Stability of algebraic solitons for nonlinear Schr\"{o}dinger equations of derivative type: variational approach

TL;DR

This paper investigates the stability of algebraic and exponential solitons in a derivative nonlinear Schrödinger equation with quintic nonlinearity, demonstrating stability of all solitons for negative parameter values using a variational approach.

Contribution

It proves the stability of all solitons, including algebraic ones, in a derivative NLS with quintic nonlinearity for the first time.

Findings

All solitons are stable when b<0.

Existence of stable algebraic solitons in this context.

Extension of stability results to equations with double power nonlinearities.

Abstract

We consider the following nonlinear Schr\"{o}dinger equation of derivative type: \begin{equation} i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \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 well-known derivative nonlinear Schr\"{o}dinger (DNLS) equation. The soliton profile of the DNLS equation satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in the equation only affects the coefficient in front of the quintic term in the elliptic equation, so the additional nonlinearity is natural as a perturbation preserving soliton profiles of the DNLS equation. If $b>-\frac{3}{16}$, the equation has algebraic solitons as well as exponentially decaying solitons. In this paper we study stability properties of solitons by variational approach, and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The existence of stable algebraic solitons shows an interesting mathematical example because stable algebraic solitons are not known in the context of the corresponding double power NLS.