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Solitary wave solutions for the KdV and mKdV equations by differential transform method. (English) Zbl 1198.35222
Summary: We aim to present a reliable algorithm in order to obtain exact and approximate solutions for the nonlinear dispersive KdV and mKdV equations with initial profile. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. We first present the definition and operation of the two-dimensional differential transform and investigate the soliton solutions of Kdv and mKdV equations are obtained by the present method. Therefore, in the present work, numerical examples are tested to illustrate the pertinent feature of the proposed algorithm. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q51 Soliton-like equations
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##### References:
 [1] Wadati, M.: Introduction to solitons, Pramana: J phys 57, No. 5 -- 6, 841-847 (2001) [2] Wadati, M.: The exact solution of the modified kortweg-de Vries equation, J phys soc jpn 32, 1681-1687 (1972) [3] Wadati, M.: The modified kortweg-de Vries equation, J phys soc jpn 34, 1289-1296 (1973) [4] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelengths, Phys rev lett 70, No. 5, 564-567 (1993) · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564 [5] Wazwaz, A. M.: Exact special solutions with solitary patterns for the nonlinear dispersive $K(m,n)$ equation, Chaos, solitons & fractals 13, 321-330 (2002) · Zbl 1028.35131 [6] Wazwaz, A. M.: New solitary-wave special solutions with compact support for the nonlinear dispersive $K(m,n)$ equation, Chaos, solitons & fractals 12, 1549-1556 (2001) · Zbl 1022.35051 [7] Wazwaz, A. M.: Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, solitons & fractals 12, 2283-2293 (2001) · Zbl 0992.35092 · doi:10.1016/S0960-0779(00)00188-0 [8] 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 · doi:10.1016/S0960-0779(00)00133-8 [9] Yan, Z. Y.: New families of solitons with compact support for Boussinesq-like $B(m,n)$ equations with fully nonlinear dispersion, Chaos, solitons & fractals 14, 1151-1158 (2002) · Zbl 1038.35082 · doi:10.1016/S0960-0779(02)00062-0 [10] Zhu, Y.; Chang, Q.; Wu, S.: Exact solitary-wave solutions with compact support for the modified KdV equation, Chaos, solitons & fractals 24, 365-369 (2005) · Zbl 1067.35099 · doi:10.1016/j.chaos.2004.09.041 [11] Zhu, Y. G.: Exact special solutions with solitary patterns for Boussinesq-like $B(m,n)$ equations with fully nonlinear dispersion, Chaos, solitons & fractals 22, 213-220 (2004) · Zbl 1062.35125 [12] &idot, I. E.; Nan; Kaya, D.: Some exact solutions to the potential Kadomtsev -- Petviashvili equation and a system of shallow water wave equations, Phys lett A 355, 314-318 (2006) · Zbl 05675862 [13] Jang, M. J.; Chen, C. L.; Liu, Y. C.: Two-dimensional differential transform for partial differential equations, Appl math comput 121, 261-270 (2001) · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3 [14] Pukhov GE. Differential transformations and mathematical modelling of physical processes. Kiev; 1986. [15] Chen, C. K.; Ho, S. H.: Solving partial differential equations by two dimensional differential transform, Appl math comput 106, 171-179 (1999) · Zbl 1028.35008 · doi:10.1016/S0096-3003(98)10115-7 [16] Ayaz, F.: On the two-dimensional differential transform method, Appl math comput 143, 361-374 (2003) · Zbl 1023.35005 · doi:10.1016/S0096-3003(02)00368-5 [17] Ayaz, F.: Solution of the system of the differential transform method, Appl math comput 147, 547-567 (2004) · Zbl 1032.35011 [18] Kurnaz, A.; Oturanç, G.; Kiriş, M. E.: N-dimensional differential transformation method for solving linear and nonlinear PDE’s, Int J comput math 82, 369-380 (2005) · Zbl 1065.35011 · doi:10.1080/0020716042000301725 [19] Hirota, R.: Direct methods in soliton theory, Solitons (1980) [20] Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave, J math phys 14, No. 7, 805-809 (1973) · Zbl 0257.35052 · doi:10.1063/1.1666399 [21] Hirota, R.: Exact N-soliton solutions of the wave equation of long waves in shallow-water and nonlinear lattices, J math phys 14, No. 7, 810-814 (1973) · Zbl 0261.76008 · doi:10.1063/1.1666400 [22] Lax, P. D.: A Hamiltonian approach to the KdV equation and others equations, Nonlinear evolution equations, 207-224 (1978) [23] Lax, P. D.: Periodic solutions of the kortweg-de Vries equation, Commun pure appl math 28, 141-188 (1975) [24] Drazin, P. G.; Johnson, R. S.: Solitons: an introduction, (1993) · Zbl 0661.35001 [25] Abdel-Halim, Hassan: Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, solitons & fractals 36, No. 1, 53-65 (2008) · Zbl 1152.65474 · doi:10.1016/j.chaos.2006.06.040