Soliton Perturbation

Ji-Huan He, Donghua University

Abstract

Glossary Soliton A soliton is a nonlinear pulse-like wave that can exist in some nonlinear systems. The isolated wave can propagate without dispersing its energy over a large region of space; collision of two solitons leads to unchanged forms, solitons also exhibit particlelike properties. Soliton perturbation theory The soliton perturbation theory is used to study the solitons that are governed by the various nonlinear equations in presence of the perturbation terms. Homotopy perturbation method The homotopy perturbationmethod is a useful tool to the search for solitons without the requirement of presence of small perturbations. In this method, a homotopy is constructed with a homotopy parameter, p. When p D 0, it becomes a nonlinear wave equation such as a KdV equation with a known soliton solution; when p D 1, it turns out to be the original nonlinear equation. To change p from zero to unity, one must only change from a trial soliton to the solved soliton. Variational iteration method The variational iteration method is a new method for obtaining soliton-type solutions of various nonlinear wave equations. The method begins with a soliton-type solution with some unknown parameters which can be determined after few iterations. The iteration formulation is constructed by a general Lagrange multiplier which can be identified optimally via variational theory. Exp-function method The exp-function method is a new method for searching for both soliton-type solutions and periodic solutions of nonlinear systems. The method assumes that the solutions can be expressed in arbitrary forms of the exp-function. Definition of the Subject The soliton is a kind of nonlinear wave. There are many equations ofmathematical physics which have solutions of the soliton type. The first observation of this kind of wave was made in 1834 by John Scott Russell [1]. In 1895, the famous KdV equation, which possesses soliton solutions, was obtained by D. J. Korteweg and H. de Vries [2], who established a mathematical basis for the study of various solitary phenomena. From a modern perspective, the soliton is used as a constructive element to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, from tornados to the Great Red Spot of Jupiter, from traffic flow to the Internet, from Tsunamis to turbulence [3]. More recently, solitary waves are of key importance in the quantum fields: on extremely small scales and at very high observational resolution equivalent to a very high energy, space–time resembles a stormy ocean and particles and their interactions have soliton-type solutions [4].