An evolution equation as the WKB correction in long-time asymptotics of Schrodinger dynamics

关于非线性薛定谔方程多相WKB分析中的非线性效应

On nonlinear effects in multiphase WKB analysis for the nonlinear Schrodinger equation

本文研究半经典极限下带外势和三次非线性的薛定谔方程，证明多相WKB初值在若干数据尺度区域中短时间内满足叠加原理，并刻画了不同数据大小下非线性干涉的出现条件。

The WKB method and geometric instability for non linear Schrodinger equations on surfaces

In this paper we are interested in constructing WKB approximations for the non linear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.

WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework

We are interested in a WKB analysis of the Logarithmic Non-Linear Schrödinger Equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross-Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to $0$ at infinity.

WKB expansion for a fractional Schrödinger equation with applications to controllability

This paper is devoted to the analysis of propagation properties for the solutions of a one-dimensional non-local Schrödinger equation involving the fractional Laplace operator $(-d_x^2)^s$, $s\in(0,1)$. We adopt a classical WKB approach and we provide a systematic procedure for building a suitable ansatz for the solutions to the problem. In this way, we can obtain quasi-solutions which are localized along the rays of geometric optics, whose group velocity can be computed explicitly in terms of the parameter $s$. Our results are then confirmed by numerical simulations, based on a finite element…

On the semi-classical limit for the nonlinear Schrodinger equation

We review some results concerning the semi-classical limit for the nonlinear Schrodinger equation, with or without an external potential. We consider initial data which are either of the WKB type, or very concentrated as the semi-classical parameter goes to zero. We sketch the techniques used according to various frameworks, and point out some open problems.

从泡利-泊松威尔/达尔文到欧拉-泊松威尔/达尔文系统的 semiclassical 极限：通过 WKB 方法

The semiclassical limit from the Pauli-Poisswell/Darwin to the Euler-Poisswell/Darwin system by WKB methods

本文通过 WKB 方法研究了从泡利-泊松威尔/达尔文到欧拉-泊松威尔/达尔文系统的 semiclassical 极限，证明了时间局部的 semiclassical 极限和弱收敛性，以及宏观密度的强收敛性。

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold $(M^d,g)$, where the metric $g$ is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.

Remarks on non-linear Schrödinger equation with magnetic fields

We study the non-linear Schrödinger equation with time depending magnetic field without smallness assumption at infinity. We obtain some results on the Cauchy problem, WKB asymptotics and instability.

Schrödinger evolution in a low-density random potential: annealed convergence to the linear Boltzmann equation for general semiclassical Wigner measures

We consider solutions of the time-dependent Schrödinger equation for a potential localised at the points of a Poisson process. We prove convergence of the phase-space distribution in the annealed Boltzmann-Grad limit to a semiclassical Wigner (or defect) measure and show that it is a solution of the linear Boltzmann equation. Our results hold for a large class of square-integrable initial data associated to Wigner measures, including Langragian states, WKB states and coherent states. This extends important previous work by Eng and Erdős.

Supercritical geometric optics for nonlinear Schrodinger equations

We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y.Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.

On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation

In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2π\Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b \in S^m (\mathbb{T}^n \times \mathbb{R}^n)$. Subsequently, we consider $\mathrm{Op}^w_\hbar(H)$ when $H=\frac{1}{2} |η|^2 + V(x)$ where $V \in C^\infty (\Bbb T^n;\Bbb R)$ and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner trans…

Wave propagation through sparse potential barriers

We prove that 3-dimensional Schrodinger operator with slowly decaying sparse potential has an a.c. spectrum that fills positive half-line. A new kind of WKB asymptotics for Green's function is found. The absense of positive eigenvalues is established as well.

WKB方法用于半经典算子：如何飞越光折线（及其他）

WKB for semiclassical operators: How to fly over caustics (and more)

本文基于微局部分析近期进展，统一阐述了改进的Maslov-WKB方法，严谨证明了半经典算子本征值的Bohr-Sommerfeld-Einstein-Brittle-Keller量子化条件，并回顾了应用与扩展。

一维Schrödinger算子带有负Coulomb型势的色散估计

Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension

本文研究了一维Schrödinger算子带有负Coulomb型衰减势的色散估计，推导了标准和正交归一Strichartz估计。通过推导WKB表达式和退化稳态相位公式，克服了快速衰减势的扰动方法不可用的困难。

关于非线性薛定谔方程在半经典极限中的傅里叶时间分裂方法 II. 解析正则性

On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity

本文研究了在半经典极限下，基于李-特罗特分裂的时间离散化方法用于非线性薛定谔方程，证明了精确解和数值解在与普朗克常数无关的时间区间内保持WKB结构，并展示了二次可观测量可以使用与普朗克常数无关的时间步长计算，通过时间依赖的解析空间框架克服了之前遇到的正则性损失现象。

Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

In the limit $\hbar\to 0$, we analyze a class of Schrödinger operators $H_\hbar = \hbar^2 L + \hbar W + V\cdot \mathrm{id}$ acting on sections of a vector bundle $\mathcal{Eh}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degenerate minimum at some point $p\in M$. We construct quasimodes of WKB-type near $p$ for eigenfunctions associated with the low lying eigenvalues of $H_\hbar$. These are obtained from eigenfunctions of the associated harmonic oscillator $H_{p, \hbar}$ at $p$, acting on smooth functi…

High-frequency limit of the Maxwell-Landau-Lifshitz system in the diffractive optics regime

We study semilinear Maxwell-Landau-Lifshitz systems in one space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times $O(1/\e)$. The leading terms of the WKB solutions solve cubic Schrödinger equations. We show that the nonlinear normal form method of Joly, Métivier and Rauch applies to this context. This implies that the Schrödinger approximation stays close to the exact solution of Maxwell-Landau-Lifshitz over its existence time. In the context of Maxwell-Landau-Lifshitz, this extends the analysis of Colin and Lannes from times $…

Self-Bound vortex states in nonlinear Schrödinger equations with LHY correction

We study the cubic-quartic nonlinear Schrödinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. A natural connection to the NLS with repulsive inverse-square potential in 2D arises, leading …

Semiclassical Analysis for Hartree equation

We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.

Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.

Loss of regularity for supercritical nonlinear Schrodinger equations

We consider the nonlinear Schrodinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse: there is a loss of regularity, in the same spirit as the result due to G.Lebeau in the case of the wave equation. We use an isotropic change of variable, which reduces the problem to a super-critical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional a la Brenier.

Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation

We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared.…

半经典极限下非线性薛定谔方程的傅里叶时间分裂方法

On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit

针对半经典极限下的薛定谔-泊松方程，在WKB近似有效时，证明了Lie-Trotter分裂算子的误差估计，并推广到弱非线性薛定谔方程。

Self-similar solutions to the derivative nonlinear Schrödinger equation

A class of self-similar solutions to the derivative nonlinear Schrödinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions.

Bloch dynamics with second order Berry phase correction

We derive the semiclassical Bloch dynamics with the second-order Berry phase correction in the presence of the slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For a uniform external electric field, the bi-characteristics system after a positional shift introduced by Berry connections agrees with the recent result in previous works. Moreover, for the case with a linear external electric field, we show that the extra terms arising in the bi-characteristics system after the positional shift are also gauge independent.

对文章“无规范不变性下低于Strauss指数的非线性薛定谔方程在加权空间中的整体适定性和散射”的更正

Correction to the article "Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance"

本文修正了原论文在二维和三维情况下映射压缩性证明中的错误，确保结论正确。