Solution of Initial Boundary Value Problems for the Nonlinear Stationary Quasi-Optical Equation with Special Gradient Term

Abstract

This study examines a fundamental problem in quantum mechanics: the initial boundary value problem for a stationary, nonlinear quasi-optical equation with a specific gradient. This complex equation, which surpasses the previously studied nonstationary linear Schrödinger equation, offers more accurate models of particle behavior at the microscopic level. The research defines the problem as a first-kind problem, then meticulously formulates the necessary and sufficient conditions for its solution. Theorems on the existence and uniqueness of the solution are proven, and an estimate of the solution is obtained. This work is significant for quantum mechanics and optics, particularly in understanding electromagnetic wave propagation when wavelengths are comparable to the optical system’s dimensions. The findings contribute to optical design and engineering applications, offering new insights beyond classical physics.