Computing stationary solutions of the two-dimensional Gross–Pitaevskii equation with deflated continuation

Abstract

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schrödinger equation with a parabolic trap and repulsive interactions. Despite the fact that this system has been studied extensively, we discover a wide variety of previously unknown branches of solutions. We analyze the stability of the newly discovered branches and discuss the bifurcations that relate them to known solutions both in the near linear (Cartesian, as well as polar) and in the highly nonlinear regimes. While deflated continuation is not guaranteed to compute the full bifurcation diagram, this analysis is a potent demonstration that the algorithm can discover new nonlinear states and provide insights into the energy landscape of complex high-dimensional Hamiltonian dynamical systems.

Two infinite families of resonant solutions for the Gross-Pitaevskii equation

“…Stability of dark rings with static ring profiles (i.e., time-independent |Ψ| 2 ) has been extensively studied in the literature on Bose-Einstein condensates [31][32][33]. There are interesting instabilities trigerring pattern formation.…”

Nonlinear waves in an experimentally motivated ring-shaped Bose-Einstein-condensate setup

“…(7), N can be identified with the mass or total number of atoms in the BEC. In what follows we find suitable starting points on a given solution branch and then vary µ using continuation methods to follow the entire branch possibly leading to bifurcations (when two solution branches collide or when new branches emanate from existing ones) as the chemical potential µ is varied [37,58]. For given chemical potential µ, we find stationary nonlinear solutions to Eq.…”

Deflation-based identification of nonlinear excitations of the three-dimensional Gross-Pitaevskii equation

“…Previous applications of deflation to the study of BECs in 2D were combined with continuation in μ, capturing solutions as they bifurcate from known ones [20,21]. This strategy is too expensive in 3D and so a different approach is taken here.…”

“…Some, especially topological ones such as skyrmions, monopoles, and Alice rings [16,17] have been of particular interest since the early exploration of BECs, while others such as knots [18] have been studied more recently. In this manuscript, we apply a powerful numerical technique called deflation [19][20][21] to identify multiple solutions of the 3D NLS/GP equation.…”