International Journal of Theoretical Physics

, Volume 37, Issue 5, pp 1547–1567 | Cite as

Exact Traveling-Wave Solutions to Bidirectional Wave Equations

  • Min Chen


In this paper, we present several systematicways to find exact traveling-wave solutions of thesystems
$$\eta _t + u_x + \left( {u\eta } \right)_x + au_{xxx} - b\eta _{xxt} = 0$$
$$u_t + \eta _x + uu_x + c\eta _{xxx} + du_{xxt} = 0$$
where a, b, c, and d are real constants. These systems,derived by Bona, Saut and Toland for describingsmall-amplitude long waves in a water channel, areformally equivalent to the classical Boussinesq systemand correct through first order with regard to asmall parameter characterizing the typicalamplitude-todepth ratio. Exact solutions for a largeclass of systems are presented. The existence of theexact traveling-wave solutions is in general extremely helpful inthe theoretical and numerical study of thesystems.


Field Theory Exact Solution Elementary Particle Quantum Field Theory Wave Equation 
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© Plenum Publishing Corporation 1998

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  • Min Chen

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