Bilinearization algorithm and KdV- type equations

Abstract

In this paper , a modification to the basic steps of the bilinearization of evolution equations by Hirota’s method is presented by writing the dependent variable u(x,t) and some of its derivatives almost as Hirota polynomials . Then , a definition to modify Peterson definition for the class of KdV-type equations is presented in addition the general work is illustrated by applications to linear equations and to three classes of the nonlinear well-known equations . (1) Introduction :- In this work a nonlinear partial differential equation and one of the most analytic methods for solving such types of equations is considered . The approach is called Hirota’s method for solving equation which has soliton solution .The soliton was coined to describe a pulse like nonlinear wave which emerges from a collision with a similar pulse having unchanged shape and speed and it was found by Scott-Russell on 1834 [15] .The fundamental idea in Hirota’s formalism is to use some dependent variable transformations to put the equation in a form where the unknown function appears bilinearly .In 1976 , the bilinear formalism was introduced by R.Hirota . Hirota’s formalism is a very powerful method [14] of simplifying algebraic calculations but one would not do it justice by looking at it only as a purely formal tool for the manipulation of complicated expressions . It also gives a great deal of insight into many problems [14] The basic symbol D of the formalism is defined