  As a first approximation beyond linearity, the nonlinear Schr\"odinger
equation (NLSE) reliably describes a broad class of physical systems. Though
numerical solutions of this model are well-established, these methods can be
computationally complex. In this paper, we showcase a code development
approach, demonstrating how computational time can be significantly reduced
with readily available graphics processing unit (GPU) hardware and a
straightforward code migration using open-source libraries. This process shows
how CPU computations with power-law scaling in computation time with grid size
can be made linear using GPUs. As a specific case study, we investigate the
Gross-Pitaevskii equation, a specific version of the nonlinear Schr\"odinger
model, as it describes in two dimensions a trapped, interacting, two-component
Bose-Einstein condensate (BEC) subject to a spatially dependent interspin
coupling, resulting in an analog to a spin-Hall system. This computational
approach lets us probe high-resolution spatial features - revealing an
interaction-dependent phase transition - all in a reasonable amount of time.
Our computational approach is particularly relevant for research groups looking
to easily accelerate straightforward numerical simulation of physical
phenomena.
