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Algebraic travelling waves for the generalized Burgers-Fisher equation. (English) Zbl 1461.34068

Summary: Consider the generalized Burgers-Fisher equation \[ \frac{\partial u}{\partial t}+bu^m\frac{\partial u}{\partial x}-a\frac{\partial^2u}{\partial x^2}-du(1-u^m)=0,\tag{\(*\)} \] where \(a,b,d\in\mathbb{R}\) with \(a\ne 0\) and \(m\ge 2\), and look for traveling wave solutions \(u= U(x-ct)\) with speed \(c\) satisfying the boundary conditions \[ \lim_{s\to-\infty}U(s)=A\quad\text{and}\quad\lim_{s\to\infty}U(s)=B, \] where \(A\) and \(B\) are solutions of \(u(1-u^m)=0\). \(U\) is a solution of the ordering differential equation \[ aU''=(bU^m-c)U'-dU(1-U^m).\tag{\(**\)} \] Since the right hand side of (\(**\)) has the form \(p(U,U')=0\), where \(p\) is a polynomial, any noncompact traveling wave of (\(*\)) is an algebraic traveling wave.
The main result of this paper presents all explicit algebraic traveling waves of (\(*\)).

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
35C07 Traveling wave solutions
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations

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