A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations. (13th February 2017) - Hong, Younghun; Sire, Yannick
Abstract: We consider the one-dimensional cubic fractional nonlinear Schrödinger equation i ∂ t u − ( − Δ ) σ u + | u | 2 u = 0, where σ ∈ ( 1 2, 1 ) and the operator ( − Δ ) σ is the fractional Laplacian of symbol | ξ | 2 σ . Despite the lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form u ( t, x ) = e − i t ( | k | 2 σ − ω 2 σ ) Q ω, k ( x − 2 t σ | k | 2 σ − 2 k ), k ∈ R, ω > 0 by a rather involved variational argument.