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
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.
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页码:1613 / 1653
页数:41
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