The KdV/KP-I limit of the Nonlinear Schrodinger equation

2008/10/21 by David Chiron, Chiron, D., Frédéric Rousset +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.0810.3868

openalex publication_date 2008/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrödinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Korteweg-de Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that there is no vortex formation for the global solution of the NLS equation in the energy space and deduce from this the convergence towards the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use an hydrodynamic reformulation of NLS and recast the problem as a singular limit for an hyperbolic system. We thus prove that smooth H^s solutions exist on a time interval independent of the small parameter. We then pass to the limit by a compactness argument and obtain the KdV/KP-I equation.

Citations

• Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

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• Ionic KdV structure in weakly collisional plasmas

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