Numerical-analytical solutions of the nonlinear Schrödinger equation

The main aim of the work is to study the numerical solutions of the focusing nonlinear Schrodinger (NLS) equation. The initial-value problem for the NLS equation
$$iu_t+2|u|^2u+u_{xx}=0,$$

$$u(x,0)=u_0(x)$$
is solved numerically using the method inverse scattering transform. Here the function $u_{0}(x)$ possesses the required smoothness and tends to its limits sufficiently rapidly as $x \to \pm \infty $. Soliton theory provides effective methods to solve nonlinear evolution partial differential equations. The inverse scattering transform method is particularly powerful in constructing soliton solutions. The inverse scattering transform method to solve the initial-value problem for the NLS equation is based on the spectral analysis of the Zakharov–Shabat system and is described in terms of the following three steps: first, solve the direct scattering problem for a Zakharov–Shabat system with initial potential $u_{0}(x)$; second, finding evolution of scattering data; third, solve the inverse scattering problem for the time evolved scattering data to arrive at the solution $u(x,t)$. The inverse scattering problem for Zakharov–Shabat system is reduced to a system of two integral equations the so-called system of Gelfand–Levitan–Marchenko (Marchenko) integral equations. This means solving the coupled system of Marchenko integral equations, associated to the scattering data. In some cases, analytical solution cannot be found for this system of integral equations. For example, in a non-reflective case. Therefore, we must apply the numerical methods for obtaining at least the approximate solutions of the system of Marchenko integral equations. By M. C. De Bonis and G. Mastroianni [17] applied Nyström method for solving systems of Fredholm integral equations on the real semiaxis. They proved that this method is stablite and convergent, and applied a specific application to an inverse scattering problem for the Schrödinger equation. In the work A. Aricò, G. Rodriguez, S. Seatzu [18] is shown system of Marchenko integral equations can be reduced to a linear system of algebraic equations. Using structured-matrix techniques the time evolved system of Marchenko integral equations is solved to arrive at the solution to the NLS equation. In this work, we have used a numerical method to obtain approximate solutions to the system of Marchenko integral equations, in the cases when the corresponding system has simple and multiple eigenvalues. It is clear that to get the best approximating solutions of the given systems, the truncation degree $N_x$ must be chosen large enough. The results are compared with the exact solution by using computer simulations.