Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schrödinger equation in the critical frequency case

Abstract

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation $$i{\partial }_{t}u+{\partial }_x^{2}u+i|u{|}^{2\sigma }{\partial }_{x}u=0.$$ The equation has a two-parameter family of solitary wave solutions of the form \begin{align*} ϕ_{ω,c}(x)=φ_{ω,c}(x)\exp{\big\{ i\frac c2 x-\frac{i}{2σ+2}\int_{-\infty}^{x}φ^{2σ}_{ω,c}(y)dy\big\}}. \end{align*} Here ${\phi }_{\omega ,c}$ is some real-valued function. It was proved in \cite{LiSiSu1} that the solitary wave solutions are stable if $-2\sqrt{\omega }<c<2z_0\sqrt{\omega }$, and unstable if $2z_0\sqrt{\omega }<c<2\sqrt{\omega }$ for some $z_0\in (0,1)$. We prove the instability at the borderline case $c=2z_0\sqrt{\omega }$ for $1<\sigma <2$, improving the previous results in \cite{Fu-16-DNLS} where $3/2<\sigma <2$.

arXiv commentThe same result was obtained independently by Miao-Tang-Xu (paper appeared on arXiv on Mar 20, 2018) by different method. They used the third derivative of the energy around the solitary wave. Our method does not require higher regularity of the energy but constructs a delicate virial identity

References in corpus (13)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$
• Well-posedness for a generalized derivative nonlinear Schrödinger equation
• A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
• Orbital stability of solitary waves for derivative nonlinear Schrödinger equation
• Stability of the traveling waves for the derivative Schrödinger equation in the energy space
• Local structure of singular profiles for a Derivative Nonlinear Schrödinger Equation
• Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
• Global Well-posedness and soliton resolution for the Derivative Nonlinear Schrödinger equation
• Instability of solitary waves for a generalized derivative nonlinear Schrödinger equation in a borderline case
• Long-Time Behavior of Solutions to the Derivative Nonlinear Schrödinger Equation for Soliton-Free Initial Data
• Stability of the sum of two solitary waves for (gDNLS) in the energy space
• Local Well-Posedness for the Derivative Nonlinear Schrödinger Equations with $L^2$ Subcritical Data

Cited by in corpus (4)

• Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations
• Microscopic conservation laws for the derivative Nonlinear Schrödinger equation
• Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case
• Instability of degenerate solitons for nonlinear Schrödinger equations with derivative