Existence and breaking property of real loop-solutions of two nonlinear wave equations
type: Item
instance of: scholarly article
title: Existence and breaking property of real loop-solutions of two nonlinear wave equations
zbMATH Open document ID: 1182.34065
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 2009-07-01T00:00:00Z
review text: Dynamical analysis has revealed that, for some nonlinear wave equations, loop- and inverted loop-solutions are actually visual artifacts. The so-called loop-soliton solution consists of three solutions, and is not a real solution. The paper answers the question as to whether or not nonlinear wave equations exist for which a ``real'' loop-solution exists, and if so, what are the precise parametric representations of these loop traveling wave solutions. The main result of the paper is the following theorem. Theorem. For any of the equations \[ u_{xxx}+\frac{A[A^3-2(3-4A^2)u^2+3(A-2)u^4]}{c(A-u^2)^4}u_t=0 \] and \[ u_{xxx}+\frac{4[B+(2B-3)u]}{c(1-u)^4}u_t=0, \] its corresponding traveling wave system has at least one family of open orbits which gives rise to a family of breaking loop-solutions of the underlying equation. In addition, these open orbits lie in an area bounded by a singular straight line and the stable and unstable manifolds of a saddle point of the system.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 74J30
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 5573912
zbMATH Keywords: planar dynamical system
zbMATH Keywords: breaking wave solution
zbMATH Keywords: loop-solution
zbMATH Keywords: nonlinear wave equation
author: Jibin Li
MaRDI profile type: Publication
full work available at URL: s10483-009-0501-4
OpenAlex ID: W1984383138
cites work: On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons
cites work: Explicit solutions of the reduced Ostrovsky equation
cites work: Some periodic and solitary travelling-wave solutions of the short-pulse equation
cites work: Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation
cites work: Dynamical understanding of loop soliton solution for several nonlinear wave equations
cites work: Solitary wave solutions of the short pulse equation
cites work: Propagation of ultra-short optical pulses in cubic nonlinear media
cites work: Q4719521
cites work: The \(N\)-soliton solution of the modified generalised Vakhnenko equation (a new nonlinear evolution equation).
cites work: The two loop soliton solution of the Vakhnenko equation
cites work: High-frequency soliton-like waves in a relaxing medium
cites work: Family of nonlinear wave equations which yield loop solutions and solitary wave solutions
cites work: ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS
cites work: TRAVELING WAVES FOR AN INTEGRABLE HIGHER ORDER KDV TYPE WAVE EQUATIONS
cites work: Q5615009
DOI: 10.1007/S10483-009-0501-4
description: scientific article; zbMATH DE number 5573912