A Symplectic Structure-Preserving Dynamical Low-Rank Method for the Stochastic Gross-Pitaevskii Equation

This paper presents an innovative Symplectic Dynamical Low-Rank Approximation (S-DLRA) technique for the stochastic Gross-Pitaevskii equation characterized by parametric uncertainty. The suggested method deals with the two problems of high stochastic dimensionality and keeping the geometric structure in computational quantum dynamics. The scheme keeps the basic geometric properties of the underlying Hamiltonian system by using a fourth-order compact spatial discretization and a symplectic time-stepping strategy based on Hamiltonian splitting. The main new idea is a symplectic projection operator that keeps the structure of low-rank manifolds intact, which allows for a big reduction in dimensionality without losing physical consistency. Theoretical analysis demonstrates the preservation of discrete symplecticity and the exact conservation of mass in the context of expected value. Numerical experiments show that fourth-order convergence works in both space and time, that it is much faster than Monte Carlo methods, and that it keeps structures strong during long simulations and nonlinear soliton interactions when there is uncertainty.