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.
Charalampidis, E. G., Kevrekidis, P. G., & Farrell, P. E. (2018). Computing stationary solutions of the two-dimensional Gross–Pitaevskii equation with deflated continuation. Communications in Nonlinear Science and Numerical Simulation, 54, 482–499. https://doi.org/10.1016/j.cnsns.2017.05.024