zbMATH — the first resource for mathematics

New solitary-wave special solutions with compact support for the nonlinear dispersive \(K(m,n)\) equations. (English) Zbl 1028.35131
Summary: The genuinely nonlinear dispersive \(K(m, n)\) equation, \(u_t+ (u^m)_x+ (u^n)_{xxx}= 0\), which exhibits compactons, solitons with compact support, is investigated. New solitary-wave solutions with compact support are developed. The specific cases, \(K(2, 2)\) and \(K(3, 3)\), are used to illustrate the pertinent features of the proposed scheme. An entirely new general formula for the solution of the \(K(m, n)\) equation is established, and the existing general formula is modified as well.

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
Full Text: DOI
[1] Dinda, P.; Remoissenet, M., Breather compactons in nonlinear klein – gordon systems, Phys. rev. E, 60, 5, 6218-6221, (1999)
[2] Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys. rev. lett., 70, 5, 564-567, (1993) · Zbl 0952.35502
[3] Rosenau, P., Nonlinear dispersion and compact structures, Phys. rev. lett., 73, 13, 1737-1741, (1994) · Zbl 0953.35501
[4] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. lett. A, 230, 5/6, 305-318, (1997) · Zbl 1052.35511
[5] Rosenau, P., On a class of nonlinear dispersive – dissipative interactions, Phys. D, 230, 5/6, 535-546, (1998) · Zbl 0938.35172
[6] 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)
[7] Ismail, M.S.; Taha, T., A numerical study of compactons, Math. comput. simulation, 47, 519-530, (1998) · Zbl 0932.65096
[8] Ismail MS. A finite difference method of Kortweg-de Vries like equation with nonlinear dispersion. Int J Comput Math 2001 [to appear]
[9] Ludu, A.; Draayer, J.P., Patterns on liquid surfaces: cnoidal waves, compactons and scaling, Phys. D, 123, 82-91, (1998) · Zbl 0952.76008
[10] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[11] Adomian, G., A review of the decomposition method in applied mathematics, J. math. anal. appl., 135, 501-544, (1998) · Zbl 0671.34053
[12] Adomian, G., Nonlinear stochastic operator equations, (1986), Academic Press San Diego · Zbl 0614.35013
[13] Adomian, G.; Rach, R., Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Comput. math. appl., 19, 12, 9-12, (1990) · Zbl 0702.35058
[14] Wazwaz, A.M., A computational approach to soliton solutions of the kadomtsev – petviashili equation, Appl. math. comput., 123, 205-217, (2001) · Zbl 1024.65098
[15] Wazwaz, A.M., Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, solitons & fractals, 12, 1549-1556, (2001) · Zbl 1022.35051
[16] Wazwaz, A.M., A first course in integral equations, (1997), World Scientific Singapore
[17] Wazwaz, A.M., Analytical approximations and Padé approximants for Volterra’s population model, Appl. math. comput., 100, 13-25, (1999) · Zbl 0953.92026
[18] 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
[19] Wazwaz, A.M., The decomposition method for solving the diffusion equation subject to the classification of mass, Int. J. appl. math., 3, 1, 25-34, (2000)
[20] Wazwaz, A.M., The modified decomposition method and Padé approximants for solving the Thomas-Fermi equation, Appl. math. comput., 105, 11-19, (1999) · Zbl 0956.65064
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.