Numerical Validation of the ESMA Framework: Stability, Propagation, and Collision Analysis

This work presents a detailed numerical validation of the ESMA (Extended Schrödinger Model Approach) framework, a nonlinear modification of the Schrödinger equation designed to support self-confined wave packet dynamics. Using the split-step Fourier method (SSFM), the model is analyzed under three validation regimes: stationary confinement, inertial propagation, and collision dynamics. The results demonstrate the emergence of stable localized structures with strong resistance to dispersion and significant structural preservation during interactions. A key limitation associated with global center-of-mass dependence is identified, leading to non-local coupling effects and interaction artifacts. This limitation motivates the development of a local nonlinear extension, which is addressed in a subsequent work (ESMA v1.2). This study establishes the foundational validation and identifies critical structural aspects of the ESMA framework.