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. Common properties of these equations are the possession of N-soliton solutions, an infinite number of conservation laws, Bäcklund transformations and so on. An interesting equation which seems to belong to the class of equations is a nonlinear integro-differential equation. This internal wave equation is reduced to the Korteweg–de Vries (KdV) equation in the shallow water limit and to the Benjamin-Ono (BO) equation in the deep water limit. This chapter presents the soliton solutions and shows that they tend to the KdV and BO solitons in their respective limits. It derives a many-body problem associated with the intermediate equation and study its relations to the already known cases. The chapter also remarks upon the Bäcklund transformation for the intermediate equation and an equation generated from its transformation.
