Resumen de Qualitative Dynamics and Homoclinic Chaos in a Stochastic Klein–Gordon–Schrödinger System
Ahmed H. Arnous
We investigate the qualitative dynamics of a Klein–Gordon–Schrödinger system subject to multiplicative Stratonovich noise, which models random perturbations in the environment or medium. A gauge transformation reduces the original stochastic partial differential equations to a deterministic mean-field system that preserves the essential nonlinear wave–field coupling. Using Lie group symmetry methods, we derive exact reductions to travelling-wave solutions and characterize the full family of solitary and periodic profiles through phase-plane and Hamiltonian analyses. The complete discriminant system method provides an explicit classification of the solution spectrum in terms of the discriminant structure of the governing cubic polynomial. To reveal how resonant perturbations break integrability, we apply the classical Melnikov method and demonstrate the existence of transverse homoclinic intersections, implying the emergence of chaotic dynamics. Numerical phase portraits and time series illustrate the persistence, modulation, and loss of coherence of localized wave structures under increasing stochastic influence and periodic forcing. The results clarify the interplay between dispersion, nonlinearity, and randomness in shaping the qualitative behavior of nonlinear dispersive systems.
