Interaction of Two Pulsatory Waves of the Korteweg-de Vries Equation in a Zigzag Hyperbolic Structure


A new exact solution for nonlinear interaction of two pulsatory waves of the Korteweg-de Vries (KdV) equation is computed by decomposition in an invariant zigzag hyperbolic tangent (ZHT) structure. A computational algorithm is developed by experimental programming with lists of equations and expressions. The structural solution is proved by theoretical programming with symbolic general terms. Convergence, tolerance, and summation of the ZHT structural approximation are discussed. When a reference level vanishes, the two-wave solution is reduced to the two-soliton solution of the KdV equation.

Share and Cite:

Miroshnikov, V. (2014) Interaction of Two Pulsatory Waves of the Korteweg-de Vries Equation in a Zigzag Hyperbolic Structure. American Journal of Computational Mathematics, 4, 254-270. doi: 10.4236/ajcm.2014.43022.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Zabusky, N.J. and Kruskal, M.D. (1965) Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States. Physical Review Letters, 15, 240-243.
[2] Hirota, R. (1971) Exact Solutions of the Korteweg-de Vries Equation for Multiple Solitons. Physical Review Letters, 27, 1192-1194.
[3] Drazin, P.G. (1983) Solitons. In: Reid, M., Ed., London Mathematical Society Lecture Note Series, No. 85, Cambridge University Press, Cambridge, 136.
[4] Izrar, B., Lusseyran, F. and Miroshnikov, V. (1995) Two-Level Solitary Waves as Generalized Solutions of the KdV Equation. Physics of Fluids, 7, 1056-1062.
[5] Varley, E. and Seymour, B.R. (1998) A Simple Derivation of the N-Soliton Solutions to the Korteweg-de Vries Equation. SIAM Journal on Applied Mathematics, 58, 904-911.
[6] Vvedensky, D.D. (1992) Partial Differential Equations with Mathematica. Addison-Wesley Publishing Company, Wokingham.
[7] Miroshnikov, V.A. (1995) Solitary Wave on the Surface of a Shear Stream in Crossed Electric and Magnetic Fields: The Formation of a Single Vortex. Magnetohydrodynamics, 31, 149-165.
[8] Miroshnikov, V.A. (1996) The Finite-Amplitude Solitary Wave on a Stream with Linear Vorticity. European Journal of Mechanics, B/Fluids, 15, 395-411.
[9] Miroshnikov, V.A. (2002) The Boussinesq-Rayleigh Approximation for Rotational Solitary Waves on Shallow Water with Uniform Vorticity. Journal of Fluid Mechanics, 456, 1-32.
[10] Miroshnikovs, V. (1996) Coupled Solitary Waves in Viscous MHD and Geophysical Flows. Comptes Rendu Académie des Sciences Paris, 323, 23-30.
[11] Keller, J.B. (1948) The Solitary Wave and Periodic Waves in Shallow Water. Communications in Pure and Applied Mathematics, 1, 323-339.
[12] Laitone, E.V. (1960) The Second Approximation to Cnoidal and Solitary Waves. Journal of Fluid Mechanics, 9, 430-444.
[13] Grimshaw, R. (1971) The Solitary Wave in Water of Variable Depth. Part 2. Journal of Fluid Mechanics, 46, 611-622.
[14] Longuet-Higgins, M.S. and Fenton, J.D. (1974) On the Mass, Momentum, Energy, and Circulation of a Solitary Wave II. Proceedings of the Royal Society A, 340, 471-493.
[15] Pennell, S.A. and Su, C.H. (1984) A Seventeenth-Order Series Expansion for the Solitary Wave. Journal of Fluid Mechanics, 149, 431-443.
[16] Pennell, S.A. (1987) On a Series Expansion for the Solitary Wave. Journal of Fluid Mechanics, 179, 557-561.
[17] Miroshnikov, G.V. (2011) Hamiltonian Modeling of Pulsar Radiation Profiles. Far East Journal of Dynamical Systems, 17, 33-47.
[18] Miroshnikov, V.A. (2012) Dual Perturbations of the Poiseuille-Hagen Flow in Invariant Elliptic Structures. Advances and Applications in Fluid Dynamics, 11, 1-58.
[19] Schwenke, T. (1979) Sensitive Chaos: The Creation of Flowing Forms in Air and Water. Schocken Books, New York.
[20] Mollison, B. (1988) Permaculture: A Designer’s Manual. Tagari Publications, Tyalgum.
[21] Thompson, D.W. (1992) On Growth and Form. The Complete Revised Edition, Dover Publications, New York.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.