论文标题
对1D聚焦随机$ l^2 $ - 临界和超批评的非线性schrödinger方程的行为与时空白噪声
Behavior of solutions to the 1D focusing stochastic $L^2$-critical and supercritical nonlinear Schrödinger equation with space-time white noise
论文作者
论文摘要
我们在$ l^2 $ - 关键的和超临界的情况下，研究了焦点随机非线性schrödinger方程，其伴奏或乘法扰动由时空白噪声驱动。与确定性的情况不同，在随机环境中，哈密顿量（或能量）并不保守，在加性情况下，质量（或$ l^2 $ norm）也不保守。因此，我们研究了这些数量的时间演变。之后，我们研究噪声对解决方案全球行为的影响。特别是，我们表明噪声可能会引起爆炸，因此停止了解决方案的全球存在，否则在确定性的环境中将是全球的。此外，我们研究了噪声对乘法和加性噪声设置中爆炸动力学的影响，并获得爆破溶液的概况和速率。我们的发现得出的结论是，爆炸参数（速率和轮廓）对噪声的类型或强度不敏感：如果发生爆炸，它具有与确定性设置相同的动力学，但是，爆破中心有一个（随机的）偏移，可以描述为随机变量正常分布。
We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that we study the influence of noise on the global behavior of solutions. In particular, we show that the noise may induce blow-up, thus, ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow-up happens, it has the same dynamics as in the deterministic setting, however, there is a (random) shift of the blow-up center, which can be described as a random variable normally distributed.
