A nonlinear Landau-Zener formula

TL;DR

This paper introduces a nonlinear extension of the Landau-Zener model relevant to Bose-Einstein Condensation, analyzing its large-time behavior and establishing a nonlinear scattering operator similar to long-range scattering in nonlinear Schrödinger equations.

Contribution

It develops a nonlinear Landau-Zener formula, demonstrating the existence of a nonlinear scattering operator and connecting it to physical models in Bose-Einstein Condensation.

Findings

Existence of a nonlinear scattering operator

Comparison with linear Landau-Zener model

Application to Bose-Einstein Condensation

Abstract

We consider a system of two coupled ordinary differential equations which appears as an envelope equation in Bose-Einstein Condensation. This system can be viewed as a nonlinear extension of the celebrated model introduced by Landau and Zener. We show how the nonlinear system may appear from different physical models. We focus our attention on the large time behavior of the solution. We show the existence of a nonlinear scattering operator, which is reminiscent of long range scattering for the nonlinear Schrodinger equation, and which can be compared with its linear counterpart.