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

Abstract

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation $$i{\partial }_{t}u-(-\Delta {)}^{\sigma }u+|u{|}^{2}u=0,$$ where $\sigma \in (\frac{1}{2},1)$ and the operator $(-\Delta {)}^{\sigma }$ is the fractional Laplacian of symbol $|\xi {|}^{2\sigma }$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k{|}^{2\sigma }-{\omega }^{2\sigma })}{Q}_{\omega ,k}(x-2t\sigma |k{|}^{2\sigma -2}k),k\in \mathbb{R},\omega >0$$ by a rather involved variational argument.

References in corpus (1)

• Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations