Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions
type: Item
instance of: scholarly article
title: Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions
zbMATH Open document ID: 1483.35146
author: Xianqiang Yang
published in: Journal of Differential Equations
publication date: 2019-03-26T00:00:00Z
review text: The Kundu-Eckhaus equation \[ iq_t+\frac{1}{2}q_{xx}+|q|^2q+2\beta |q|^4q-2i\beta(|q|^2)_x+q=0 \] is considered, with the initial condition \[ q(x,0)\sim Ae^{i(\mu x+\theta_\pm)} \text{ when } x\to\pm\infty. \] Here \(\beta, A(>0),\mu\), and \(\theta_\pm\) are real constants. The long-time asymptotics of the solution is established in three different sectors depending on the magnitude \(\xi=x/t\): plane wave sector \(\xi<\xi_1=\mu-2\beta A^2-2\sqrt{2}A\), plane wave sector \(\xi>\xi_2=\mu-2\beta A^2+2\sqrt{2}A\), and modulated genus 1 elliptic wave sector \(\xi_1<\xi<\xi_2\).
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 37K40
Mathematics Subject Classification ID: 37K15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 7041833
zbMATH Keywords: long-time asymptotics
zbMATH Keywords: Kundu-Eckhaus equation
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: nonlinear steepest descent method
reviewed by: Ilya M. Spitkovsky
MaRDI profile type: Publication
full work available at URL: j.jde.2018.10.053
OpenAlex ID: W2902576395
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Wikidata QID: Q128898387
DOI: 10.1016/J.JDE.2018.10.053
description: scientific article; zbMATH DE number 7041833