Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation

2017/10/26 by Bonheure, Denis, Castéras, Jean-Baptiste, Santos, Ederson Moreira dos +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.09775

Abstract

We study the mixed dispersion fourth order nonlinear Schrödinger equation %\protect4NLS i ∂_t ψ-γΔ² ψ+βΔψ+|ψ|^2σ ψ=0 in \R ×\R^N, where γ,σ>0 and β∈ \R. We focus on standing wave solutions, namely solutions of the form ψ(x,t)=e^iαtu(x), for some α∈ \R. This ansatz yields the fourth-order elliptic equation %\protect* γΔ² u -βΔu +αu =|u|^2σ u. We consider two associated constrained minimization problems: one with a constraint on the L²-norm and the other on the L^2σ+2-norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

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