Rogue Wave Patterns for the Degenerate Three-Wave Resonant Interaction Equations: Spectral Jump and Deep Learning

Three-wave resonant interaction equations can model nonlinear dynamics in many fields, e.g., fluids, optics, and plasma. Rogue waves, i.e., modes algebraically localized in both space and time, are obtained analytically. The aim of this paper is to study degenerate three-wave resonant interaction equations, where two out of the three interacting wave packets have identical group velocities. Physically, degenerate resonance typically occurs for dispersion relation, possessing many branches, e.g., internal waves in a continuously stratified fluid. Here, the Nth-order rogue wave solutions for this dynamical model are presented. Based on these solutions, we examine the effects of the group velocity on the width and structural profiles of the rogue waves. The width of the rogue waves exhibit a linear increase as the group velocity increases, a feature well-correlated with the prediction made using modulation instability. In terms of structural profiles, first-order rogue waves display ‘four-petal’ and ‘eye-shaped’ patterns. Second-order rogue waves can reveal intriguing configurations, e.g., ‘butterfly’ patterns and triplets. To ascertain the robustness of these modes, numerical simulations with random initial conditions were performed. Sequences of localized modes resembling these analytical rogue waves were observed. A spectral jump was observed, with the jump broadening in the case of rogue wave triplets. Furthermore, we predict new rogue waves based on information from two existing ones obtained using the deep learning technique in the context of rogue wave triplets. This predictive model holds potential applications in ocean engineering.