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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P. L. Bhatnagar, Nonlinear waves in one-dimensional dispersive systems, Clarendon Press, Oxford, 1979
R. Hirota, Exact N-soliton solution of the wave equation of long waves in shallow-water and in nonlinear lattices, J. Math. Phys. 14, 810–814 (1973)
H. P. Mckean, Boussinesq’s equation on the circle, Comm. Pure Appl. Math. 34, 599–691 (1981)
P. Prasad and R. Ravindran, A theory of nonlinear waves in multi-dimensions: with special reference to surface waves, J. Inst. Math. and its Appl. 20, 9–20, (1977)
M. Toda, and M. Wadati, A soliton and two solitons in an exponential lattice and related equations, J. Phys. Soc. Japan 34, 18–25, (1973)
N. J. Zabusky, A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, 223–258, Nonlinear Partial Differential Equations (ed. by W. F. Ames), Academic Press, New York, 1967
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Article copyright:
© Copyright 1984
American Mathematical Society
