Soliton structures, modulational instability, and chaotic dynamics of the coupled Schrödinger–Boussinesq equation

We investigate the coupled Schrödinger–Boussinesq (SB) system, a nonlinear model describing resonant interactions between short- and long-wave components in optics, plasma physics, and fluid mechanics. Using a traveling-wave reduction, we transform the governing PDEs into a canonical nonlinear ODE and derive a broad family of exact solutions, including solitary and singular solitons, finite-background localized states, and Jacobi elliptic periodic waves. We analyze the modulational instability of continuous-wave states, identifying parameter regimes where uniform wave trains destabilize into localized excitations and elucidating the interplay between dispersion, coupling, and nonlinearity. Recasting the reduced dynamics in phase space, we classify equilibria, phase portraits, and connecting orbits, thereby characterizing the conditions for solitary and periodic patterns. With weak external periodic forcing, we apply the Melnikov method to derive explicit thresholds for homoclinic orbit splitting and rigorously predict the onset of chaos. Together, these results establish a unified analytical framework connecting soliton formation, modulational instability, and chaotic dynamics in the SB system, thereby advancing the broader understanding of nonlinear wave phenomena in multiscale physical media.