Mathematical Prerequisites

Abstract

Before entering in the field of nonlinear waves on closed contours and surfaces we need to recall some useful mathematical concepts. The cnoidal waves, solitary waves, and solitons are solutions of nonlinear equations that could be partial differential (PDE), integro-differential, finite difference-differential, or even functional equations. They describe the evolution of the wave solutions in space and time. These nonlinear equations are usually coupled with linear or nonlinear boundary conditions (BC), initial conditions, or asymptotic conditions. The properties of solutions are dependent on the topological and geometrical structure of the space on which they are defined. In the following we assume for the reader to be familiar with the general concept of group, Abelian group, quotient group, rank of a group, and group homomorphism.