A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations

Abstract

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation $$i\partial_tu-(-Δ)^σu+|u|^{2}u=0,$$ where $σ\in (\frac12,1)$ and the operator $(-Δ)^σ$ is the fractional Laplacian of symbol $|ξ|^{2σ}$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k|^{2σ}-ω^{2σ})}Q_{ω,k}(x-2tσ|k|^{2σ-2}k),\quad k\in\mathbb{R},\ ω>0$$ by a rather involved variational argument.