A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations. (13th February 2017)

• Record Type: Journal Article
• Authors: Hong, Younghun; Sire, Yannick
• Is Part Of: Nonlinearity. Volume 30:Number 4(2017:Apr.)
• Page Start: 1262
• Page End: 1286
• Publication Date: 2017-02-13
• Subjects: nonlinear fractional Schrödinger equation -- soliton -- Gagliardo–Nirenberg inequality; 35Q41 -- 35Q55 -- 35J20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa5b12
• Languages: English
• ISSNs: 0951-7715

Abstract

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.