Geodesic
On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$
Ufa mathematical journal, Tome 7 (2015) no. 2, pp. 3-16 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, nonlinear Schrödinger equation, stationary modes.
Mots-clés : singular solutions 
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G. L. Alfimov; M. E. Lebedev. On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$. Ufa mathematical journal, Tome 7 (2015) no. 2, pp. 3-16. http://geodesic.mathdoc.fr/item/UFA_2015_7_2_a0/

[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor”, Science (New Series), 269 (1995), 198–201 | DOI

[2] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance”, Phys. Rev. Lett., 93 (2004), 123001 | DOI

[3] Y. V. Kartashov, B. A. Malomed, L. Torner, “Solitons in nonlinear lattices”, Rev. Mod. Phys., 83 (2011), 247–305 | DOI

[4] Pitaevskii L. P., “Kondensaty Boze–Einshteina v pole lazernogo izlucheniya”, UFN, 176:4 (2006), 345–364 | DOI

[5] V. A. Brazhnyi, V. V. Konotop, “Theory of nonlinear matter waves in optical lattices”, Modern Phys. Lett. B, 18 (2004), 627–651 | DOI

[6] G. L. Alfimov, A. I. Avramenko, “Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential”, Physica D, 254 (2013), 29–45 | DOI | MR | Zbl

[7] B. G. Pachpatte, Inequalities for Differential and Integral Equation, Mathematics in Science and Engineering, 197, 1998 | MR

[8] G. L. Alfimov, D. A. Zezyulin, “Nonlinear modes for the Gross–Pitaevskii equation – a demonstrative computational approach”, Nonlinearity, 20 (2007), 2075–2092 | DOI | MR | Zbl

[9] Kuznetsov A. P., Kuznetsov S. P., Ryskin N. M., Nelineinye kolebaniya, Fizmatlit, M., 2005

[10] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964 | MR | Zbl

[11] W. A. Coppel, Stability and asymptotic behaviour of differential equations, Heath and Co., Boston, 1965 | MR