# American Institute of Mathematical Sciences

• Previous Article
A note on the global stability of an SEIR epidemic model with constant latency time and infectious period
• DCDS-B Home
• This Issue
• Next Article
Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation
January  2013, 18(1): 163-172. doi: 10.3934/dcdsb.2013.18.163

## Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations

 1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received  March 2012 Revised  July 2012 Published  September 2012

For the system of KP like equation coupled to a Schrödinger equation, a corresponding four-dimensional travelling wave systems and a two-order linear non-autonomous system are studied by using Congrove's results and dynamical system method. For the four-dimensional travelling wave systems, exact explicit homoclinic orbit families, periodic and quasi-periodic wave solution families are obtained. The existence of homoclinic manifolds to four kinds of equilibria including a hyperbolic equilibrium, a center-saddle and an equilibrium with zero pair of eigenvalues is revealed. For the two-order linear non-autonomous system, the dynamical behavior of the bounded solutions is discussed.
Citation: Jibin Li, Fengjuan Chen. Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 163-172. doi: 10.3934/dcdsb.2013.18.163
##### References:
 [1] X. B. Hu, The higher order KdV equation with a source and nonlinear superposition formula, Chaos, Solitonsand Fract., 7 (1996), 211-215.  Google Scholar [2] S. Z. Rida and M. Khalfallah, New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Commun. nonlinear Sci. Numer. Simulat., 15 (2010), 2818-2827.  Google Scholar [3] M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.  Google Scholar [4] A. M. Wazwaz, Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182 (2006), 283-300.  Google Scholar [5] A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.  Google Scholar [6] J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.  Google Scholar [7] L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.  Google Scholar

show all references

##### References:
 [1] X. B. Hu, The higher order KdV equation with a source and nonlinear superposition formula, Chaos, Solitonsand Fract., 7 (1996), 211-215.  Google Scholar [2] S. Z. Rida and M. Khalfallah, New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Commun. nonlinear Sci. Numer. Simulat., 15 (2010), 2818-2827.  Google Scholar [3] M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.  Google Scholar [4] A. M. Wazwaz, Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182 (2006), 283-300.  Google Scholar [5] A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.  Google Scholar [6] J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.  Google Scholar [7] L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.  Google Scholar
 [1] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [2] Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623 [3] Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249 [4] Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268 [5] A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97 [6] Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213 [7] Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873 [8] Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161 [9] Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615 [10] Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241 [11] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [12] Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171 [13] Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121 [14] Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [15] Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007 [16] Gennadiy Burlak, Salomon García-Paredes. Matter-wave solitons with a minimal number of particles in a time-modulated quasi-periodic potential. Conference Publications, 2015, 2015 (special) : 169-175. doi: 10.3934/proc.2015.0169 [17] Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure & Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421 [18] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487 [19] Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467 [20] Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75

2019 Impact Factor: 1.27