $\Phi_0(s)$ which stabilizes rapidly with exponent $\lambda_-<-1$. Such a monotone solution of the nonlinear equation (1.2) can exist for an exceptional value $\gamma_0>2$ in the case when the function $\Phi_*(s)$ (solution for $\gamma=2$) is not monotone and the corresponding wave is unstable [18]. The value $\gamma_0$ corresponds to the minimum velocity $V_0$ of a stable wave. The corrections $\Phi_n(s)$ are expressed in terms of integrals of basis solutions, and the construction is possible in the class of rapidly stabilizing functions
$$
\begin{equation*}
\Phi_n(s)\approx s^n\exp(\lambda_-s),\qquad s\to+\infty.
\end{equation*}
\notag
$$
Numerical experiments confirm the asymptotics of this type for solutions $\phi(s,t)$ that stabilize not too slowly at the initial moment of time, namely, where the exponent satisfies $\lambda<\lambda_+<0$. This result on convergence to a wave with minimum velocity $V_0$ resembles the result by Kolmogorov, Petrovskii, and Piskunov, with the difference that there is no rigorous substantiation of such an asymptotic behaviour. The stability of the solution $\Phi_0(s)$ with respect to initial perturbations of various types was analyzed in detail in [6]. One of the aims of this paper is to prepare a basis for further studies (begun in [30]–[34]) of perturbations of travelling waves for problems with a small parameter. The results on the universal part of the asymptotic expansions obtained here open up the possibility of describing effectively the leading front of a perturbed wave. 
Citation in format AMSBIB
\Bibitem{Kal25} |
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