Multiple branches of travelling waves for the Gross-Pitaevskii equation

Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited...

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Bibliographic Details
Published inNonlinearity Vol. 31; no. 6; pp. 2809 - 2853
Main Authors Chiron, David, Scheid, Claire
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.06.2018
Subjects
Analysis of PDEs
Kadomtsev-Petviashvili equation
Mathematics
multi-lump solitary waves
nonlinear Schrödinger equation
travelling waves
vortex
Kadomtsev-Petviashvili equation
vortex
Non Linear Schrodinger equation
travelling waves
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Summary:Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross-Pitaevskii (GP) equation, which in some long wave regime converges to the KP-I equation. Numerical simulations have already shown that a branch of travelling waves of GP converges to a ground state of KP-I, expected to be the lump. In this work, we perform numerical simulations showing that other explicit solitary waves solutions to the KP-I equation give rise to new branches of travelling waves of GP corresponding to excited states.
Bibliography:NON-102204.R2
London Mathematical Society
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aab4cc