The stochastic fractional nonlinear Schrödinger equations in H^α and structure-preserving algorithm

2024/01/28 by Zhang, Ao, Ao Zhang, Zhang, Yanjie +6
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2401.15608

openalex publication_date 2024/01/28 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28

Abstract

In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schrödinger equation with radially symmetric initial data in a suitable energy space H^α. We then show that the stochastic fractional nonlinear Schrödinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schrödinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory.

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