zbMATH — the first resource for mathematics

A study of nonlinear dispersive equations with solitary-wave solutions having compact support. (English) Zbl 0999.65109
The nonlinear \(K(m,n)\) equation \(u_{t} + (u^{m})_{x} +(u^{n})_{xxx} = 0\) and its exact solution by the Adomian decomposition method are discussed. Then two numerical illustrations \(K(2,2)\) and \(K(3,3)\) are investigated to illustrate the pertinent features of the proposed scheme. The technique can be used for any nonlinear dispersive equations.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994. · Zbl 0802.65122
[2] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1988) · Zbl 0671.34053
[3] Ismail, M.S.; Taha, T., A numerical study of compactons, Math. comput. simulation, 47, 519-530, (1998) · Zbl 0932.65096
[4] M.S. Ismail, A finite difference method of Kortweg – de Vries like equation with nonlinear dispersion, Int. J. Comput. Math., in press.
[5] Olver, P.J.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. rev. E, 53, 2, 1900-1906, (1996)
[6] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys. rev. lett., 70, 5, 564-567, (1993) · Zbl 0952.35502
[7] Rosenau, P., Nonlinear dispersion and compact structures, Phys. rev. lett., 73, 13, 1737-1741, (1994) · Zbl 0953.35501
[8] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. lett. A, 230, 5/6, 305-318, (1997) · Zbl 1052.35511
[9] Rosenau, P., On a class of nonlinear dispersive – dissipative interactions, Phys. D, 230, 5/6, 535-546, (1998) · Zbl 0938.35172
[10] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997. · Zbl 0924.45001
[11] Wazwaz, A.M., Analytical approximations and padé’ approximants for volterra’s population model, Appl. math. comput., 100, 13-25, (1999) · Zbl 0953.92026
[12] Wazwaz, A.M., The modified decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Appl. math. comput., 110, 251-264, (2000) · Zbl 1023.65109
[13] Wazwaz, A.M., The decomposition method for solving the diffusion equation subject to the classification of mass, Ijam, 3, 1, 25-34, (2000)
[14] A.M. Wazwaz, Approximate solutions to boundary value problems of higher-order by the modified decomposition method, Comput. Math. Appl. 40 (2000) 679-691. · Zbl 0959.65090
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.