DYNAMICS OF INTERACTING ALGEBRAIC SOLITONS

Abstract

A survey is made which highlights recent topics on the dynamics of algebraic solitons, which are exact solutions to a certain class of nonlinear integrodifferential evolution equations. The model equations that we consider here are the Benjamin-Ono (BO) and its higher-order equations together with the BO-Burgers equation, a model equation for deep-water waves, the sine-Hilbert (sH) equation and a damped sH equation. While these equations have their origin either in physics or in mathematics, each equation exhibits a novel type of algebraic soliton solution and hence its characteristic is worth studying in its own right. After deriving these equations, we are concerned with each equation separately. We first present explicit N-soliton solutions and then summarize related mathematical properties of the equation. Subsequently, a detailed description is given to the interaction process of two algebraic solitons using the pole expansion of the solution. Particular attention is paid to investigating the effects of small perturbations on the overtaking collision of two BO solitons by employing a direct multisoliton perturbation theory. It is shown that the dynamics of interacting algebraic solitons reveal new aspects which have never been observed in the interaction process of usual solitons expressed in terms of exponential functions.