Solitonic Solutions of the One-Dimensional Harmonically Trapped Repulsive Bose-Einstein Condensate via Neural Network Quantum States

We demonstrate the existence of bright solitons in a repulsively interacting, harmonically trapped quasi-one-dimensional Bose-Einstein condensate described by the Gross-Pitaevskii equation. Using a neural-network quantum state (NNQS) approach, we parametrize the initial wavefunction and optimize it to find solutions that recur after one trap period, effectively balancing repulsion with trap-induced attraction. Aside from the bright solitonic solution, we also report double bright and dark soliton states. Perturbing the initial state with multiplicative phase and amplitude noise confirms that these periodic orbits are orbitally stable. Our results indicate that NNQS provides a powerful framework for uncovering coherent structures in nonlinear wave systems.