A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations

초록

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation i∂tu−(−Δ)σu + |u|2u=0, where $\sigma \in \left(\frac{1}{2},1\right)$ and the operator ${(-\Delta )}^{\sigma }$ is the fractional Laplacian of symbol $|\xi {|}^{2\sigma }$ . Despite the 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−ω2σ)Qω,k(x−2tσ|k|2σ−2k),k∈R, ω>0 by a rather involved variational argument.

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