BILINEARIZATION METHOD FOR SOLITON EQUATIONS - A NONLINEAR VERSION OF FOURIER'S METHOD.

This paper is a compilation of the present author's recent works on soliton equations. In the first place, it is shown that solutions describing solitons in the background of ripples (i. e. , the generalized soliton solutions) or various soliton equations can be constructed through their bilinear forms. It is then shown that solutions of initial value problems for various soliton equations can be constructed using their generalized soliton solutions. Moreover, it is also clarified that certain types of the Painleve equations can also be solved by the similar technique, where the Painleve equations are the typical examples of equations without movable critical points and describe asymptotic solutions of soliton equations.