The stochastic fractional nonlinear Schrödinger equations in $H^α$ and structure-preserving algorithm
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913324470370304 |
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| author | Zhang, Ao Zhang, Yanjie Wang, Pengde Wang, Xiao Duan, Jinqiao |
| author_facet | Zhang, Ao Zhang, Yanjie Wang, Pengde Wang, Xiao Duan, Jinqiao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stochastic fractional nonlinear Schrödinger equations in $H^α$ and structure-preserving algorithm Zhang, Ao Zhang, Yanjie Wang, Pengde Wang, Xiao Duan, Jinqiao Numerical Analysis Analysis of PDEs 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. |
| title | The stochastic fractional nonlinear Schrödinger equations in $H^α$ and structure-preserving algorithm |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2401.15608 |