随机参数激励非线性薛定谔方程中的孤立波
Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation
AI总结：
本文研究了随机参数激励非线性薛定谔方程中孤立波的稳定性及正则性保持，提出新的相位滞后方法并给出噪声强度二次项精度的显式表达式。
摘要：
我们研究了一个由乘法平移不变噪声驱动的参数激励非线性薛定谔（PFNLS）方程。我们证明了在随机方程中，孤立波在指数于噪声强度倒数平方的时间尺度上是轨道稳定的。我们给出了相位偏移和围绕偏移波的波动的显式表达式，这些表达式在噪声强度的二次项精度内是准确的。这是通过发展一种新的视角来介绍Krüger和Stannat提出的相位滞后方法实现的。此外，我们证明了方程在分数贝塞尔空间H^s中对于任何s∈[0,∞)都是良好的，展示了正则性的持续性。
英文摘要：
We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space $H^{s}$ for any $s \in [0,\infty)$, demonstrating persistence of regularity.
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