Solitary waves in Bose-Einstein condensates — dark solitons, vortex rings, solitonic vortices — are usually studied in specific parameter regimes: the Thomas-Fermi limit (strong interactions, where the condensate is large and the kinetic energy negligible) or the linear limit (weak interactions, where the condensate behaves almost like a noninteracting gas). These regimes are treated as fundamentally different, with different approximation methods and different intuitions about which structures are possible.
Wang (arXiv:2603.17996) shows they are connected. Starting from exact solutions in the linear limit of a two-dimensional BEC with anisotropic harmonic trapping, solitary wave patterns can be continuously tracked — numerically, with full control — through the Thomas-Fermi regime and even into isotropic traps. The continuation is smooth: no bifurcations destroy the patterns, no phase transitions intervene. The solitary wave in the weakly interacting gas evolves continuously into a solitary wave in the strongly interacting condensate.
The method works by treating the interaction strength as a continuation parameter. At zero interaction, the BEC is a quantum harmonic oscillator with known eigenstates. Small interactions perturb these states into nonlinear solitary waves. Increasing the interaction further, the solver tracks how each wave's shape, energy, and stability evolve. The anisotropy of the trap matters because it lifts degeneracies that would otherwise cause bifurcations — the asymmetry acts as a regularizer, keeping each branch of solutions distinct and trackable.
Many wave patterns discovered near the linear regime survive into the Thomas-Fermi regime. The catalog of solitary waves in strongly interacting condensates is larger than previously known, because the continuation from the linear limit reveals structures that had no obvious Thomas-Fermi construction. The physics doesn't change at a boundary — it changes continuously, and the mathematical continuation follows it.