Solitary Wave Solutions to the General Class of Nonlocal Nonlinear Coupled Wave Equations
Abstract
In this paper, we study a general class of nonlocal nonlinear coupled wave equations that includes the convolution operation with kernel functions. For appropriate selections of the kernel functions, the system becomes well-known nonlinear coupled wave equations, for instance Toda lattice system, coupled improved Boussinesq equations. A numerical scheme is proposed for the solitary wave solutions of the system using the Pethiashvili method. Using the different kernels, the validity of the numerical method has been tested.
Keywords
Coupled Boussinesq equations, Petviashvili's iteration method, solitary wave solutions.
References
- [1] A.C. Eringen, “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves,” Journal of Applied. Physics, vol. 54, pp. 4703–4710, 1983.
- [2] J.A.D. Wattis, “Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods,” Physics Letters A, vol. 284, pp. 16–22, 2001.
- [3] P.L. Christiansen, P.S. Lomdahl, V. Muto, “On a Toda lattice model with a transversal degree of freedom,” Nonlinearity, vol. 4, pp. 477–501, 1991.
- [4] K.R. Khusnutdinova, A.M. Samsonov, A.S. Zakharov, “Nonlinear layered lattice model and generalized solitary waves in imperfectly bonded structures,” Physical Review E, vol. 79, Article ID 056606, 2009.
- [5] S.K. Turitsyn, “On a Toda lattice model with a transversal degree of freedom. Sufficient criterion of blow-up in the continuum limit,” Physics Letters A, vol. 267, pp. 173-267, 1993.
- [6] A. De Godefroy, “Blow up of solutions of a generalized Boussinesq equation,” IMA Journal of Applied Mathematics, vol. 60, pp. 123–138, 1998.
- [7] S. Wang, M. Li, “The Cauchy problem for coupled IMBq equations,” IMA Journal of Applied Mathematics, vol. 74, pp. 726–740, 2009.
- [8] M. Lazar, G.A. Maugin, and E.C. Aifantis, “On a theory of nonlocal elasticity of bi-Helmholtz type and some applications,” International Journal of Solids and Structures., 43, pp. 1404–1421, 2006.
- [9] P. Rosenau, “Dynamics of dense discrete systems,” Progress of Theoretical Physics, vol. 79, pp. 1028–1042, 1988.
- [10] N. Duruk, A. Erkip, and H.A. Erbay, “A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity,” IMA Journal of Applied Mathematics, vol. 74, pp. 97– 106, 2009.