Nonlinear coupled-mode framework for coupled systems: From exact Hamiltonian models to improved reduced-order formulations

Accurate modeling of coupled nonlinear phenomena is essential for predicting energy transfer in many physical systems; however, conventional coupled-mode theory often relies on approximations that limit its validity under moderate and strong coupling. In this study, a nonlinear coupled-mode framework is developed that systematically incorporates key nonlinear interactions and counter-propagating wave effects, which are typically neglected in traditional formulations. To illustrate the approach, two nonlinear LC circuits containing Josephson junctions are analyzed. Starting from an exact Hamiltonian description, a reduced-order formulation is derived that retains the essential nonlinear and self-coupling contributions, including rapidly oscillating terms. Comparative analysis demonstrates that the conventional coupled-mode model captures only qualitative trends and exhibits significant phase and amplitude errors outside the weak-coupling regime. In contrast, the proposed reduced-order formulation achieves a close quantitative agreement with the exact Hamiltonian dynamics across the weak-to-strong coupling regimes. These findings clarify the limitations of existing phenomenological models and establish a systematic pathway for constructing accurate reduced-order descriptions of coupled nonlinear systems, with implications for circuit dynamics, photonics, and related areas involving energy transfer in complex media.