No abstract

We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. 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.2000 Mathematics Subject Classification. 35J40; 58E05. Key words and phrases. Quasi-linear Schrödinger equations, orbital stability, orbital instability by blow-up, ground state solutions, variational methods.The third author was partially supported by the Italian PRIN Research Project 2007 Metodi variazionali e topologici nello studio di fenomeni non lineari.

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e 2iωt ϕ, 0, 0), (0, e 2iωt ϕ, 0), (0, 0, e 2iωt ϕ), where ϕ is a ground state of the scalar nonlinear Schrödinger equation. RésuméDans cet article, on s'intéresse à un système de trois équations de Schrödinger qui modélise le phénomène d'amplification Raman dans les plasmas. On étudie la stabilité orbitale de solutions scalaires de la forme (e 2iωt ϕ, 0, 0), (0, e 2iωt ϕ, 0), (0, 0, e 2iωt ϕ), où ϕ est l'état fondamental d'une équation de Schrödinger scalaire.