ABSTRACT
This chapter discusses the underlying structure of solitons that is embodied in a mathematical method known as inverse scattering. This technique not only provide with a quantitative description of transient soliton dynamics, it guide in understanding soliton collisions and the re-emergence of solitons after a nonlinear collision. The inverse scattering method will allow to determine the number and amplitude/speed/width of solitons that emerge from an arbitrary initial condition. The chapter describes the mechanics of finding the solution to an initial value problem using the results of the inverse scattering method for some common initial profiles. It examines a familiar math problem that has apparently nothing to do with goal of solving the Korteweg-de Vries (KdV) equation, the linear eigenvalue problem. The chapter describes in depth the transient solution of the KdV equation using the inverse scattering method. It looks at several different methods for examining the unique dynamics of solitons: Hirota's Direct Method, ISM Reflectionless Potential, and ISM Non-reflectionless Potential.
