Abstract:
The paper is devoted to the equation uxx+Q(x)u+P(x)u3=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.
\Bibitem{AlfLeb15}
\by G.~L.~Alfimov, M.~E.~Lebedev
\paper On regular and singular solutions for equation $u_{xx}+Q(x)u+P(x)u^3=0$
\jour Ufa Math. J.
\yr 2015
\vol 7
\issue 2
\pages 3--16
\mathnet{http://mi.mathnet.ru/eng/ufa275}
\crossref{https://doi.org/10.13108/2015-7-2-3}
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\elib{https://elibrary.ru/item.asp?id=24188341}
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Linking options:
https://www.mathnet.ru/eng/ufa275
https://doi.org/10.13108/2015-7-2-3
https://www.mathnet.ru/eng/ufa/v7/i2/p3
This publication is cited in the following 8 articles:
Mikhail E. Lebedev, Georgy L. Alfimov, “Numerical Evidence of Hyperbolic Dynamics and Coding of Solutions for Duffing-Type Equations with Periodic Coefficients”, Regul. Chaotic Dyn., 29:3 (2024), 451–473
M. V. Gasanov, A. G. Gulkanov, “A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation”, Rus. J. Nonlin. Dyn., 19:4 (2023), 575–584
G. L. Alfimov, M. E. Lebedev, “Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient”, Rus. J. Nonlin. Dyn., 19:4 (2023), 473–506
G.L. Alfimov, A.P. Fedotov, N.A. Kutsenko, D.A. Zezyulin, “Stationary modes for vector nonlinear Schrödinger-type equations: A numerical procedure for complete search and its mathematical background”, Physica D: Nonlinear Phenomena, 454 (2023), 133858
G. L. Alfimov, I. V. Barashenkov, A. P. Fedotov, V. V. Smirnov, D. A. Zezyulin, “Global search for localised modes in scalar and vector nonlinear Schrodinger-type equations”, Physica D, 397 (2019), 39–53
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
G. L. Alfimov, P. P. Kizin, “On solutions of Cauchy problem for equation uxx+Q(x)u−P(u)=0 without singularities in a given interval”, Ufa Math. J., 8:4 (2016), 24–41
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
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