New exact solutions for the KdV equation with higher order nonlinearity by using the variational method. (English) Zbl 1236.35156

Summary: The Korteweg-de Vries (KdV) equation with higher order nonlinearity models the wave propagation in one-dimensional nonlinear lattice. A higher-order extension of the familiar KdV equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface. The variational approximation method is applied to obtain the solutions for the well-known KdV equation. Explicit solutions are presented and compared with the exact solutions. Very good agreement is achieved, demonstrating the high efficiency of variational approximation method. The existence of a Lagrangian and the invariant variational principle for the higher order KdV equation are discussed. The simplest version of the variational approximation, based on trial functions with two free parameters is demonstrated. The jost functions by quadratic, cubic and fourth order polynomials are approximated. Also, we choose the trial jost functions in the form of exponential and sinh solutions. All solutions are exact and stable, and have applications in physics.


35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
35A15 Variational methods applied to PDEs
Full Text: DOI


[1] Jin, L., Application of variational iteration method to the fifth-order KdV equation, Int. J. Contemp. Math. Sci., 3, 213-221 (2008) · Zbl 1146.35079
[2] Baldwin, D.; Goktas, U.; Hereman, W.; Hong, L.; Martino, R. S.; Miller, J. C., Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbolic Comput., 37, 669-705 (2004) · Zbl 1137.35324
[3] Goktas, U.; Hereman, W., Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbolic Comput., 24, 591-621 (1997) · Zbl 0891.65129
[4] Hereman, W.; Nuseir, A., Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulation, 43, 13-27 (1997) · Zbl 0866.65063
[5] Ito, M., An extension of nonlinear evolution equations of the KdV (mKdV) type to higher orders, J. Phys. Soc. Japan, 49, 771-778 (1980) · Zbl 1334.35282
[6] Khater, A. H.; Hassanb, M. M.; Temsaha, R. S., Cnoidal wave solutions for a class of fifth-order KdV equations, Math. Comput. Simulation, 70, 221-226 (2005) · Zbl 1125.35403
[7] Kupershmidt, B. A., A super KdV equation: an integrable system, Phys. Lett. A, 102, 213-215 (1984)
[8] Kawahara, T., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, 260-264 (1972)
[9] Long, R. R., Solitary waves in one- and two-fluids systems, Tellus, 8, 460-471 (1956)
[10] Benney, D. J., Long nonlinear waves in fluid flows, J. Math. Phys., 45, 52-63 (1966) · Zbl 0151.42501
[11] Lee, C.; Beardsley, R. C., The generation of long nonlinear internal waves in a weakly stratified shear flow, J. Geophys. Res., 79, 453-462 (1974)
[12] Kakutani, T.; Yamasaki, N., Solitary waves in a two-layer fluid, J. Phys. Soc. Japan, 45, 674-679 (1978)
[13] Koop, C. G.; Butler, G., An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., 112, 225-251 (1981) · Zbl 0479.76036
[14] Gear, J. A.; Grimshaw, R., A second-order theory for solitary waves in shallow fluids, Phys. Fluids, 26, 14-29 (1983) · Zbl 0508.76035
[15] Drazin, P. G.; Johnson, R. S., Solutions: An Introduction (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0661.35001
[16] Karpman, V. I., Lyapunov approach to the soliton stability in highly dispersive systems. II. KdV-type equations, Phys. Lett. A, 2, 257-259 (1996) · Zbl 0972.35518
[17] Liu, X. Q.; Bai, C. L., Exact solutions of some fifth-order nonlinear equations, Appl. Math. Scr. B (Engl. Ed.), 15, 28-32 (2000) · Zbl 0954.35143
[18] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98, 288-300 (1996) · Zbl 0948.76595
[19] Wazwaz, A. M., Compactons and solitary patterns solutions to fifth-order KdV-like equations, Physica A, 371, 273-279 (2006)
[20] Wazwaz, A. M., Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182, 283-300 (2006) · Zbl 1107.65092
[21] Biswas, A.; Konar, S., Soliton perturbation theory for the compound KdV equation, Internat. J. Theoret. Phys., 46, 237-243 (2007) · Zbl 1147.35346
[22] Biswas, A.; Konar, S.; Zerrad, E., Soliton perturbation theory for the generalized modified Degasperis-Procesi Camassa-Holm equation, Int. J. Mod. Math., 2, 35-40 (2007) · Zbl 1141.35441
[23] Biswas, A.; Zerrad, E., Soliton perturbation theory for the generalized fifth-order nonlinear equation, Contemp. Eng. Sci., 1, 63-69 (2008) · Zbl 1156.35454
[24] Kawamoto, S., Solitary wave solutions of the Korteweg-de Vries equation with higher order nonlinearity, J. Phys. Soc. Japan, 53, 3729-3731 (1984)
[25] Khater, A. H.; Moussa, M. H.; Abdul-Aziz, S. F., Invariant variational principles and conservation laws for some nonlinear partial differential equations with variable coefficients part II, Chaos Solitons Fractals, 15, 1-13 (2003) · Zbl 1032.35018
[26] Tonti, E., Variational formulations of nonlinear differential equations I, Acad. Roy. Belg. Bull. Cl. Sci., 55, 137-165 (1969) · Zbl 0182.11402
[27] Tonti, E., Variational formulations of nonlinear differential equations II, Acad. Roy. Belg. Bull. Cl. Sci., 55, 262-278 (1969) · Zbl 0186.14301
[28] Kaup, D. J.; Malomed, B. A., Variational principle for the Zakharov-Shabat equations, Physica D, 84, 319-338 (1995) · Zbl 0900.35390
[29] Khater, A. H.; Callebaut, D. K.; Helal, M. A.; Seadawy, A. R., Variational principle for the nonlinear dynamics of an elliptic magnetic stagnation line, Eur. Phys. J. D, 39, 237-245 (2006)
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.