On the Classical and Deformed Korteweg-de Vries Equation

Abstract

Given the general nonlinear partial differential equations and the importance of the Korteweg-de Vries equation (KdV) in physics, this chapter presents a basic survey of the two-dimensional Korteweg-de Vries model. We begin by examining various symmetries of systems, and then explore the concept of integrability through two different methods: the Hamiltonian formalism and the existence of conserved quantities. By introducing the concept of q-deformation, we construct the corresponding q-deformation integrable model and the integrability of the resulting system is guaranteed by the existence of Lax pairs. We also present the KdV equation in the Moyal space of moments in its noncommutative version, we present the algebraic structure of the system and we study the integrability using the notion of Lax pair.