Soliton Interactions in Two Dimensions

Publisher Summary This chapter focuses on soliton interactions in two dimensions. The word “soliton” was coined around 1965 by Zabusky and Kruskal to describe solitary wave pulses, which they observed while numerically integrating a nonlinear partial differential equation—the so-called Kortewegde Vries (K-de V) equation. The solitary wave solution of this equation has been known for many years. The derivation of one of the fundamental equations of soliton theory was given by Korteweg and de Vries in their famous paper of 1895. The equation describes the amplitude of long one-dimensional waves on the surface of a fluid. In physical terms, the effects included in the equation are weak, nonlinearity, and dispersion. The extension of this equation to motions in more than one dimension were given by Kadomtsev and Petviashvili, who generalized the dispersion relation to give an extra term in the equation due to the extra dimension. This chapter elaborates Korteweg-de Vries equation and two-soliton interactions. Inverse Scattering theory is described. A discussion on positive dispersion and the Kadomtsev–Petviashvili equation is also presented.

[1]  G. Whitham.  Linear and non linear waves , 1974 .

[2]  A. Scott,et al.  The soliton: A new concept in applied science , 1973 .

[3]  R. Johnson,et al.  A solution of the inverse scattering problem for the Kadomtsev-Petviashvili equation by the method of separation of variables , 1978 .

[4]  John W. Miles,et al.  Resonantly interacting solitary waves , 1977, Journal of Fluid Mechanics.

[5]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[6]  R. Miura.  The Korteweg–deVries Equation: A Survey of Results , 1976 .

[7]  J. Fenton.  A ninth-order solution for the solitary wave , 1972, Journal of Fluid Mechanics.

[8]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[9]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[10]  Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf) , 1972 .

[11]  G. Simons,et al.  Formation of vortex rings in a stratified atmosphere , 1974 .

[12]  R. Hirota.  Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .

[13]  A. Degasperis,et al.  Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation , 1978 .

[14]  Ljudmila A. Bordag,et al.  Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction , 1977 .

[15]  Junkichi Satsuma,et al.  N-Soliton Solution of the Two-Dimensional Korteweg-deVries Equation , 1976 .

[16]  D. J. Kaup,et al.  Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[17]  E. Laitone,et al.  The second approximation to cnoidal and solitary waves , 1960, Journal of Fluid Mechanics.

[18]  John W. Miles,et al.  Obliquely interacting solitary waves , 1977, Journal of Fluid Mechanics.

[19]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[20]  J. Satsuma.  Solitons and Rational Solutions of Nonlinear Evolution Equations (Theory of Nonlinear Waves) , 1978 .

[21]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[22]  N. C. Freeman,et al.  Interpretation of three-soliton interactions in terms of resonant triads , 1978, Journal of Fluid Mechanics.