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 . 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 . 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