The method of inverse scattering and Bäcklund transformations

Solving integrable nonlinear partial differential equations by the method of inverse scattering and Bäcklund transformations are very effective. Both of these methods depend on the existence of Lax pair. The Bäcklund transformations provide us some particular solutions of the integrable nonlinear partial differential equations but the method of inverse scattering solves the initial value problem of these equations. In this work, we use these methods to solve the Korteweg-de Vries equation, the Korteweg-de Vries hierarchy and AKNS system.