Core Concepts
The stochastic fractional nonlinear Schrödinger equation exhibits global existence of solutions in the energy space Hα and possesses a stochastic multi-symplectic structure.
Abstract
The paper investigates the global existence and stochastic symplectic structure of the stochastic fractional nonlinear Schrödinger equation (SFNSE) with multiplicative noise in the energy space Hα.
Key highlights:
The global existence of a unique solution to the SFNSE with radially symmetric initial data in Hα is established under suitable assumptions on the nonlinearity and noise.
It is shown that the SFNSE in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity.
A stochastic midpoint scheme is developed for the SFNSE from the perspective of symplectic geometry, and it is proved that the scheme satisfies the corresponding symplectic law in the discrete sense.
A numerical example is conducted to validate the efficiency of the proposed theory.
The authors first introduce the necessary notations and definitions, including the fractional Laplacian operator and the Hα space. They then prove the local well-posedness of the SFNSE with radially symmetric initial data in Hα using a fixed point argument.
To establish the global existence, the authors derive a priori estimates on the mass and energy of the solution. This involves carefully analyzing the stochastic integral terms and utilizing the fractional chain rule and Gagliardo-Nirenberg inequality. The global existence is then obtained by combining the local well-posedness and the a priori estimates.
Next, the authors demonstrate that the SFNSE in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system. They provide a suitable decomposition of the fractional Laplacian operator and show that the phase flow of the system preserves symplecticity.
Finally, a stochastic midpoint scheme is developed for the SFNSE, and it is proved that the scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is presented to validate the efficiency of the proposed theory.
Stats
The paper does not contain any explicit numerical data or statistics. The focus is on the theoretical analysis of the global existence and stochastic symplectic structure of the stochastic fractional nonlinear Schrödinger equation.
Quotes
The paper does not contain any striking quotes that support the key logics. The content is primarily focused on the mathematical analysis and development of the theoretical framework.