zbMATH — the first resource for mathematics

Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. (English) Zbl 1022.35051
Summary: We present a reliable algorithm to study the known model of nonlinear dispersive waves proposed by Boussinesq. The modified algorithm of the Adomian decomposition method is used with an emphasis on the single soliton solution. New exact periodic solutions and polynomial solutions are obtained. The results of numerical examples are presented and only few terms are required to obtain accurate solutions.

35Q35 PDEs in connection with fluid mechanics
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
35B10 Periodic solutions to PDEs
PDF BibTeX Cite
Full Text: DOI
[1] Hirota, R., Direct methods in soliton theory, () · Zbl 0124.21603
[2] Hirota, R., Exact envelope-soliton solutions of a nonlinear wave, J math phys, 14, 7, 805-809, (1973) · Zbl 0257.35052
[3] Hirota, R., Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices, J math phys, 14, 7, 810-814, (1973) · Zbl 0261.76008
[4] Debnath, L., Nonlinear partial differential equations for scientists and engineers, (1998), Birkhauser Berlin
[5] Debnath, L., Nonlinear water waves, (1994), Academic Press Boston · Zbl 0793.76001
[6] Bhatnagar, P.L., Nonlinear waves in one-dimensional dispersive systems, (1976), Clarendon Press Oxford · Zbl 0487.35001
[7] Drazin PG, Johnson RS. Solitons: an introduction. Cambridge, New York, 1993
[8] Lamb, G.L., Elements of soliton theory, (1980), Wiley New York · Zbl 0445.35001
[9] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
[10] Ablowitz, M.; Segur, H., Solitons and inverse scattering transform, (1981), SIAM Philadelphia, PA · Zbl 0472.35002
[11] Freeman, N.C., Soliton interactions in two dimensions, Adv appl mech, 20, 1-37, (1980) · Zbl 0477.35077
[12] Freeman, N.C., Soliton solutions of non-linear evolution equations, IMA J appl math, 32, 125-145, (1984) · Zbl 0542.35079
[13] Nimmo, J.J.C.; Freeman, N.C., A method of obtaining the N-soliton solutions of the Boussinesq equation in terms of a Wronskian, Phys lett, 95A, 4-6, (1983) · Zbl 0588.35077
[14] Nimmo, J.J.C.; Freeman, N.C., The use of Bäcklund transformations in obtaining the N-soliton solutions in Wronskian form, J phys A: math general, 17, 1415-1424, (1984) · Zbl 0552.35071
[15] Kaptsov, O.V., Construction of exact solutions of the bousseniseq equation, J appl mech and tech phys, 39, 3, 389-392, (1998)
[16] Andreev, V.K.; Kaptsov, O.V.; Pukhnachov, V.V.; Rodionov, A.A., Applications of group-theoretical methods in hydrodynamics, (1998), Kluwer Academic Publishers Boston · Zbl 0912.35001
[17] Bratsos, A.G., The solution of the Boussinesq equation using the method of lines, Comput methods appl mech eng, 157, 33-44, (1998) · Zbl 0952.76068
[18] Wazwaz, A.M., A first course in integral equations, (1997), World Scientific Singapore
[19] Wazwaz, A.M., Analytical approximations and Padé’ approximants for Volterra’s population model, Appl math and comput, 100, 13-25, (1999) · Zbl 0953.92026
[20] Wazwaz, A.M., A reliable modification of Adomian decomposition method, Appl math and comput, 102, 77-86, (1999) · Zbl 0928.65083
[21] Wazwaz, A.M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl math and comput, 111, 53-69, (2000) · Zbl 1023.65108
[22] Wazwaz, A.M., The modified decomposition method and Padé approximants for solving the Thomas-Fermi equation, Appl math and comput, 105, 11-19, (1999) · Zbl 0956.65064
[23] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston · Zbl 0802.65122
[24] Adomian, G., A review of the decomposition method in applied mathematics, J math anal appl, 135, 501-544, (1988) · Zbl 0671.34053
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.