Traveling Wave Solutions of a Two-Component Dullin–Gottwald–Holm System

Abstract

In this paper, we investigate the traveling wave solutions of a two-component Dullin–Gottwald–Holm (DGH) system. By qualitative analysis methods of planar systems, we investigate completely the topological behavior of the solutions of the traveling wave system, which is derived from the two-component Dullin–Gottwald–Holm system, and show the corresponding phase portraits. We prove the topological types of degenerate equilibria by the technique of desingularization. According to the dynamical behaviors of the solutions, we give all the bounded exact traveling wave solutions of the system, including solitary wave solutions, periodic wave solutions, cusp solitary wave solutions, periodic cusp wave solutions, compactonlike wave solutions, and kinklike and antikinklike wave solutions. Furthermore, to verify the correctness of our results, we simulate these bounded wave solutions using the software maple version 18.

Analyzing Sensitivity and Solitonic Behavior Using the Dullin-Gottwald-Holm Model in Shallow Water Waves

“…Xiao et al [21] discussed exact results by employing the traveling-wave transformation and the exp-function technique. The use of qualitative planar systems to describe limited exact traveling wave solutions was examined [22]. Periodic wave solutions were discovered by Meng et al [23] using integral bifurcation and semi-inverse techniques.…”

Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin–Gottwald–Holm system and modulation instability analysis

“…In Meng et al, 34 periodic wave solutions were obtained with the aid of integral bifurcation and semi‐inverse methods. Peakon–antipeakon interaction with direct computation has been investigated in Zhou et al 35 The qualitative technique of planar systems to discuss the bounded exact traveling wave solutions was studied 36 . In Yu, 37 the dynamical behaviour of traveling wave solutions and its bifurcations were presented in different parameter regions.…”