Focusing NLS Equation: Long-Time Dynamics of Step-Like Initial Data

被引：55

作者：

de Monvel, Anne Boutet ^{[1
]}

Kotlyarov, Vladimir P. ^{[2
]}

Shepelsky, Dmitry ^{[2
]}

机构：

[1] Univ Paris 07, Inst Math Jussieu, F-75013 Paris, France

[2] Inst Low Temp Phys, Div Math, UA-61103 Kharkov, Ukraine

来源：

INTERNATIONAL MATHEMATICS RESEARCH NOTICES | 2011年 / 2011卷 / 07期

关键词：

NONLINEAR SCHRODINGER-EQUATION; PERIODIC BOUNDARY-CONDITION; STEEPEST DESCENT METHOD; ASYMPTOTICS;

D O I：

10.1093/imrn/rnq129

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We consider the initial value problem for the focusing nonlinear Schrodinger equation with "step-like" initial data: q(x, 0) = 0 for x <= 0 and q(x, 0) = Aexp (-2iBx) for x > 0, where A> 0 and B is an element of R are constants. The paper aims at studying the long-time asymptotics of the solution to this problem. We show that there are three regions in the halfplane -infinity < x < infinity, t > 0, where the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x < -4Bt, a modulated elliptic wave for -4Bt < x < -4(B -A root 2)t, and a plane wave for x > -4(B -A root 2)t. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem.

引用

页码：1613 / 1653

页数：41

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