Dispersive Wave Coupling and Nonlinear Energy Transfer in the Riemann System: Exact Solitary Structures and Chaotic Regimes

This work provides a comprehensive analytical and computational investigation of the Riemann coupled wave system, which serves as a prototypical model for the propagation and interaction of nonlinear waves in dispersive and dissipative media. A systematic bifurcation analysis is carried out to elucidate the intrinsic nonlinear dynamics of the system, with particular emphasis on the emergence of chaotic regimes and quasi periodic oscillations in the corresponding phase space. Two complementary exact solution methodologies are employed, namely the Khater III approach and a modified Kudryashov scheme. These frameworks enable the construction of explicit traveling wave solutions and persistent coherent wave forms that characterize the dominant dynamical features of the model. To evaluate the accuracy and stability of the derived solutions, He’s variational iteration method is adopted as an independent semi analytical benchmark, allowing for a detailed quantitative comparison between closed form expressions and numerically generated approximations. The Riemann coupled wave system provides a canonical setting for the study of localized wave packet interactions and nonlinear energy exchange in complex continua, with direct implications for fluid dynamics, plasma physics, and acoustic wave propagation. Its structural relationship with classical integrable evolution equations, including the Korteweg de Vries and Kadomtsev Petviashvili hierarchies, places the present analysis within a broader class of coupled nonlinear wave models of current interest in mathematical physics. By integrating symbolic computation with carefully designed numerical experiments, the study demonstrates a high level of agreement between exact analytical solutions and their variational approximations, thereby confirming the robustness and consistency of the proposed methodologies. The coherent wave structures identified herein persist across both analytical and numerical frameworks, underscoring the effectiveness of hybrid analytical computational strategies for advancing the theoretical understanding of nonlinear wave phenomena in applied mathematics and mathematical physics.