A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems

The Inverse scattering transform fourier analysis for nonlinear problems

Dust -acoustic waves in dusty plasmas

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

/pdf/backlund-and-darboux-transformations-geometry-and-modern-4k80wq4g1i.pdf

Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

Korteweg-devries equation and generalizations. VI. methods for exact solution

Propagation of ion acoustic solitary waves of small amplitude.

The self-trapping and the modulational instability of nonlinear hydromagnetic $n$ waves of right-hand polarization in a cold plasma are discussed on the basis of a non-linear dispersive equation, which enables us to directly apply the results obtained in the nonlinear optics.

Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma

: It is shown that under certain approximations the basic system of equations governing the propagation of ion acoustic waves can be reduced to the Kortweg-deVries equation. (Author)

On the propagation of ion acoustic solitary waves of small amplitude.

The self-focusing of the transverse waves (comprising the whistler, Alfvenic and electromagnetic modes) propagating along an applied magnetic field in a plasma of finite temperature is discussed. The method of stretching which has been used by Taniuti and Washimi 5) is applied. It is shown that behaviours of the wave amplitude are described by the nonlinear Schrodinger equation and that the threshold power for the self-focusing becomes very small not only at the special frequency for the whistler wave which was already obtained in the previous paper 5) but also at a frequency for the Alfven wave. The expression of the coefficient of the nonlinear term of the nonlinear Schrodinger equation is revised.

Self-Focusing of Transverse Waves in a Magnetoplasma

It is shown that at frequencies approximately equal to one-half the electron cyclotron frequency, the threshold power for the self-focusing of the whistler mode becomes sufficiently small to be available for laboratory experiments.

Haruichi Washimi

Papers

Propagation of ion acoustic solitary waves of small amplitude.

Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma

On the propagation of ion acoustic solitary waves of small amplitude.

Self-Focusing of Transverse Waves in a Magnetoplasma

Self-focusing of a plasma wave along a magnetic field.