Integrable systems and nonlinear evolution equations are essential for modelling dynamic systems and physical phenomena across various disciplines, including physics, engineering, and quantum mechanics. These systems are characterized by infinite conserved quantities and soliton solutions, which provide deep insights into stability and wave interactions. Ordinary and partial differential equations (ODEs and PDEs) serve as foundational tools for describing these phenomena. Analytical and numerical methods play a pivotal role in solving nonlinear equations, offering both exact and approximate solutions. Qualitative analysis provides a comprehensive examination of equations, their properties, and solution behaviours, exploring the stability, equilibrium, and global dynamics of systems governed by ODEs and PDEs, with a focus on asymptotic trends and bifurcations. By studying geometric and topological features, it predicts system behaviors and identifies critical patterns, such as chaos and wave interactions. Numerical simulations further enhance understanding by exploring soliton stability, wave collisions, and the intricate dynamics of these systems.