Nonlinear Klein-Gordon and Sine-Gordon Equations

Abstract

This chapter deals with the theory and applications of nonlinear Klein-Gordon (KG) and Sine-Gordon (SG) equations. Special emphasis is given to various methods of solutions of these equations. The Green function method combined with integral transforms is employed to solve the linear Klein-Gordon equation. The Whitham averaging procedure and the Whitham averaged Lagrangian principle are used to discuss solutions of the nonlinear Klein-Gordon equation. Included are different ways of finding general and particular solutions of the sine-Gordon equation. Special attention is given to solitons, antisolitons, breather solutions and the energy associated with them, interaction of solitons, Bäcklund transformations, similarity solutions, and the inverse scattering method. Significant features of these methods and solutions are described with other ramifications.