Soliton Dynamics, Chaos, Sensitivity and Energy Analysis to Nonlinear Models

Abstract

The Schwarz-Kortewegde Vries (SKdV) equation, a form of the Kortewegde Vries (KdV) equation, is a partial differential equation modeling wave propagation in shallow water. It plays a fundamental role in understanding nonlinear wave behavior, particularly soli- tonsstable, self-reinforcing waves that maintain shape and speed over long distances due to a balance between nonlinearity and dispersion. Originally applied to shallow water waves, the KdV equation has broader applications, including plasma physics, optical fiber transmission, and traffic flow modeling. A soliton is a stable, self-reinforcing wave that maintains its shape and speed over time, resulting from a balance between nonlinearity and dispersion in a medium. It is observed in systems like water waves, optical fibers, and plasma. This project investigates the Complete Discrimination System for Polynomial Methods (CDSPM), initially proposed by Liu. The trial equation technique further aids in finding solutions such as periodic, singular, kink, dark, and bright solitons by simplifying complex equations. We’ll also study the chaotic and quasi-periodic behavior of the SKdV equa- tion, using time series, phase portraits, and Poincar maps to examine solution geometries and the sensitivity analysis which reveals how small changes in initial conditions affect the system’s dynamics. We’ll also study energy balance analysis (EBA), which provides ap- proximate periodic solutions of nonlinear oscillatory systems. The method is based on the principle of energy conservation, where the total energy, combining both kinetic and po- tential energy remain invariant. Analytical soliton solutions for the SKdV model will also be explored, emphasizing their significance in applications like fiber optics communication and nonlinear optical signal processing. We’ll also study 2D/3D graphs and Poincar maps, providing insights into quasi-periodic dynamics under various initial values. Key words: Complete discriminant system; qualitative analysis; quasi periodic behaviour; sensitivity analysis; solitons.