论文标题
用空间相关的噪声的1D聚焦随机非线性schrödinger方程的解决方案的行为
Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise
论文作者
论文摘要
我们在一个空间维度上研究了聚焦随机的非线性schrödinger方程，并在$ l^2 $ - 关键的和超临界的情况下，由Wiener工艺进行白色，在空间上进行颜色，并在空间上进行乘法。由于通过Stratonovich积分定义的乘法噪声，质量（$ l^2 $ -norm）是保守的，不能保留能量（汉密尔顿）。我们首先研究了能量如何受到各种空间相关的随机扰动的影响。然后，我们研究噪声对根据相关核的各种参数的爆炸概率与散射行为的概率与散射行为的概率的影响。最后，我们研究了空间相关的噪声对爆炸行为的影响，并得出结论，这种随机扰动不会影响爆破动力学，除了爆破中心位置的移动。这与我们在[32]中观察到的时空白色驾驶噪声相似。
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 how the energy is affected by various spatially correlated random perturbations. We then study the influence of the noise on the global dynamics measuring the probability of blow-up versus scattering behavior depending on various parameters of correlation kernels. Finally, we study 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 in [32] for a space-time white driving noise.
