×

zbMATH — the first resource for mathematics

Traveling wave solutions of the generalized nonlinear evolution equations. (English) Zbl 1170.35514
Summary: Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C05 Solutions to PDEs in closed form
PDF BibTeX Cite
Full Text: DOI
References:
[1] Korteweg, D.J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. mag., 39, 422-443, (1895) · JFM 26.0881.02
[2] Zabuski, N.J.; Kruskal, M.D., Integration of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. rev. lett., 15, 240-242, (1965) · Zbl 1201.35174
[3] Gardner, C.S.; Green, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the korteweg – de Vries equations, Phys. rev. lett., 19, 1095-1097, (1967) · Zbl 1061.35520
[4] Lax, P.D., Integrals of nonlinear equations of evolution and solitary waves, Commun. pure appl. math., 21, 467-490, (1968) · Zbl 0162.41103
[5] Ablowitz, M.J.; Clarkson, P.A., Solitons nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press · Zbl 0762.35001
[6] Kudryashov, N.A., Analytical theory of nonlinear differential equations, (2004), Institute of Computer Investigations Moskow - Igevsk, (in Russian)
[7] Kuramoto, Y.; Tsuzuki, T., Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. theor. phys., 52, 2, 356-369, (1976)
[8] Benney, D.J., Long waves on liquid films, J. math. phys., 45, 150, (1966) · Zbl 0148.23003
[9] Topper, J.; Kawahara, T., Approximate equation for long nonlinear waves on a viscous fluid, J. phys. soc. jpn., 44, 663-666, (1978) · Zbl 1334.76054
[10] Shkadov, V.Y., Solitary waves in layer of viscous fluid, Fluid dynam., 1, 63-66, (1977)
[11] Cohen, B.I.; Krommers, J.A.; Tang, W.M.; Rosenbluth, M.N., Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nucl. fusion, 16, 6, 971-992, (1976)
[12] Michelson, D., Elementary particles as solutions of the Sivashinsky equation, Physica D, 44, 502-556, (1990) · Zbl 0702.34007
[13] Lu, D.; Hong, B.; Tian, L., New solitary wave and periodic wave solutions for general types of KdV and kdv – burgers equations, Commun. nonlinear sci. numer. simul., 14, 77-84, (2009) · Zbl 1221.35352
[14] Peng, Y.Z., Exact travelling wave solutions for the zakharov – kuznetsov equation, Appl. math. comput., 199, 397-405, (2008) · Zbl 1138.76026
[15] Zhu, Z.N., Exact solutions to two-dimensional generalized fifth order kuramoto – sivashinsky type equation, Chin. J. phys., 84, 2, 85-90, (1996)
[16] Qin, M.; Fan, G., An effective method for finding special solutions of nonlinear differential equations with variable coefficients, Phys. lett. A, 372, 3240-3242, (2008) · Zbl 1220.35099
[17] Kudryashov, N.A., Solitary and periodic solutions of the generalized kuramoto – sivashinsky equation, Regul. chaotic dynam., 13, 3, 234-238, (2008) · Zbl 1229.35229
[18] Kudryashov, N.A., Exact soliton solutions of the generalized evolution equation of wave dynamics, J. appl. math. mech., 52, 3, 360-365, (1988)
[19] Conte, R.; Museette, M., Painleve analysis and backlund transformation in the kuramoto – sivashinsky equation, J. phys. A: math. gen., 22, 169-177, (1989) · Zbl 0687.35087
[20] Kudryashov, N.A., Exact solutions of the generalized kuramoto – sivashinsky equation, Phys. lett. A, 147, 287-291, (1990)
[21] Kudryashov, N.A., On types of nonlinear nonintegrable equations with exact solutions, Phys. lett. A, 155, 269-275, (1991)
[22] Kudryashov, N.A.; Zargaryan, E.D., Solitary waves in active – dissipative dispersive media, J. phys. A: math. gen., 29, 8067-8077, (1996) · Zbl 0901.35090
[23] Berloff, N.G.; Howard, L.N., Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. appl. math., 99, 1-24, (1997) · Zbl 0880.35105
[24] Davalos-Orozco, L.A.; Busse, F.N., Instability of a thin film flowing on a rotating horizontal or inclined plane, Phys. rev. E, 65, 026312-2-026312-10, (2002)
[25] Fu, S.; Liu, S.; Liu, S., New exact solutions to the kdv – burgers – kuramoto equation, Chaos solitons fract., 23, 609-616, (2005) · Zbl 1069.35069
[26] Zhang, S., New exact solutions of the kdv – burgers – kuramoto equation, Phys. lett. A, 358, 414-420, (2006) · Zbl 1142.35592
[27] Khuri, S.A., Traveling wave solutions for nonlinear differential equations: a unified ansatse approach, Chaos solitons fract., 32, 252-258, (2007) · Zbl 1137.35422
[28] Nickel, J., Traveling wave solutions to the kuramoto – sivashinsky equation, Chaos solitons fract., 33, 1376-1382, (2007) · Zbl 1137.35063
[29] Kudryashov, N.A., Exact solitary waves of the Fisher equation, Phys. lett. A, 342, 99-106, (2005) · Zbl 1222.35054
[30] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos solitons fract., 24, 1217-1231, (2005) · Zbl 1069.35018
[31] Kudryashov, N.A.; Demina, M.V., Polygons of differential equation for finding exact solutions, Chaos solitons fract., 33, 1480-1496, (2007) · Zbl 1133.35084
[32] Kudryashov, N.A.; Loguinova, N.B., Extended simplest equation method for nonlinear differential equations, Appl. math. comput., 205, 1, 396-402, (2008) · Zbl 1168.34003
[33] Kudryashov, N.A.; Loguinova, N.B., Be careful with exp-function method, Commun. nonlinear sci. numer. simul., (2008) · Zbl 1221.35344
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.