Analytic methods for soliton systems

Abstract

The study of soliton systems continues to be a highly rewarding exercise in nonlinear dynamics, even though it has been almost thirty years since the introduction of the soliton concept by Zabusky & Kruskal. Increasingly sophisticated mathematical concepts are being identified with integrable soliton systems, while newer applications are being made frequently. In this pedagogical review, after introducing solitons and their (2+1)-dimensional generalizations, we give an elementary discussion on the various analytic methods available for investigation of the soliton possessing nonlinear evolution equations. These include the inverse scattering transform method and its generalization, namely the d-bar approach, for solving the Cauchy initial value problem, as well as direct methods for obtaining N-soliton solutions. We also indicate how the Painleve singularity structure analysis is useful for the detection of soliton systems.