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

TL;DR

This paper uses deflated continuation to find new steady-state solutions of the 2D Gross-Pitaevskii equation, revealing previously unknown branches and analyzing their stability and bifurcations.

Contribution

It demonstrates the effectiveness of deflated continuation in discovering new nonlinear solutions in high-dimensional Hamiltonian systems.

Findings

Discovery of new solution branches in the 2D Gross-Pitaevskii equation.

Analysis of stability and bifurcation structure of solutions.

Validation of deflated continuation as a tool for complex bifurcation analysis.

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 Schr\"odinger 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.