zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 35Q53 KdV-like (Korteweg-de Vries) equations 35C07 Traveling wave solutions of PDE 35A24 Methods 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 $\varphi$xxx+6q$\varphi$x+6r$\varphi =\lambda \varphi$, 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).