Nonlinear time-dependent Schrödinger equations: the Gross-Pitaevskii equation with double-well potential (Q1762619)

The Gross-Pitaevskii equation (GPE) is a nonlinear Schrödinger equation with cubic nonlinearity, which also includes a linear term with a given external potential. The GPE has recently gained great popularity and significance as a fundamental model for the description of Bose-Einstein condensation. A case quite important to the latter application is a situation when the external potential consists of two separated deep and sufficiently narrow identical potential wells. In this case, a natural conjecture is, assuming that the nonlinear term is weak, to approximate the wave function as a superposition of two wave functions of the corresponding linear Schrödinger equation, each being a ground state of the deep potential well, taken in isolation. The objective is to derive a system of two ordinary differential equations (ODE) for the complex amplitudes in front of the two terms. This approach being already developed before, the objective of the present paper is to develop a rigorous justification of the approximation. Using a priori estimates for solutions of the GPE in the one- and three-dimensional cases, the authors prove theorems about the proximity of the approximation to an exact solution, under certain assumptions concerning the shape of the potential wells. They also consider a special situation when the weak nonlinearity is of the same order of magnitude as a small splitting between the two energy levels of the corresponding linear Schrödinger equation, produced by the weak overlap between the wells. In the latter case, the issue is an estimate for the time during which linear beatings between the two basic ground-state wave functions will persist. As concerns the ODE approximation, it is an integrable system of two first-order linearly coupled equations for the complex amplitudes, with cubic nonlinear terms. An explicit general solutions for the ODE system is given, in terms of periodic elliptic functions.