On the Cauchy problem of defocusing mKdV equation with finite density initial data: Long time asymptotics in soliton-less regions

被引：5

作者：

Xu, Taiyang ^{[1
]}

Zhang, Zechuan ^{[1
]}

Fan, Engui ^{[1
]}

机构：

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

来源：

JOURNAL OF DIFFERENTIAL EQUATIONS | 2023年 / 372卷

基金：

美国国家科学基金会;

关键词：

Defocusing mKdV equation; Riemann-Hilbert problem; partial differential steepest descent method; Long time asymptotics; Soliton-less regions; NONLINEAR SCHRODINGER-EQUATION; NONZERO BOUNDARY-CONDITIONS; RIEMANN-HILBERT PROBLEMS; STEEPEST DESCENT METHOD; EVOLUTION-EQUATIONS; INVERSE SCATTERING;

D O I：

10.1016/j.jde.2023.06.038

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We investigate the long time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv :2108 .03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector {(x, t) : |& xi;| < 6} with & xi; := x/t. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the solitonless regions {(x, t) : | & xi; | > 6, | & xi; | = O(1)} for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics is in compatible with the "background solution" and the error terms are derived via rigorous analysis.& COPY; 2023 Elsevier Inc. All rights reserved.

引用

页码：55 / 122

页数：68

共 50 条

[11] Long-time behavior of the solution to the mKdV equation with step-like initial data
Minakov, Alexander
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 2011, 44 (08)
[12] Long-time asymptotics for a mixed nonlinear Schrodinger equation with the Schwartz initial data
Cheng, Qiaoyuan
Fan, Engui
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2020, 489 (02)
[13] Long-time Asymptotic Behavior for the Derivative Schrodinger Equation with Finite Density Type Initial Data
Yang, Yiling
Fan, Engui
CHINESE ANNALS OF MATHEMATICS SERIES B, 2022, 43 (06) : 893 - 948
[14] The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics
Xu, Kai
Yang, Yiling
Fan, Engui
JOURNAL OF DIFFERENTIAL EQUATIONS, 2024, 380 : 24 - 91
[15] Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space
Deift, P
Zhou, X
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2003, 56 (08) : 1029 - 1077
[16] Long-time Asymptotic Behavior for the Derivative Schr?dinger Equation with Finite Density Type Initial Data
Yiling YANG
Engui FAN
ChineseAnnalsofMathematics,SeriesB, 2022, (06) : 893 - 948
[17] Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data
Deift, Percy
Park, Jungwoon
INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2011, 2011 (24) : 5505 - 5624
[18] Long-time Asymptotic Behavior for the Derivative Schrödinger Equation with Finite Density Type Initial Data
Yiling Yang
Engui Fan
Chinese Annals of Mathematics, Series B, 2022, 43 : 893 - 948
[19] The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data
Martynenko, A. V.
Tedeev, A. F.
Shramenko, V. N.
IZVESTIYA MATHEMATICS, 2012, 76 (03) : 563 - 580
[20] Asymptotics of the Cauchy Problem for the One-Dimensional Schrodinger Equation with Rapidly Oscillating Initial Data and Small Addition to the Smooth Potential
Dobrokhotov, S. Yu.
RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS, 2023, 30 (04) : 466 - 479

← 1 2 3 4 5 →