×

New exact solutions for modified nonlinear dispersive equations \(mK(m,n)\) in higher dimensions spaces. (English) Zbl 1073.35052

Summary: With the use of some proper transformations and symbolic computation, we present a general and unified method for investigating the general modified nonlinear dispersive equations \(mK(m,n)\) in higher dimensions spaces. The work formally shows how to construct the general solutions and some special exact-solutions for \(mK(m,n)\) equations in higher dimensional spatial domains. The general solutions not only contain the solutions by A. M. Wazwaz [Math. Comput. Simul. 59, No. 6, 519–531 (2002; Zbl 0996.35065)] but also contain many new compact and noncompact solutions.

MSC:

35C05 Solutions to PDEs in closed form
35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 0996.35065
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M.J. Ablowitz, P.A. Clarkson. Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991.; M.J. Ablowitz, P.A. Clarkson. Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. · Zbl 0762.35001
[2] C.H. Gu, et al., Soliton Theory and its Application, Zhejiang Science and Technology Press, Zhejiang, 1990.; C.H. Gu, et al., Soliton Theory and its Application, Zhejiang Science and Technology Press, Zhejiang, 1990.
[3] Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelengths, Phys. Rev. Lett., 70, 5, 564-567 (1993) · Zbl 0952.35502
[4] 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)
[5] Rosenau, P., Nonlinear dispersion and compact structures, Phys. Rev. Lett., 73, 13, 1737-1741 (1994) · Zbl 0953.35501
[6] Rosenau, P., On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A, 230, 5-6, 305-318 (1997) · Zbl 1052.35511
[7] Rosenau, P., On a class of nonlinear dispersive-dissipative interactions, Physica D, 230, 5-6, 535-546 (1998) · Zbl 0938.35172
[8] Ismail, M. S.; Taha, T., A numerical study of compacts, Math. Comput. Simulation, 47, 519-530 (1998) · Zbl 0932.65096
[9] Ludu, A.; Draayer, J. P., Patterns on liquid surfaces: cnoidal waves, compactons and scaling, Physca D, 123, 82-91 (1998) · Zbl 0952.76008
[10] Dinda, P. T.; Remoissenet, M., Breather compactons in nonlinear Klein-Gordon systems, Phys. Rev. E, 60, 5, 6218-6221 (1999)
[11] Wazwaz, A. M., New solitary-wave special solutions with compact support for the nonlinear dispersive \(K(m,n)\) equations, Chaos Solitons Fractals, 13, 2, 321-330 (2002) · Zbl 1028.35131
[12] Wazwaz, A. M., Exact special solutions with solitary patterns for the nonlinear dispersive \(K(m,n)\) equatons, Chaos Solitons Fractals, 13, 1, 161-170 (2001) · Zbl 1027.35115
[13] Wazwaz, A. M., Compactons dispersive structures for variants of the \(K(n,n)\) and the KP equations, Chaos Solitons Fractals, 13, 5, 1053-1062 (2002) · Zbl 0997.35083
[14] Wazwaz, A. M., A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simulation, 56, 269-276 (2001) · Zbl 0999.65109
[15] Wazwaz, A. M., General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations \(mK (n,n)\) in higher dimensional spaces, Math. Comput. Simulation, 59, 519-531 (2002) · Zbl 0996.35065
[16] Wazwaz, A. M., A computational approach to soliton solutions of the Kadomtsev-Petviashili equation, Appl. Math. Comput., 123, 2, 205-217 (2001) · Zbl 1024.65098
[17] Fan, E. G.; Zhang, H. Q., New exact solutions for a system of coupled Kdv equation, Phys. Lett. A, 245, 389-392 (1998) · Zbl 0947.35126
[18] Fan, E. G.; Zhang, H. Q., A note on the homogeneous balance method, Phys. Lett. A, 246, 403-406 (1998) · Zbl 1125.35308
[19] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 212, 277 (2000)
[20] Fan, E., Travelling wave solutions for two generalized Hirota-Satsuma coupled KdV systems, Z. Naturforsch. A, 56, 312-318 (2001)
[21] Fan, E.; Zhang, J.; Hon, B. Y.C., A new complex line soliton for the two-dimensional KdV-Burgers equations, Phys. Lett. A, 291, 376-380 (2001) · Zbl 0979.35126
[22] Yan, Z. Y., New explicit travelling wave solutions for two new intgrable coupled nonlinear evolution equations, Phys. Lett. A, 292, 100-106 (2001) · Zbl 1092.35524
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.