Novel conformal structure-preserving schemes for the linearly damped nonlinear Schrödinger equation

In this work, by utilizing the Strang splitting technique, some advanced conformal structure-preserving schemes are developed for solving the damped nonlinear Schrödinger equation (DNLSE). The proposed innovative numerical approaches, namely the high-order compact conformal multi-symplectic method and the conformal momentum-preserving method, have two key computational advantages. Firstly, these two approaches excel in conserving local structures, and more importantly, they are capable of maintaining exact energy dissipation rates in any time-space region, particularly under periodic boundary conditions. To validate the theoretical properties and demonstrate the efficacy and stability of these approaches in long-time integrations, we conduct through extensive numerical simulations involving both bright and dark solitons.