Local Nonlinear Extension of the ESMA Framework: Density-Gradient Dynamics and Removal of Global Coupling

This work introduces a local nonlinear extension of the ESMA (Extended Schrödinger Model Approach) framework, addressing limitations identified in its initial formulation. In the original model (ESMA v1.1), the dependence on the global center of mass introduced non local coupling effects, leading to physically inconsistent behavior in multi packet interactions. To resolve this issue, a density gradient based nonlinear term is proposed, resulting in a fully local formulation of the model. The extended framework preserves the ability to generate stable localized wave structures while eliminating global dependencies and improving the physical consistency of interactions. Numerical simulations using the split step Fourier method confirm that the system maintains stability over extended time scales and exhibits high structural fidelity during interactions. This work represents a structural evolution of the ESMA framework toward a fully local nonlinear field description.