Nonlinear Bohmian Schr¨odinger Equation for Oceanic Waves with Saturable Curvature Effects, Ermakov–Lewis Invariant, Feynman Propagator and Solitonic Solutions

A nonlinear Bohmian Schr¨odinger equation for oceanic wave dynamics is proposed using the Madelung–de Broglie–Bohm hydrodynamic formalism. The probability density is interpreted as the oceanic wave intensity. The model incorporates saturable nonlinear effects, curvature corrections and nonlinear gradient contributions. The associated continuity equation and Euler-type equation are derived. A generalized Ermakov–Lewis invariant is developed together with a Bohmian Feynman propagator. Exact bright soliton solutions are obtained in the weak saturation regime. The model may describe coherent oceanic structures, rogue-wave precursors and nonlinear hydrodynamic wave packets.