Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On periodic traveling wave solutions of Boussinesq equation

Authors: Bao Ping Liu and C. V. Pao
Journal: Quart. Appl. Math. 42 (1984), 311-319
MSC: Primary 35Q20; Secondary 35B10
DOI: https://doi.org/10.1090/qam/757169
MathSciNet review: 757169
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Abstract: This paper is concerned with periodic traveling wave solutions of a generalized Boussinesq equation in the form $ {u_{tt}} = \alpha {u_{xxxx}} + {({f_0}(u))_{xx}}$. The basic approach to this problem is to establish an equivalence relation between a corresponding periodic boundary value problem for the traveling wave solution and a Hammerstein type integral equation. This integral representation generates a compact operator in the space of continuous periodic functions of the given period. It is shown by restricting the integral operator on a suitable domain that the Boussinesq equation has the trivial solution as well as a nonconstant periodic traveling wave solution. Special attention is given to the traditional Boussinesq equation where $ {f_0}(u) = a{u^2}$. Both of the so called ``good'' and ``bad'' Boussinesq equation are treated and the existence of nonconstant traveling wave solution is discussed.

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DOI: https://doi.org/10.1090/qam/757169
Article copyright: © Copyright 1984 American Mathematical Society

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