Exact solution for nonlinear deep-water waves

In this paper, we developed a generalized analytical framework for nonlinear deep-water waves by incorporating time-varying background flows $(s_1(t),s_2(t))$, effectively extending the classical Gerstner wave theory to unsteady environments. Within the framework of nonlinear partial differential equations, exact solutions for particle trajectories, velocity fields, and vorticity were rigorously derived using a time-varying Lagrangian mapping, while volume conservation throughout the evolutionary process was ensured via Jacobian analysis. A self-consistent pressure function was constructed to satisfy the nonlinear dynamic boundary conditions at the free surface. The core contribution of this study lies in the derivation of a modulated dispersion relation, which reveals how the coupling mechanism between wave dynamics and background acceleration induces an effective gravity shift, thereby explicitly regulating the phase velocity. By integrating these time-varying background flow terms, this work provides a robust theoretical foundation for characterizing transient wave-current interactions and nonlinear evolutionary processes in complex, non-steady ocean systems.