Global search for localised modes in scalar and vector nonlinear Schrodinger-type equations

PHYSICA D-NONLINEAR PHENOMENA

卷:397

Article

Alfimov, G. L.^1,2  Barashenkov, I., V^3,4,5  Fedotov, A. P.¹  Smirnov, V. V.¹  Zezyulin, D. A.⁶ 

[1] Natl Res Univ Elect Technol MIET, Moscow 124498, Russia

[2] Russian Acad Sci, Ufa Sci Ctr, Inst Math Comp Ctr, Chernyshevskii Str 112, Ufa 450008, Russia

[3] Univ Cape Town, Dept Math, ZA-7701 Rondebosch, South Africa

[4] Univ Bath, Dept Phys, Bath BA2 7AY, Avon, England

[5] Joint Inst Nucl Res, Dubna 141980, Russia

[6] ITMO Univ, St Petersburg 197101, Russia

关键词: Nonlinear mode;    Soliton;    Blow up;    Defocusing nonlinear Schrodinger equation;    Gross-Pitaevskii equation;    Lugiato-Lefever equation;   

DOI  :  10.1016/j.physd.2019.03.003

来源: Elsevier

【 摘 要 】

We present a new approach to the search for coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrodinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions - and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to the asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances X-+/- to the nearest forward and backward singularities; large X+ or X- indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a PT-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion. (C) 2019 Elsevier B.V. All rights reserved.

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