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

TL;DR

This work studies the stochastic fractional nonlinear Schrödinger equation in the energy space $H^{\alpha}$ with multiplicative Stratonovich noise and a fractional Laplacian $(-\Delta)^{\alpha}$, establishing global well-posedness for radially symmetric initial data and revealing an infinite-dimensional stochastic Hamiltonian, stochastic multi-symplectic structure. It develops structure-preserving numerical schemes, including a stochastic midpoint method and generalized stochastic multi-symplectic discretizations based on Fourier pseudospectral spatial discretization, and proves discrete mass conservation and stochastic symplecticity. Numerical experiments validate mass conservation across fractional powers and illustrate energy behavior under noise, highlighting the method's robustness and geometric fidelity. The results advance understanding of stochastic geometric properties of SFNSE and provide robust, structure-preserving tools for simulating fractional dispersive waves under randomness.

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.

Figures (7)

• Figure 1: The evolution of the solution $u(x,t)$ with the initial condition $u(x,0)=\mathrm{sech}(x)\mathrm{exp}(2ix)$, where $\epsilon=0.01$, $\sigma=1, \lambda=1$ and different $\alpha=0.6,0.7,0.8,0.95$.
• Figure 2: The evolution of the solution $u(x,t)$ with the initial condition $u(x,0)=\mathrm{sech}(x)\mathrm{exp}(2ix)$, where $\epsilon=0.01$, $\sigma=1$, $\lambda=-1$ and different $\alpha=0.6,0.7,0.8,0.95$.
• Figure 3: The evolution of the solution $u(x,t)$ with the initial condition $u(x,0)=\mathrm{sech}(x)\mathrm{exp}(2ix)$, where $\epsilon=0.1$, $\sigma=1$, $\lambda=1$ and different $\alpha=0.6,0.7,0.8,0.95$.
• Figure 4: The evolution of the solution $u(x,t)$ with the initial condition $u(x,0)=\mathrm{sech}(x)\mathrm{exp}(2ix)$, where $\epsilon=0.1$, $\sigma=1$, $\lambda=-1$ and different $\alpha=0.6,0.7,0.8,0.95$.
• Figure 5: The evolution of the solution $u(x,t)$ with the initial condition $u(x,0)=\mathrm{sech}(x)\mathrm{exp}(2ix)$, where $\epsilon=1$, $\sigma=1$, $\lambda=1$ and different $\alpha=0.6,0.7,0.8,0.95$.

Theorems & Definitions (26)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• proof
• Remark 1
• Proposition 1
• Theorem 2
• proof