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 Ufimsk. Mat. Zh., 2015, Volume 7, Issue 2, Pages 3–18 (Mi ufa275)

On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$

G. L. Alfimov, M. E. Lebedev

National Research University of Electronic Technology, 4806 av., 5, 124498, Moscow, Zelenograd, Russia

Abstract: The paper is devoted to the equation $u_{xx}+Q(x)u+P(x)u^3=0$. The equations of such kind have been used to describe stationary modes in the models of Bose–Einstein condensate. It is known that under some conditions for $P(x)$ and $Q(x)$, the “most part” of solutions for such equations are singular, i.e. tend to infinity at some point of the real axis. In some situations this fact allows us to apply the methods of symbolic dynamics to describe non-singular solutions of this equation and to construct comprehensive classification of these solutions. In the paper we present (i) necessary conditions for existence of singular solutions as well as conditions for their absence; (ii) the results of numerical study of the case when $Q(x)$ is a constant and $P(x)$ is an alternate periodic function. Basing on these results, we formulate a conjecture that all the non-singular solutions of the equation can be coded by bi-infinite sequences of symbols of a countable alphabet.

Keywords: ODE with periodic coefficients, singular solutions, nonlinear Schrödinger equation, stationary modes.

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English version:
Ufa Mathematical Journal, 2015, 7:2, 3–16 (PDF, 952 kB); https://doi.org/10.13108/2015-7-2-3

Bibliographic databases:

UDC: 517.9
MSC: 34L30, 34C11, 35Q55, 37B10

Citation: G. L. Alfimov, M. E. Lebedev, “On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$”, Ufimsk. Mat. Zh., 7:2 (2015), 3–18; Ufa Math. J., 7:2 (2015), 3–16

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. L. Alfimov, P. P. Kizin, “On solutions of Cauchy problem for equation $u_{xx}+Q(x)u-P(u)=0$ without singularities in a given interval”, Ufa Math. J., 8:4 (2016), 24–41
2. M. E. Lebedev, G. L. Alfimov, B. A. Malomed, “Stable Dipole Solitons and Soliton Complexes in the Nonlinear Schrodinger Equation With Periodically Modulated Nonlinearity”, Chaos, 26:7 (2016), 073110
3. G. L. Alfimov, P. P. Kizin, D. A. Zezyulin, “Gap Solitons For the Repulsive Gross-Pitaevskii Equation With Periodic Potential: Coding and Method For Computation”, Discrete Contin. Dyn. Syst.-Ser. B, 22:4 (2017), 1207–1229
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