Some mathematical aspects in determining the 3D controlled solutions of the Gross-Pitaevskii equation

TL;DR

This paper explores a mathematical decomposition method for the 3D Gross-Pitaevskii equation, enabling solutions to be constructed from coupled 2D and 1D Schrödinger equations under specific conditions.

Contribution

It introduces the controlling potential method (CPM) that decomposes the 3D GPE into coupled lower-dimensional equations based on a variational principle.

Findings

CPM allows solutions to be expressed as products of transverse and longitudinal components.

The method is based on a variational principle minimizing energy effects.

Conditions for the decomposition are mathematically established.

Abstract

The possibility of the decomposition of the three dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schr\"{o}dinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, solutions of the 3D controlled GPE can be constructed from the solutions of a 2D linear Schr\"{o}dinger equation (transverse component of the GPE) coupled with a 1D nonlinear Schr\"{o}dinger equation (longitudinal component of the GPE). Such a decomposition, called the 'controlling potential method' (CPM), allows one to cast the above solutions in the form of the product of the solutions of the transverse and the longitudinal components of the GPE. The coupling between these two equations is the functional of both the transverse and the longitudinal profiles. The analysis shows that the CPM is based on the variational principle that sets up a condition on the controlling potential well, and whose physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control operation.