Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces
type: Item
instance of: scholarly article
title: Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces
zbMATH Open document ID: 1093.35005
author: A. H. Khater
author: D. K. Callebaut
author: S. M. Sayed
published in: Journal of Computational and Applied Mathematics
publication date: 2006-03-24T00:00:00Z
review text: Bäcklund transformations are used in the calculation of soliton solutions of certain nonlinear evolution equations (NLEEs) [see \textit{R. K. Dodd}, J. Phys. A 11, 81--92 (1978; Zbl 0368.35058)]. For some nonlinear evolution equations which describe pseudo-spherical surfaces two new exact solution classes are generated. Exact travelling wave and solitary wave solutions for Burgers' equation \(2u_t-2uu_x-u_{xx}=0\) and also for a fifth-order evolution equation \(u_t+au^2u_x+bu_xu_{xx}+ huu_{xxx}+du_{xxxxx} =0\), where \(a,b,h\), and \(d\) are constants, containing the fifth-order Korteweg-de Vries equation and also the Sawada-Kotera equation [\textit{E. J. Parkes} and \textit{B. R. Duffy}, Comput. Phys. Commun. 98, No. 3, 288--300 (1996; Zbl 0948.76595)] are obtained by using an improved sine-cosine method and the Wu's elimination method. Employing the Bäcklund transformations involving explicitly the wave solutions, new solutions are generated for Burgers' equation. The obtained results are illustrated.
reviewed by: Valery V. Karachik
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 58J90
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K25
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 35Q51
zbMATH DE Number: 5015141
zbMATH Keywords: Bäcklund transformations
zbMATH Keywords: travelling wave
zbMATH Keywords: solitary wave
zbMATH Keywords: Burgers' equation
zbMATH Keywords: Sawada-Kotera equation
describes a project that uses: ATFM
MaRDI profile type: Publication
full work available at URL: j.cam.2005.10.007
OpenAlex ID: W2018632782
cites work: The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
cites work: Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations
cites work: Conservation laws for nonlinear evolution equations
cites work: Pseudospherical Surfaces and Evolution Equations
cites work: The NLS-equation and itsSL(2,R) structure
cites work: On differential systems describing surfaces of constant curvature
cites work: Gauge equivalence of differential equations describing surfaces of constant Gaussian curvature
cites work: Method for Solving the Korteweg-deVries Equation
cites work: On differential equations describing pseudo-spherical surfaces
cites work: Exact solutions for self-dual Yang-Mills equations
cites work: Bäcklund transformation and exact soliton solutions for some nonlinear evolution equations of the ZS/AKNS system
cites work: Backlund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere
cites work: Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces
cites work: Two new classes of exact solutions for the KdV equation via Bäcklund transformations
cites work: Exact solutions for self-dual \(\text{SU}(2)\) and \(\text{SU}(3)\) Yang--Mills fields
cites work: Self-dual solutions for SU(2) and SU(3) gauge fields on Euclidean space
cites work: Simple derivation of Bäcklund transformation from Riccati form of inverse method.
cites work: The homogeneous balance method, Lax pair, Hirota transformation and a general fifth-order KdV equation
cites work: An automated \(\tanh\)-function method for finding solitary wave solutions to nonlinear evolution equations
cites work: Pseudo-spherical surfaces and integrability of evolution equations
cites work: Some geometric aspects of integrability of differential equations in two independent variables
cites work: Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces
cites work: Geometric integrability of the Camassa-Holm equation
cites work: On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces
cites work: Transformations of solutions for equations and hierarchies of pseudo-spherical type
cites work: Q4537527
cites work: Bäcklund transformations and their applications
cites work: Prolongation structures of nonlinear evolution equations
cites work: Prolongation structures of nonlinear evolution equations. II
cites work: Exact solutions for a compound KdV-Burgers equation
cites work: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics
cites work: Q4525678
cites work: A simple transformation for nonlinear waves.
cites work: New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics
cites work: Q4536092
DOI: 10.1016/J.CAM.2005.10.007
description: scientific article; zbMATH DE number 5015141