Orbital stability of solitary waves for Kundu equation
type: Item
instance of: scholarly article
title: Orbital stability of solitary waves for Kundu equation
zbMATH Open document ID: 1179.35280
author: Yinghao Qin
author: Yan Zhao
author: Wei-Guo Zhang
author: Boling Guo
published in: Journal of Differential Equations
publication date: 2009-08-12T00:00:00Z
review text: The Kundu equation \[ i u_t + u_{xx} + c_1 |u|^2 u + c_2 |u|^4 u + c_3 i (|u|^2 u)_x + c_4 i u (|u|^2)_x = 0 \] includes several nonlinear evolution equations integrable by inverse scattering such the derivative nonlinear Schrödinger equation, the Chen--Lee--Lin equation, and the Gerdjikov--Ivanov equation. Solitary waves in all these equations can be obtained in the explicit form by integrating a system of ordinary differential equations. The authors study orbital stability of solitary waves in the Kundu equation. Since the Kundu equation can not be formulated in a standard symplectic Hamiltonian form, it is difficult to apply the Grillakis--Shatah--Strauss theory on orbital stability of solitary waves in Hamiltonian systems. The authors overcome this difficulty with the spectral stability analysis and the construction of three invariants of motion. As a result, they derive sufficient conditions for orbital stability of solitary waves, which generalize the previously found conditions of \textit{B. Guo} and \textit{Y. Wu} [J. Differ. Equations 123, No. 1, 35--55 (1995; Zbl 0844.35116)] and \textit{M. Colin} and \textit{M. Ohta} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 5, 753--764 (2006; Zbl 1104.35050)].
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K40
Mathematics Subject Classification ID: 35C08
zbMATH DE Number: 5593950
zbMATH Keywords: Kundu equation
zbMATH Keywords: derivative nonlinear Schrödinger equation
zbMATH Keywords: solitary wave
zbMATH Keywords: orbital stability
zbMATH Keywords: spectral analysis
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
full work available at URL: j.jde.2009.05.008
OpenAlex ID: W2057249990
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DOI: 10.1016/J.JDE.2009.05.008
description: scientific article; zbMATH DE number 5593950