Homoclinic bifurcation and chaotic dynamics in Schrödinger equations with higher-order dispersion and cubic–quintic nonlinearity

We investigate the persistence of solitary waves and the emergence of chaotic dynamics in a class of nonlinear Schrödinger equations with high-order dispersion and cubic–quintic nonlinearities subjected to periodic perturbations and dissipation. By applying a traveling-wave reduction together with a novel transformation, the governing equation is reduced to a four-dimensional dynamical system admitting a nontrivial integral of motion, which allows a geometric characterization of its invariant manifolds. For the unperturbed system, phase-space analysis reveals that heteroclinic connections are generic, whereas homoclinic connections arise only under specific parameter constraints and are essential for the persistence of localized structures. For the perturbed non-Hamiltonian system, a multiscale Melnikov method is developed by exploiting the special structure of the motion integral. The first-order Melnikov analysis shows that weak perturbations do not destroy the homoclinic geometry, while the second-order Melnikov function yields explicit criteria for transverse intersections of stable and unstable manifolds, signaling the onset of Smale horseshoe chaos. Numerical simulations, including largest Lyapunov exponent calculations and Poincaré sections diagrams, are in good agreement with the analytical predictions and confirm the transition from stable solitary-wave propagation to chaotic behavior.