Davide Proment

Robert-Jones solitary waves on the two-dimensional sphere

Robert–Jones solitary waves, also known as Robert–Jones solitons, are fully nonlinear, localised traveling wave solutions of the Gross–Pitaevskii equation (a defocusing nonlinear Schrödinger equation) in two and three dimensions [1]. At low speeds, these waves appear as vortex dipoles in two dimensions and vortex rings in three dimensions, producing topological phase excitations. As the wave speed approaches a critical threshold, the vortical structure disappears, leaving a localised depletion in the field’s amplitude.

In this work, we investigate the existence and stability of Robert–Jones solitary waves on curved spatial manifolds, focusing on the two-dimensional sphere. We analyse how these structures compare to delocalised Bogoliubov excitations and the classical point vortex model on the sphere [2]. Our results reveal how spatial curvature affects the structure and dynamics of nonlinear excitations in quantum fluids, with implications for recent experiments on ultracold atomic gases confined to spherical shells.

[1] Jones, C. A., & Roberts, P. H. (1982). Motions in a Bose condensate. IV. Axisymmetric solitary waves. Journal of Physics A: Mathematical and General, 15(8), 2599.
[2] Newton, P. K. (2001). The N-vortex problem: analytical techniques. Applied Mathematical Sciences, Springer.