The Convergence of Symmetric Discretization Models for Nonlinear Schrödinger Equation in Dark Solitons’ Motion

The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study convergence symmetric discretization models for nonlinear dark solitons’ motion and verify theoretical results through numerical experiments. Via second-order difference, can obtain two popular space-symmetric motion: direct-discrete model Ablowitz–Ladik model. Furthermore, applying midpoint scheme with symmetry to space models, time–space models: Crank–Nicolson method new difference method. Secondly, demonstrate that solutions converge solution equation. Additionally, prove order O(h2+τ2) discrete L2-norm error estimates. Finally, present some experiments show our agree well proven results.