Abstract
Integrability serves as a cornerstone in understanding a wide range of dynamical systems, from classical mechanics to quantum field theories. This work begins by exploring Liouville integrability in classical systems, which guarantees the existence of sufficient conserved quantities for exact solvability. Central to this framework is the Lax representation, where the evolution of a system is encapsulated in a pair of matrices whose compatibility condition encodes the equations of motion. The introduction of the r-matrix formalism reveals an elegant algebraic structure that links integrability to the symmetries of the system. Arising naturally in this context, the Yang-Baxter equation provides a fundamental criterion for constructing integrable systems and lays the groundwork for their quantization. We then extend these ideas to field theories, focusing on the Classical Nonlinear Schrödinger equation, a paradigmatic integrable system in 1+1 dimensions. By treating the NLS equation as a field-theoretic counterpart to finite-dimensional integrable systems, we explore its Lax representation and the corresponding r-matrix formalism. The inverse scattering transform emerges as a powerful method for solving the NLS equation, drawing parallels with classical integrability concepts. This journey—from classical oscillatory systems to field theories—highlights the unifying principles of integrability and underscores the relevance of these mathematical structures in solving nonlinear dynamical problems.