Collapse in the nonlocal nonlinear Schr\"odinger equation

TL;DR

This paper investigates the conditions under which collapse occurs in the nonlocal nonlinear Schrödinger equation, showing that nonlocality can prevent collapse in various dimensions and kernel types, with implications for Bose-Einstein condensates.

Contribution

It provides a comprehensive analysis of collapse scenarios in nonlocal nonlinear Schrödinger equations, including proofs for non-occurrence of collapse in certain cases and numerical studies for Bose-Einstein condensates.

Findings

Collapse does not occur for nonsingular attractive nonlocal interactions.

Collapse is prevented for nonlocal kernels with lpha<2 in dimensions n.

Adding local repulsive interactions can prevent collapse in Bose-Einstein condensates.

Abstract

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoff's method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $n\geq2$ and singular kernel $\sim 1/r^\alpha$, no collapse takes place if $\alpha<2$, whereas collapse is possible if $\alpha\ge2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $\sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases.