ABSTRACT
One of the recent advances in the study of nonlinear partial differential equations is the development of analytical tools for solving a certain new class of such equations. In 1967, Gardner, Greene, Kruskal, and Miura [1] discovered a method of solving the Korteweg-deVries (KdV) equation, which describes weakly nonlinear and dispersive one-dimensional systems (e.g., shallow water waves, ion acoustic waves in a plasma, etc.). The method, now referred to as the inverse scattering transform (IST), is very powerful. After this discovery, it was shown that the method is also applicable to many other physically important one-dimensional (e.g., nonlinear Schrödinger, Sine-Gordon, modified KdV), differential difference, partial difference, and multi-dimensional equations [2]. Common properties of these equations are the possession of N-soliton solutions, an infinite number of conservation laws, Bäcklund transformations, etc. (Moreover, it has been recently revealed that there is a deep connection between nonlinear partial differential equations solvable by IST and nonlinear ordinary differential equations without movable critical points [3].)
