Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two

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• 作者：David Chiron ; Claire Scheid
• 关键词：Nonlinear Schrödinger equation ; Travelling wave ; Kadomtsev–Petviashvili equation ; Constrained minimization ; Gradient flow ; Continuation method ; 35B38 ; 35C07 ; 35J20 ; 35J61 ; 35Q40 ; 35Q55 ; 35J60
• 刊名：Journal of Nonlinear Science
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：26
• 期：1
• 页码：171-231
• 全文大小：6,230 KB
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• 作者单位：David Chiron (1)
  Claire Scheid (1) (2)
  
  1. Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 02, France
  2. INRIA Sophia Antipolis-Méditerranée Research Center, NACHOS Project-Team, 06902, Sophia Antipolis Cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy–momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross–Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and “one-dimensional spreading” phenomena. Keywords Nonlinear Schrödinger equation Travelling wave Kadomtsev–Petviashvili equation Constrained minimization Gradient flow Continuation method