Existence of Chaos for a Singularly Perturbed NLS Equation

一种求解局部势极大值处奇异摄动NLS的新方法

A New Approach to Solving Singularly Perturbed NLS at Local Potential Maxima

本文针对具有局部势极大值、满足Berestycki-Lions条件的奇异摄动NLS方程，提出通过推导梯度范数精细下界来求解的新方法。

Fractional NLS equations with magnetic field, critical frequency and critical growth

The paper is devoted to the study of a singularly perturbed fractional Schrödinger equations involving critical frequency and critical growth in the presence of a magnetic field. By using variational methods, we obtain the existence of mountain pass solutions $u_{\varepsilon}$ which tend to the trivial solution as $\varepsilon\rightarrow0$. Moreover, we get infinitely many solutions and sign-changing solutions for the problem in absence of magnetic effects under some extra assumptions.

The instability of Bourgain-Wang solutions for the L^2 critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schrödinger equation $i\pa_tu+Δu+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with the pseudo conformal speed $$\|\nabla u(t)\|_{L^2}\sim \frac{1}{T-t},$$ and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical $L^2$ mass lies on the boundary of both $H^1$ open sets of global solutio…

Propagation of chaos and the higher order statistics in the wave kinetic theory

This manuscript continues and extends in various directions the result in arXiv:2104.11204, which gave a full derivation of the wave kinetic equation (WKE) from the nonlinear Schrödinger (NLS) equation in dimensions $d\geq 3$. The wave kinetic equation describes the effective dynamics of the second moments of the Fourier modes of the NLS solution at the kinetic timescale, and in the kinetic limit in which the size of the system diverges to infinity and the strength of the nonlinearity vanishes asymptotically according to a specified scaling law. Here, we investigate the behavior of the joint d…

Continuations of the nonlinear Schrödinger equation beyond the singularity

We present four continuations of the critical nonlinear \schro equation (NLS) beyond the singularity: 1) a sub-threshold power continuation, 2) a shrinking-hole continuation for ring-type solutions, 3) a vanishing nonlinear-damping continuation, and 4) a complex Ginzburg-Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that leads to a loglog collapse, the sub-threshold power limit is a Bourgain-Wang solution, both before and after the singularity, and the vanishing …

低正则性高斯测度对一维五次非线性薛定谔方程的定性准不变性

Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

研究一维五次非线性薛定谔方程在环面上的高斯测度的准不变性，证明当s>9/10时测度在方程流下保持绝对连续性，并扩展了已知阈值至s>9/10。