Instability of degenerate solitons for nonlinear Schr\"odinger equations with derivative

TL;DR

This paper investigates the instability of degenerate solitons in a derivative nonlinear Schrödinger equation with quintic nonlinearity, revealing conditions under which these solitons are unstable, especially for positive parameter values.

Contribution

It extends instability analysis to a perturbed DNLS equation with quintic nonlinearity, focusing on degenerate solitons and their dynamic behavior.

Findings

Degenerate solitons with zero momentum and energy are unstable for certain parameters.

The study provides a qualitative framework for understanding soliton instability in derivative NLS equations.

Results include a large set of initial data leading to instability when parameter b > 0.

Abstract

We consider the following nonlinear Schr\"{o}dinger equation with derivative: \begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of the well-known derivative nonlinear Schr\"{o}dinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in our equation only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional nonlinearity is natural as a perturbation preserving soliton profiles of DNLS. When $b\ge 0$, the equation has degenerate solitons whose momentum and energy are zero, and if $b=0$, they are algebraic solitons. Inspired from the works on instability theory of the $L^2$-critical generalized KdV equation, we study the instability of degenerate solitons in a qualitative way, and when $b>0$, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case $b=0$ in exactly the same way, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.