Long time asymptotics for the focusing nonlinear Schrodinger equation in the solitonic region with the presence of high-order discrete spectrum

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:505

Article

Fan, Engui¹ 

[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

关键词: Focusing nonlinear Schrodinger equation;    Non-generic initial data;    High-order discrete spectrum;    Riemann-Hilbert problem;    (partial derivative)over-bar Steepest descent method;    Soliton resolution;   

DOI  :  10.1016/j.jmaa.2021.125635

来源: Elsevier

【 摘 要 】

In this paper, we study the initial value problem for focusing nonlinear Schrodinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we show how to characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; Further the initial value problem is formulated into an appropriate enlarged RH problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: S(x(1), x(2), v(1), v(2)) := {(x,t) is an element of R-2 : x = x(0) + vt, x(0) is an element of [x(1), x(2]), v is an element of [v(1),v(2])}. Our result is a verification of the soliton resolution conjecture for the fNLS equation in the solitonic region with the presence of high-order discrete spectrum. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term is up to O(t(-3/4)) which comes from the corresponding (partial derivative) over bar equation. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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