Dynamics of the generalized derivative resonant NLS equation: bifurcation, exact solutions, and the transition from periodic to chaotic waves

This work presents a inclusive inspection of the nonlinear dynamics of the generalized derivative resonant nonlinear Schrödinger equation (GD-RNLS), a fundamental model for wave phenomena in dispersive media. We use a tailored transformation to recast the GD-RNLS as a two-dimensional conservative dynamic system, which is equivalent to a Hamiltonian system with a single degree of freedom. Leveraging the Hamiltonian framework, we perform a detailed bifurcation analysis. The corresponding phase portraits are constructed, and we studied the degeneracy of the phase plane trajectories as the bifurcation parameter varies. Within this dynamic systems approach, we derive a family of novel exact solutions, categorizing them as periodic, super-periodic, kink (and anti- kink), and solitary wave solutions; selected solutions are illustrated graphically to clarify their properties. Furthermore, we examine a perturbed variant of the GD-RNLS, incorporating an external periodic forcing term modeled by a Jacobi elliptic function. The influence of this perturbation is explored numerically through two-dimensional (2D) and three- dimensional (3D) phase portraits, as well as the time series. Our simulations reveal a sequence of behavioral transitions: the system's initially periodic state gives way to quasi-periodic dynamics as the forcing frequency increases. Upon a further increase in frequency, chaotic behavior emerges. This transition to chaos is quantitatively confirmed by calculating the largest Lyapunov exponent and qualitatively visualized by the irregular structure of the Poincaré surface of section, both of which underscore the complex, stochastic nature of the system's dynamics under strong forcing.