Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation in the endpoint case

TL;DR

This paper investigates the stability of solitary wave solutions for a generalized derivative nonlinear Schrödinger equation, demonstrating their instability at the critical endpoint case where the wave speed equals twice the square root of the frequency.

Contribution

It proves the instability of solitary wave solutions specifically at the endpoint case c=2√ω, extending previous stability results to this critical scenario.

Findings

Solitary waves are unstable at the endpoint case c=2√ω.

Previous stability results do not hold at the endpoint.

The paper establishes instability in the critical case.

Abstract

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0, $$ where $1<\sigma<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{\omega,c}(t,x)=e^{i\omega t+i\frac c2(x-ct)-\frac{i}{2\sigma+2}\int_{-\infty}^{x-ct}\varphi^{2\sigma}_{\omega,c}(y)dy}\varphi_{\omega,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrt{\omega}$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrt{\omega}$.