Exact Traveling-Wave Solutions to Bidirectional Wave Equations

In this paper, we present several systematicways to find exact traveling-wave solutions of thesystems\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\eta _t + u_x + \left( {u\eta } \right)_x + au_{xxx} - b\eta _{xxt} = 0$$ \end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u_t + \eta _x + uu_x + c\eta _{xxx} + du_{xxt} = 0$$ \end{document}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.