Stability and instability results for standing waves of quasi-linear Schr\"odinger equations

TL;DR

This paper investigates the existence, stability, and instability of standing wave solutions in quasi-linear Schrödinger equations relevant to superfluid films in plasma physics, using variational and gauge transform methods.

Contribution

It provides new results on the stability and instability of standing waves in a class of quasi-linear Schrödinger equations, including existence proofs under certain conditions.

Findings

Existence of standing wave solutions established.

Orbital stability of certain standing waves demonstrated.

Instability results for other classes of solutions obtained.

Abstract

We study a class of quasi-linear Schr\"odinger equations arising in the theory of superfluid film in plasma physics. First, using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.