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Exact traveling-wave solutions to bidirectional wave equations. (English) Zbl 1097.35115
Summary: In this paper, we present several systematic ways to find exact traveling-wave solutions of the systems
\[ \eta _t + u_x + \left( {u\eta } \right)_x + au_{xxx} - b\eta _{xxt} = 0 \]
\[ u_t + \eta _x + uu_x + c\eta _{xxx} + du_{xxt} = 0 \]
where \(a, b, c,\) and \(d\) are real constants. These systems, derived by Bona, Saut and Toland for describing small-amplitude long waves in a water channel, are formally equivalent to the classical Boussinesq system and correct through first order with regard to a small parameter characterizing the typical amplitude-to depth ratio. Exact solutions for a large class of systems are presented. The existence of the exact traveling-wave solutions is in general extremely helpful in the theoretical and numerical study of the systems.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
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