On positive travelling wave solutions for a general class of KdV-Burger type equation

Volume 12, Issue 7, pp 485--502 http://dx.doi.org/10.22436/jnsa.012.07.06

Publication Date: March 18, 2019 Submission Date: November 23, 2018 Revision Date: January 17, 2019 Accteptance Date: January 23, 2019

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Authors

Gilberto Arenas-Díaz - Escuela de Matematicas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia. José R. Quintero - Departamento de Matematicas, Universidad del Valle, A.A. 25360, Cali, Colombia.

Abstract

In this paper, we establish the existence of positive traveling waves solutions for the third order differential equation \(u_{t}+\alpha u_{xx}+\beta u_{xxx}+\left(f\left(x,u(x)\right)\right)_{x}=0\), where \(t,x\in\bf R\), \(f\) is a non-negative continuous function with some properties. The result is a consequence of the characterization of the travelling wave solutions as fixed points of some functional, defined using the Green's function associated to the linear problem, and the Krasnosel'skii fixed point theorem on cone expansion and compression of norm type.

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Keywords

• Travelling wave solutions
• Green function
• Krasnosel'skii fixed point theorem

MSC

•  34G20
•  35C07
•  74J35

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