Chapters authored
On the Classical and Deformed Korteweg-de Vries Equation
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.
Part of the book: Optimization Algorithms
Topological String Theory: A Source for Conformal Field Theories
In this paper, we first recall the approach to extract Liouville field theories from the action of worldsheets superstring in the standard case and also in the presence of gauge fields. In order to extend this formalism, we treat the case of Toda field equations based on a special ansatz of our superfields. Then, we study the integrability of the system by the Lax formulation based on the structure of the associated Lie superalgebra.
Part of the book: Recent Topics on Topology - From Classical to Modern Applications [Working title]