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Research Papers

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

[+] Author and Article Information
Jiyu Zhong

Department of Mathematics,
Lingnan Normal University,
Zhanjiang 524048, China
e-mail: matzhjy@sina.com

Shengfu Deng

Department of Mathematics,
Lingnan Normal University,
Zhanjiang 524048, China
e-mail: sf_deng@sohu.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 4, 2016; final manuscript received October 18, 2016; published online December 5, 2016. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 12(3), 031006 (Dec 05, 2016) (8 pages) Paper No: CND-16-1003; doi: 10.1115/1.4035194 History: Received January 04, 2016; Revised October 18, 2016

## 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.

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## Figures

Fig. 5

The graphs of f for Δ > 0,d > c, and m1 < c < m2: (a) f(m1) < 0 and f(m2) < 0, (b) f(m1) < 0 and f(m2) = 0, (c) f(m1) < 0 and f(m2) > 0, (d) f(m1) = 0 and f(m2) > 0, (e) f(m1) = 0 and f(m2) = 0, (f) f(m1) = 0 and f(m2) < 0, (g) f(m1) > 0 and f(m2) < 0, (h) f(m1) > 0 and f(m2) = 0, and (i) f(m1) > 0 and f(m2) > 0

Fig. 9

The simulations of bounded wave solutions of system (1): (a) solitary wave solution, (b) periodic wave solution, (c) cusp solitary wave solution, (d) compactonlike wave solution, (e) periodic cusp wave solution, (f) kinklike wave solution, (g) antikinklike wave solution, and (h) compactonlike wave solution

Fig. 2

The graphs of f for Δ > 0,d > c, and c≤m1 (or c≥m2): (a) c≤m1 and f(m2) > 0, (b) c≥m2 and f(m1) > 0, (c) c≤m1 and f(m2)=0, (d) c≥m2 and f(m1)=0, (e) c≤m1 and f(m2) < 0, and (f) c≥m2 and f(m1) < 0

Fig. 1

The phase portraits of system (5) for d > c if one of the following conditions holds: (1)Δ≤0; (2)Δ > 0,c < m1, and f(m2) > 0; (3)Δ > 0,c > m2, and f(m1) > 0; and (4)Δ > 0,m1 < c < m2,f(m1) > 0, and f(m2) > 0

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