Conservative Dissipative Bohm–Schrödinger Equation: A Hydrodynamic Nonlinear Model with Solitonic Oceanic Interpretation

In this work we propose a nonlinear dissipative Schrödinger equation preserving both the continuity equation and the Euler equation within the de Broglie–Bohm hydrodynamic formalism. The model introduces a structural dissipative term acting on the density gradients without violating global probability conservation. We derive the corresponding Madelung equations, analyze solitonic solutions, obtain a dissipative dispersion relation, and discuss an effective hydrodynamic interpretation for oceanic solitary waves. The equation naturally reduces to nonlinear Schrödingertype and Korteweg–de Vries-type dissipative equations under suitable approximations.