• 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 and 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.

[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.

[3]

M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.

[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.

[5]

A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.

[6]

J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.

[7]

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.

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.

[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.

[3]

M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.

[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.

[5]

A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.

[6]

J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.

[7]

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.

[1]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and 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 and 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 and Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[4]

A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97

[5]

Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268

[6]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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]

Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure and Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487

[18]

Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure and Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421

[19]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467

[20]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]