Traveling Wave Solutions of Some $abcd$-Water Wave Models Describing Small Amplitude, Long Wavelength Gravity Waves on the Surface of Water (Journal of Nonlinear Modeling and Analysis, 2025)
For some $abcd$-water wave models describing small amplitude, longwavelength gravity waves on the surface of water, in this paper, by using themethod of dynamical systems to analyze corresponding traveling wave systemsand find the bifurcations of phase portraits, the dynamical behavior of systemscan 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 exactexplicit parametric representations of the traveling wave solutions are given.
Keywords: Pseudo-peakon; solitary wave; kink and anti-kink wave; compacton family; planar dynamical system; $abcd$-water wave models.