The Lie group and integrability of the Fisher type travelling wave equation
type: Item
instance of: scholarly article
title: The Lie group and integrability of the Fisher type travelling wave equation
zbMATH Open document ID: 1191.34002
author: Minghui Liu
author: Keying Guan
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2010-01-18T00:00:00Z
review text: The authors consider reaction-diffusion equations of the form \[ u_t = u_{xx} + u (1-u) (1 + \gamma u) \tag{*} \] with \(\gamma\) a real parameter and \(u (x,t) \in {\mathbb R}\). The travelling wave solutions \(u(x,t) = v (x- c t) \equiv v (z)\) are obtained as solutions to the ODE \[ v'' + c v' + v (1-v) (1 + \gamma v) = 0. \tag{**} \] The paper studies the symmetry properties of (\(**\)), by establishing that it admits a two-parameter Lie group of symmetries provided either \(\{ c^2 = 9/2 , \gamma = 1 \}\) or \(\{ c^2 = 25/6 , \gamma = 0 \}\). Under these conditions, they use a method developed in previous work of themselves to give two independent first integrals which permit that these the corresponding travelling wave solutions can be expressed by elementary functions.
reviewed by: Giuseppe Gaeta
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34B40
zbMATH DE Number: 5659764
zbMATH Keywords: Reaction-diffusion equations
zbMATH Keywords: travelling wave
zbMATH Keywords: symmetry
zbMATH Keywords: first integrals
MaRDI profile type: Publication
full work available at URL: s10255-007-7106-6
OpenAlex ID: W1977583397
Wikidata QID: Q115384963
cites work: ON THE INTEGRABILITY, BÄCKLUND TRANSFORMATION AND SYMMETRY ASPECTS OF A GENERALIZED FISHER TYPE NONLINEAR REACTION–DIFFUSION EQUATION
cites work: Symmetries and differential equations
cites work: Burgers-Korteweg-de Vries equation and its traveling solitary waves
cites work: Exact solution in terms of elliptic functions for the Burgers --- Korteweg --- de Vries equation
cites work: On explicit exact solutions to the compound Burgers-KdV equation
cites work: On travelling wave solutions of the Burgers–Korteweg–de Vries equation
cites work: An attempt to differential Galois theory of second order polynomial system and solvable subgroup of Möbius transformations
cites work: Q3150306
cites work: Q4514347
cites work: Solitary wave solutions to some classes of nonlinear evolution type equations using inverse variational methods
cites work: Q4391582
cites work: An exact solution to two-dimensional Korteweg-de Vries-Burgers equation
cites work: Q3715336
cites work: Liouvillian First Integrals of Differential Equations
cites work: On the integrability and exact solutions of the nonlinear diffusion equation with a nonlinear source
cites work: An analytic solution of Burgers-KdV equation
DOI: 10.1007/S10255-007-7106-6
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description: scientific article; zbMATH DE number 5659764