On collapse in the nonlinear Schrodinger equation with time dependent nonlinearity. Application to Bose-Einstein condensates

TL;DR

This paper proves that in the nonlinear Schrödinger equation with time-dependent, sign-definite nonlinearity, collapse can occur regardless of oscillation frequency, with increased frequency accelerating collapse, relevant to Bose-Einstein condensates.

Contribution

It rigorously demonstrates collapse conditions in 2D and 3D nonlinear Schrödinger equations with periodic, sign-definite nonlinearity, extending understanding of collapse dynamics.

Findings

Collapse occurs at any oscillation frequency of nonlinearity.

Increasing oscillation frequency accelerates collapse.

A sufficient condition for collapse in 3D is established.

Abstract

It is proven that periodically varying and sign definite nonlinearity in a general case does not prevent collapse in two- and three-dimensional nonlinear Schrodinger equations: at any oscillation frequency of the nonlinearity blowing up solutions exist. Contrary to the results known for a sign alternating nonlinearity, increase of the frequency of oscillations accelerates collapse. The effect is discussed from the viewpoint of scaling arguments. For the three-dimensional case a sufficient condition for existence of collapse is rigorously established. The results are discussed in the context of the meanfield theory of Bose-Einstein condensates with time dependent scattering length.