On the Existence and Approximation of Solutions to a Nonlinear Schrodinger-Type Equation

We investigate the mathematical structure and approximate solutions of a nonlinear equation arising in quantum mechanics. Specifically, we study a stationary nonlinear Schr\"odinger equation with a cubic nonlinearity, which is relevant in models of Bose--Einstein condensates and nonlinear optics. We establish existence results under suitable boundary conditions and develop a perturbative approximation scheme for small coupling. Numerical experiments illustrate the validity of the approximation and the emergence of localized states.

کلیدواژه ها:

Bose-Einstein condensates ، Perturbative expansion ، Harmonic oscillator potential