Unveiling Solitary Waves and Chaotic Dynamics in the Non-Standard Nonlinear Schr\"{o}dinger Equation via Advanced Analytical Techniques

In this paper, we study the dynamical behaviour and exact travelling wave solutions of a non-standard nonlinear Schrödinger equation. By using two powerful and complementary analytical techniques, the [Formula: see text]-expansion method and the Newly Improved Kudryashov method, a wide family of exact solutions are constructed. From a dynamical systems perspective the equation is further examined to get deeper insight into the underlying physical mechanisms. A planar dynamical system is derived and its qualitative behaviour is analyzed using phase portrait analysis, time-series analysis, Poincare section and three-dimensional phase-space trajectories. We investigate the bifurcation structure of the state variable [Formula: see text] under small perturbations and perform a bifurcation analysis to demonstrate the transition between periodic, quasi-periodic and chaotic regimes as the system parameters are varied. The obtained bifurcation diagrams confirm the sensitivity of the system to the variation of parameters and exhibit the coexistence of stable and unstable dynamical structures. Graphical simulations are provided to visualize the physical meaning of the obtained solutions. The results of this work can be useful for further understanding the phenomena of wave propagation in oceans, plasma physics, and other areas of mathematical physics.