Numerical behaviour of stable and unstable solitary waves
type: Item
instance of: scholarly article
title: Numerical behaviour of stable and unstable solitary waves
zbMATH Open document ID: 1002.65134
DOI: 10.1016/S0168-9274(01)00144-1
author: A. Durán
author: M. A. López-Marcos
published in: Applied Numerical Mathematics
publication date: 2002-08-22T00:00:00Z
review text: The authors discuss the numerical solution of the generalised Benjamin-Bona-Mahony (BBM) equation \(u_t+u_x+(u^p)_x-u_{txx}=0\) with an integer \(p>1\). In contrast to the classical BBM equation (p=2), it is in general not a completely integrable system. However, it is still a Hamiltonian system invariant under a one-parameter group of translations and admits a two-parameter family of solitary wave solutions that may be interpreted as relative equilibria associated with this symmetry group. For \(p>5\) the (orbital) stability of these solitary waves depends on the velocity: in a certain range the waves are linearly unstable. The main goal of the article is to show that numerical methods that preserve some conserved quantities are more appropriate for this problem. The authors compare the use of two different Runge-Kutta methods for the time integration: the implicit midpoint rule preserving the first integral \(I\) and a third order diagonally implicit method which preserves neither the Hamiltonian \(H\) nor \(I\). For the spatial discretisation a Fourier pseudospectral method is applied so that essentially all numerical errors are due to the time integration. The results show clearly that the conservative method is superior despite its lower order. The phase shift is much smaller and the error in the amplitude is bounded; the integration error grows only linearly in contrast to a quadratic growth for the third order method. The main part of the article is devoted to a detailed error analysis in order to explain this behaviour. The analysis is based on the construction of an asymptotic expansion in which a so-called modified solitary wave appears which is perturbed in the basic parameters: amplitude, velocity and phase. It turns out that the classical approach already applied to a number of other equations does not well explain the experimental observations. The authors derive a new modified solitary wave that better explains the observations.
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1787725
zbMATH Keywords: generalised Benjamin-Bona-Mahony equation
zbMATH Keywords: solitary waves
zbMATH Keywords: relative equilibria
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: geometric integration
zbMATH Keywords: stability
zbMATH Keywords: error propagation
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: Fourier pseudospectral method
reviewed by: Werner M. Seiler
MaRDI profile type: Publication
cites work: Error propagation in the numerical integration of solitary waves. The regularized long wave equation
cites work: Q4692311
cites work: Model equations for long waves in nonlinear dispersive systems
cites work: Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation
cites work: On the stability of solitary-wave solutions of model equations for long waves
cites work: Q4748596
cites work: Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems
cites work: The numerical integration of relative equilibrium solutions. Geometric theory
cites work: The numerical integration of relative equilibrium solutions. The nonlinear Schrodinger equation
cites work: Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation
cites work: The life-span of backward error analysis for numerical integrators
cites work: Solving Ordinary Differential Equations I
cites work: Q4318503
cites work: Asymptotic stability of solitary waves for the regularized long-wave equation
cites work: Q4273954
cites work: Euler operators and conservation laws of the BBM equation
cites work: Eigenvalues, and instabilities of solitary waves
cites work: Q4358703
cites work: Q4315882
cites work: Fourier pseudospectral solution of the regularised long wave equation
cites work: Instability of a class of dispersive solitary waves
cites work: Variable steps for reversible integration methods
cites work: Modulational Stability of Ground States of Nonlinear Schrödinger Equations
cites work: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation
cites work: Q4151274
full work available at URL: s0168-9274(01)00144-1
OpenAlex ID: W2034086080
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description: scientific article; zbMATH DE number 1787725