Approximations for KdV equation as a Hamiltonian system
type: Item
instance of: scholarly article
title: Approximations for KdV equation as a Hamiltonian system
zbMATH Open document ID: 0952.65104
DOI: 10.1016/S0898-1221(00)00040-7
author: Pingfu Zhao
author: Qin Meng Zhao
publication date: 2000-08-20T00:00:00Z
review text: Several Fourier spectral approximations and collocation approximations for the Korteweg-de Vries (KdV) equation are considered paying special attention to the preservation of its Hamiltonian character. In the first part some basic facts on (infinite dimensional) Hamiltonian systems are briefly reviewed. Also the authors observe that the KdV equation \(u_t + u_{xxx}+6 u u_x=0\) can be written in Hamiltonian form by using two different Hamiltonian functions, and therefore there are two conserved quantities. Now Fourier spectral methods are applied to each of these formulations in order to get semidiscrete approximations for the standard basis \(\{ 1/2, \cos x, \sin x, \ldots, \cos (Nx), \sin (Nx) \}\), showing that for \(N=1\) both formulations give the same finite dimensional approximation. However for \( N>1\), only one of these discretizations retains its Hamiltonian nature. Next, the authors consider collocation methods applied to both Hamiltonians for the periodic KdV equation. Once more they find that for \(N=1\) both spatial discretizations are Hamiltonian but for \(N>1\) only one of them retains this character.
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37M05
zbMATH DE Number: 1471574
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: KdV equation
zbMATH Keywords: Fourier spectral methods
zbMATH Keywords: semidiscrete approximations
zbMATH Keywords: collocation methods
zbMATH Keywords: Korteweg-de Vries equation
reviewed by: Manuel Calvo
MaRDI profile type: Publication
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1471574