Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Approximation of the Korteweg-de Vries equation by the nonlinear Schrödinger equation
zbMATH Open document ID: 0940.35179
author: Guido Schneider
published in: Journal of Differential Equations
publication date: 2000-07-19T00:00:00Z
review text: In 1968, Zakharov derived the nonlinear Schrödinger equation as an approximation equation for the water wave problem. Naturally, one wonders if solutions of the water wave problem can actually be rigorously approximated by solutions of the nonlinear Schrödinger equation. This paper considers solutions of the Korteweg-de Vries equation, which describes water waves with small amplitude in a long and narrow canal. Let \(\psi\) be the approximation wave constructed from a fixed \(C([0, T], H^{s+5}(\mathbb{R}, \mathbb{C}))\)-solution of the nonlinear Schrödinger equation, where \(s\geq 3\) and \(T>0\). Then, the following result is shown: There are \(C\), \(\varepsilon_0> 0\) such that for each \(\varepsilon\in (0,\varepsilon_0)\), the Korteweg-de Vries equation has a solution \(u\) satisfying \[ \sup_{t\in [0, T/\varepsilon^2]}\|u- \psi\|_{H^s(\mathbb{R}, \mathbb{R})}\leq C\varepsilon^{3/2}. \]
reviewed by: Hongyou Wu
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 1212348
zbMATH Keywords: water wave problem
zbMATH Keywords: KdV equation
zbMATH Keywords: NLS equation
zbMATH Keywords: approximation
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3417
OpenAlex ID: W2047012233
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DOI: 10.1006/JDEQ.1998.3417
description: scientific article; zbMATH DE number 1212348