We study the focusing stochastic nonlinear Schrödinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the \(L^2\)-critical and supercritical cases. The mass (\(L^2\)-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate both theoretically and numerically how the energy is affected by various spatially correlated random perturbations and its dependence on the discretization parameters and the schemes. We then perform numerical investigation of the noise influence on the global dynamics measuring the probability of blow-up versus scattering behavior depending on parameters of correlation kernels. Finally, we study numerically the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed for a space-time white driving noise in Millet et al. (Numerical study of solutions behavior to the 1d stochastic \(L^2\)-critical and supercritical nonlinear Schrödinger equation, 2020. arXiv:2006.10695).

我们研究了在（L ^ 2 \）临界和超临界情况下，维纳过程在时间上呈白色且在空间上呈彩色的情况，在一维空间上具有乘性噪声的情况下研究了随机非线性Schrödinger方程的聚焦问题。质量（\（L ^ 2 \）-范数是守恒的，因为通过Stratonovich积分定义了乘法噪声，所以不保留能量（哈密顿量）。我们首先在理论和数值上都研究了能量如何受到各种空间相关的随机扰动及其对离散化参数和方案的依赖性的影响。然后，我们根据相关内核的参数对噪声对整体动力学的影响进行数值研究，从而测量爆炸和散射行为的概率。最后，我们从数值上研究了空间相关噪声对爆炸行为的影响，并得出结论，这种随机扰动除了影响爆炸中心位置的移动之外，不会影响爆炸动力学。这类似于我们在Millet等人的论文中观察到的时空白色驾驶噪声。（一维随机变量解行为的数值研究\（L ^ 2 \）-临界和超临界非线性Schrödinger方程，2020。arXiv：2006.10695）。