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New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations. (English) Zbl 1185.35225
Summary: We consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By means of scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By means of scaling, new exact solutions to general Ito equation are formally derived.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35C07Traveling wave solutions of PDE
35A24Methods of ordinary differential equations for PDE
References:
[1]Hirota, R.: Direct methods in soliton theory, (1980)
[2]Ablowitz, M. J.; Clarkson, P. A.: Soliton, nonlinear evolution equations and inverse scattering, (1991) · Zbl 0762.35001
[3]Baldwin, D.; Goktas, U.; Hereman, W.; Hong, L.; Martino, R. S.; Miller, J. C.: Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear pdfs, J. symbolic comput. 37, No. 6, 669-705 (2004) · Zbl 1137.35324 · doi:10.1016/j.jsc.2003.09.004
[4]Fan, E.: Extended tanh-function method and its applications to nonlinear equations, Phys. lett. A 227, 212-218 (2000) · Zbl 1167.35331 · doi:10.1016/S0375-9601(00)00725-8
[5]Conte, R.; Musette, M.: Link betwen solitary waves and projective Riccati equations, J. phys. A – math. 25, 5609-5623 (1992) · Zbl 0782.35065 · doi:10.1088/0305-4470/25/21/019
[6]Yan, Z.: The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equation, Comput. phys. Commun. 152, No. 1, 1-8 (2003) · Zbl 1196.35068 · doi:10.1016/S0010-4655(02)00756-7
[7]Olver, P. J.: Applications of Lie group to differential equations, (1980) · Zbl 0599.58050
[8]Wazwaz, A. M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. math. Comput. 84-2, 1002-1014 (2007) · Zbl 1115.65106 · doi:10.1016/j.amc.2006.07.002
[9]Kaup, D. J.: On the inverse scattering problem for cubic eingevalue problems of the class ϕxxx+6qϕx+6rϕ=λϕ, Stud. appl. Math. 62, 189-216 (1980) · Zbl 0431.35073
[10]Sawada, K.; Kotera, T.: A method for finding N-soliton solutions for the KdV equation and KdV-like equation, Prog. theory phys. 51, 1355-1367 (1974) · Zbl 1125.35400 · doi:10.1143/PTP.51.1355
[11]Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves, Commun. pure appl. Math. 62, 467-490 (1968) · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
[12]Gomez, C. A.: Special forms of the fkdv equations with new periodic and soliton solutions, Appl. math. Comput. 189, 1066-1077 (2007) · Zbl 1122.65393 · doi:10.1016/j.amc.2006.11.158
[13]Gomez, C. A.: Special solutions for a new fifth-order integrable system, Rev. col. Mat. 40, 119-125 (2006) · Zbl 1189.35274 · doi:emis:journals/RCM/revistas.art810.html
[14]Gomez, C. A.; Salas, A.: Exact solutions for the generalized shallow water wave equation by the general projective Riccati equations method, Bol. mat. 1, 50-56 (2006) · Zbl 1203.35220
[15]C.A. Gomez, A. Salas, New exact solutions for the combined sinh – cosh-Gordon equation, Lec. Mat. (2006) 87 – 93 (special issue).