American Institute of Mathematical Sciences

Advanced
  • 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

Jibin Li 1, and  Fengjuan Chen 1,

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.
Keywords: quasi-periodic wave solution, homoclinic manifold, center manifold, integrable system., Exact solution.
Mathematics Subject Classification: 34A26, 34C15, 34C23, 34C37, 35C07, 35C08, 37J35, 37K4.
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, 7 (1996), 211.   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.   Google Scholar

[3]

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

[5]

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

[6]

J. K. Hale, "Ordinary Differential Equation,", second Edition, (1980).   Google Scholar

[7]

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations,, Springer-Verlag, (1959).   Google Scholar

show all references

References:
[1]

X. B. Hu, The higher order KdV equation with a source and nonlinear superposition formula,, Chaos, 7 (1996), 211.   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.   Google Scholar

[3]

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

[5]

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

[6]

J. K. Hale, "Ordinary Differential Equation,", second Edition, (1980).   Google Scholar

[7]

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations,, Springer-Verlag, (1959).   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 - A, 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 - A, 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]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[7]

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
[8]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[9]

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
[10]

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
[11]

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
[12]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
[13]

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
[14]

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
[15]

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
[16]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[17]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[18]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[19]

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
[20]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences