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.

References

References
1.
Hasegawa
,
A.
, and
Tappert
,
F.
,
1973
, “
Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers—I: Anomalous Dispersion
,”
Appl. Phys. Lett.
,
23
(
3
), pp.
142
144
.
2.
Hasegawa
,
A.
, and
Tappert
,
F.
,
1973
, “
Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers—II: Normal Dispersion
,”
Appl. Phys. Lett.
,
23
(
4
), pp.
171
172
.
3.
Heimburg
,
T.
, and
Jackson
,
A.
,
2005
, “
On Soliton Propagation in Biomembranes and Nerves
,”
Proc. Natl. Acad. Sci. U.S.A.
,
102
(
28
), pp.
9790
9795
.
4.
Kosevich
,
A.
,
Gann
,
V.
,
Zhukov
,
A.
, and
Voronov
,
V.
,
1998
, “
Magnetic Soliton Motion in a Nonuniform Magnetic Field
,”
J. Exp. Theor. Phys.
,
87
(
2
), pp.
401
407
.
5.
Deng
,
S.
,
Guo
,
B.
, and
Wang
,
T.
,
2011
, “
Travelling Wave Solutions of a Generalized Camassa–Holm–Degasperis–Procesi Equation
,”
Sci. China Ser. A
,
54
(
3
), pp.
555
572
.
6.
Li
,
J.
,
2012
, “
Bifurcations and Exact Traveling Wave Solutions of the Generalized Two-Component Camassa–Holm Equation
,”
Int. J. Bifurcation Chaos
,
22
(
12
), p.
1250305
.
7.
Meng
,
Q.
,
He
,
B.
,
Long
,
Y.
, and
Li
,
Z.
,
2011
, “
New Exact Periodic Wave Solutions for the Dullin–Gottwald–Holm Equation
,”
Appl. Math. Comput.
,
218
(
8
), pp.
4533
4537
.
8.
Zhu
,
M.
, and
Xu
,
J.
,
2012
, “
On the Wave-Breaking Phenomena for the Periodic Two-Component Dullin–Gottwald–Holm System
,”
J. Math. Anal. Appl.
,
391
(
2
), pp.
415
428
.
9.
Han
,
Y.
,
Guo
,
F.
, and
Gao
,
H.
,
2013
, “
On Solitary Waves and Wave-Breaking Phenomena for a Generalized Two-Component Integrable Dullin–Gottwald– Holm System
,”
J. Nonlinear Sci.
,
23
(
4
), pp.
617
656
.
10.
Guo
,
F.
,
Gao
,
H.
, and
Liu
,
Y.
,
2012
, “
On the Wave-Breaking Phenomena for the Two-Component Dullin–Gottwald–Holm System
,”
J. London Math. Soc.
,
86
(
2
), pp.
810
834
.
11.
Chen
,
Y.
,
Gao
,
H.
, and
Liu
,
Y.
,
2013
, “
On the Cauchy Problem for the Two-Component Dullin–Gottwald–Holm System
,”
Discrete Contin. Dyn. Syst.
,
33
(
8
), pp.
3407
3441
.
12.
Liu
,
X.
, and
Yin
,
Z.
,
2013
, “
Local Well-Posedness and Stability of Solitary Waves for the Two-Component Dullin–Gottwald–Holm System
,”
Nonlinear Anal.
,
88
, pp.
1
15
.
13.
Zhu
,
M.
, and
Xu
,
J.
,
2013
, “
Wave-Breaking Phenomena and Global Solutions for Periodic Two-Component Dullin–Gottwald–Holm Systems
,”
Electron. J. Differ. Equations
,
2013
(
44
), pp.
1
27
.http://ejde.math.txstate.edu/
14.
Zhang
,
Z.
,
Ding
,
T.
,
Huang
,
W.
, and
Dong
,
Z.
,
1991
,
Qualitative Theory of Differential Equations
,
American Mathematical Society
,
Providence, RI
.
15.
Tang
,
Y.
, and
Zhang
,
W.
,
2004
, “
Generalized Normal Sectors and Orbits in Exceptional Directions
,”
Nonlinearity
,
17
(
4
), pp.
1407
1426
.
16.
Li
,
J.
, and
Qiao
,
Z.
,
2013
, “
Bifurcations and Exact Traveling Wave Solutions for a Generalized Camassa–Holm Equation
,”
Int. J. Bifurcation Chaos
,
23
(
3
), p.
1350057
.
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