Traveling Wave Solutions of Some $abcd$-Water Wave Models Describing Small Amplitude, Long Wavelength Gravity Waves on the Surface of Water

Abstract

For some $abcd$-water wave models describing small amplitude, long wavelength gravity waves on the surface of water, in this paper, by using the method of dynamical systems to analyze corresponding traveling wave systems and find the bifurcations of phase portraits, the dynamical behavior of systems can be derived. Under some given parameter conditions, for a wave component, the existence of periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions as well as compacton families can be proved. Possible exact explicit parametric representations of the traveling wave solutions are given.