Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case

Abstract

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves $\Q\sts{x-ct}\e^{ıωt}$ of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case $c=2z_0\sqrt{\omega }$, where $z_0=z_0\stsσ $ is the unique zero point of $F\sts{z;~σ}$ in $\sts{-1, ~ 1}$. The new ingredients in the proof are the refined modulation decomposition of the solution near $\Q$ according to the spectrum property of the linearized operator $\Scal_{ω, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

33 pages, 2 figures, all comments are welcome

References in corpus (3)

• Well-posedness for a generalized derivative nonlinear Schrödinger equation
• A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
• Stability of multi-solitons for the derivative nonlinear Schr{ö}dinger equation

Cited by in corpus (3)

• Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case
• The stability of degenerate solitons for derivative nonlinear Schrodinger equations
• Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case