The Cauchy problem for Schrodinger flows into Kahler manifolds

Remarks on solitary waves and Cauchy problem for a Half-wave-Schrödinger equations

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

关于 Gelfand-Shilov 空间中 Schrödinger 型方程 Cauchy 问题的几点注记

Some remarks on the Cauchy Problem for Schrödinger type equations in Gelfand-Shilov spaces

本文研究 Schrödinger 型算子的 Cauchy 问题，在一阶系数虚部满足适当衰减假设下证明了其在 Gelfand-Shilov 类中的适定性，并通过实例讨论了结果的优化性。

对数薛定谔方程的柯西问题再探

The Cauchy problem for the logarithmic Schrödinger equation revisited

本文重新探讨对数薛定谔方程的柯西问题，在$H^1$、能量空间和$H^2$-能量空间中构造强解，采用不依赖紧性论证的建设性方法，通过近似解序列的柯西性证明强收敛。

Global Cauchy problems for the Klein-Gordon, wave and fractional Schrödinger equations with Hartree nonlinearity on modulation spaces

We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …

Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case

In this paper and the companion work \cite{LIZE2}, we prove that the Schrödinger map flows from $\Bbb R^d$ with $d\ge 2$ to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper with the companion work \cite{LIZE2} solves the open problem …

Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i\partial_t u=-\epsΔu+Δ^2 u+P((\partial_x^αu)_{\absα\ls 2}, (\partial_x^α\bar{u})_{\absα\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in $H^1$. The well-posedness theory in $H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $β=(4-2α)/(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.

Kähler流形上的周期Schrödinger映射流

Periodic Schrödinger map flow on Kähler manifolds

本文通过推导新的div-curl型引理并结合能量与动量平衡律获得时空估计，证明了从圆到Kähler流形的Schrödinger映射流全局正则，从而完全解决了Wei-Yue Ding猜想。

The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension,the defocusing case

This paper has been withdrawn by the author

带卷积势的非线性薛定谔方程的柯西问题

The Cauchy problem for the nonlinear Schrödinger equation with a convolution potential

本文研究带奇异卷积势的非线性薛定谔方程的柯西问题，证明其在修正Sobolev空间中的局部适定性，并给出解在能量空间中一致有界的条件，从而与驻波解的存在性建立联系。

The Cauchy problem for the quasilinear Schrodinger equatio

These are lectures notes for a 6 hour course given at PCMI, Park City, Utah, in the summer of 2003. The notes are based on a series of joint works of Kenig-Ponce-Vega. Each lecture had problems assigned with it. The lecture notes were prepared with the help of Justin Holmer. Justin Holmer also provided solutions to the problems, in an appendix to the notes.

On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger equations with potential

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast …

On Schrödinger maps

We study the question of well-posedness of the Cauchy problem for Schrödinger maps from $\rone \times \rtwo$ to the sphere $\stwo$ or to ${\mathbb H^2}$, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schrödinger map itself from it. In proving we…

On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity

We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $α$, $γ$, $λ$, $ψ$. We also remark on finite time blowup of solutions when $λ= -1$.

环面上半线性 Schrödinger 方程 Cauchy 问题的适定性与不适定性

Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

本文研究环面 $\mathbb T$ 上半线性 Schrödinger 方程的 Cauchy 问题，利用能量估计、规范变换及解不存在性证明，给出 $s>5/2$ 时其在 $H^s(\mathbb T)$ 中适定的多项式非线性项充要条件。

Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data

We consider the inverse boundary value problem of determining the potential $q$ in the equation $Δu + qu = 0$ in $Ω\subset\mathbb{R}^n$, from local Cauchy data. A result of global Lipschitz stability is obtained in dimension $n\geq 3$ for potentials that are piecewise linear on a given partition of $Ω$. No sign, nor spectrum condition on $q$ is assumed, hence our treatment encompasses the reduced wave equation $Δu + k^2c^{-2}u=0$ at fixed frequency $k$.

Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.

环面上导数分数阶非线性薛定谔方程柯西问题的适定性与不适定性

Well- and Ill-posedness of the Cauchy problem for derivative fractional nonlinear Schrödinger equations on the torus

本文将环面上半线性薛定谔方程适定性的非线性项充要条件推广到导数分数阶薛定谔方程，并用修正能量方法证明适定性，同时借助柯西–黎曼型算子证明不适定性。

The Cauchy Problem for nonlinear Quadratic Interactions of the Schrödinger type in one dimensional space

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for data {in Sobolev spaces} with low regularity. {To obtain} the local theory, I prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, I establish the global well-posedness in $H^s(\mathbb{R})\times H^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the global result …

The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data

We set-up and solve the Cauchy problem for Schrödinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.

Sharp weights in the Cauchy problem for nonlinear Schrodinger equations with potential

We review different properties related to the Cauchy problem for the (nonlinear) Schrodinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well-posed. The characterization of the minimal decay is different in the case of super-quadratic potentials.

The Cauchy Problem of the Schrödinger-Korteweg-de Vries System

We study the Cauchy problem of the Schrödinger-Korteweg-de Vries system. First, we establish the local well-posedness results, which improve the results of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results, which show that they are sharp in some well-posedness thresholds. Particularly, we obtain the local well-posedness for the initial data in $H^{-{3/16}+}(\R)\times H^{-{3/4}+}(\R)$ in the resonant case, it is almost the optimal except the endpoint. At last we establish the global well-posedness results in $H^s(\R)\times H^s(\R)$ when $s>\dfrac{1}{2}$ no matter in the res…

Uniqueness of Schrodinger flow via energy inequality

In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrodinger flow in the whole space $R^n$ provided there is a smooth solution in the energy class.

Semiclassical Cauchy Estimates and Applications

In this note, we study solutions to semiclassical Schrodinger equations on a real analytic manifold with a real analytic potential and prove the semiclassical version of Cauchy estimates on derivatives. As an application, we use Donnelly and Fefferman's method to prove the upper and lower bounds for (n-1)-dimensional Hausdorff measure of the nodal sets of the solutions to semiclassical Schrodinger equations.

The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $H^{s}(\mathbb R^{n})$

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge3$, $1\le s<\frac{n}{2} $, $0<b<2$ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^{σ} u$ with $λ\in \mathbb C$ and $σ=\frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $H^{s} (\mathbb R^{n} )$ with $1\le s<\frac{n}{2}$ for the critical INLS equation under some assumption on $b$. To this end, we first establi…

On the Cauchy Problem for nonlinear Schrödinger equations with rotation

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11,12]. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case.

Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s> \max \le…

Long wave limit for Schrodinger maps

We study long wave limits for general Schrodinger maps systems into Kahler manifolds with a constraining potential vanishing on a Lagrangian submanifold. We obtain KdV type systems set on the tangent space of the submanifold. Our general theory is applied to study the long wave limit of the Gross-Pitaevskii equation, and of the Landau-Lifshitz systems for ferromagnetic and antiferromagnetic chains.

Continuous Dependence of Cauchy Problem For Nonlinear Schrödinger Equation in $H^{s}$

We consider the Cauchy problem for the nonlinear Schrödinger equation $i \partial_{t}u+ Δu=λ_{0}u+λ_{1}|u|^αu$ in $\mathbb{R}^{N}$, where $λ_{0},λ_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical case: $0<α\leq\frac{4}{N-2s}$ when $1<s<\frac{N}{2}$ and $0<α<+\infty$ when $s\geq\frac{N}{2}$. We show that the solution depends continuously on the initial value in the standard sense in $H^{s}(\mathbb{R}^{N})$ if $α$ satisfies certain assumptions.

Almost sure local wellposedness of energy critical fractional Schrödinger equations with hartree nonlinearity

We consider a Cauchy problem of energy-critical fractional Schrödinger equation with Hartree nonlinearity below the energy space. Using a method of randomization of functions on $\mathbb{R}^d$ associated with the Wiener decomposition, introduced by Á. Bényi, T. Oh, and O. Pocovnicu \cite{beohpo1,beohpo2}, we prove that the Cauchy problem is almost surely locally well-posed. Our result includes Hartree Schrödinger equation ($α= 2$).

On the global well-posedness of the one-dimensional Schrodinger map flow

We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u=λ|x|^{-b} |u|^{σ} u,\; u(0)=u_{0} \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $λ=\pm1$, $0<b<2$, $σ=\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first prove the local well-posedness as well as small data global well-posedness and scattering in $H^{1}$ for $c>-\frac{(d+2-2b)^{2}-4}{(d+2-2b)^{2}}c(d)$ and $0<b<\frac{4}{d}$, by using the contraction mapping principle based on the Strich…

通过局部柯西数据同时恢复各向异性薛定谔型方程中两个系数的Lipschitz稳定性估计

Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data

本文研究各向异性薛定谔型方程中两个系数同时识别的反问题，在无符号和谱条件假设下，利用奇异解与唯一延拓定量估计，证明了维数n≥3时的全局Lipschitz稳定性。

Random data Cauchy problem for supercritical Schrödinger equations

In this paper we consider the Schrödinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$. Here we use the randomisation method of the inital conditions, introduced by N. Burq-N. Tzvetkov and we are able to show that the equation admits strong solutions for data in $\H^{s}$ for some $s<s_{c}$. In the appendix we prove the equivalence between the smoothing effect for a Schrödinger operator …

带逆幂势的非齐次非线性薛定谔方程的柯西问题研究

On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential

本文研究带逆幂势的非齐次非线性薛定谔方程的柯西问题，借助索伯列夫-洛伦兹空间的施特里哈茨估计与压缩映射原理建立局部适定性，还探究了H¹解的全局存在性与爆破，推广了已有相关结果。

Lipschitz stability for the Finite Dimensional Fractional Calderón Problem with Finite Cauchy Data

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $q \in L^{\infty}(Ω) $ in the equation $((-Δ)^s+ q)u = 0 \mbox{ in } Ω\subset \mathbb{R}^n$ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $L^{\infty}(Ω)$. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $q$. We allow for the …

Manton的Chern-Simons-Schrödinger方程的Cauchy问题

The Cauchy problem for Manton's Chern-Simons-Schrödinger equation

该论文在DeTurck规范下证明了Manton的Chern-Simons-Schrödinger方程的全局适定性，并利用几何Littlewood-Paley理论获得自对偶涡旋的轨道稳定性。

Solution theory to semilinear stochastic equations of Schrödinger type on curved spaces I -- Operators with uniformly bounded coefficients

We study the Cauchy problem for Schrödinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.

Calderon inverse Problem for the Schrodinger Operator on Riemann Surfaces

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $\mc{N}$) of the Schrödinger operator $Δ+V$ with $V\in C^2(M_0)$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n} )$

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation \[iu_{t} +Δu=|x|^{-b} f(u),\;u(0)\in H^{s} (\mathbb R^{n} ),\] where $n\in \mathbb N$, $0<s<\min \{ n,\; 1+n/2\} $, $0<b<\min \{ 2,\;n-s,\;1+\frac{n-2s}{2} \} $ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^{σ} u$ with $σ>0$ and $λ\in \mathbb C$. Recently, An--Kim \cite{AK21} proved the local existence of solutions in $H^{s}(\mathbb R^{n} )$ with $0\le s<\min \{ n,\; 1+n/2\}$. However even though the solution is constructed by a fixed point technique, continuous dependence in the …

Regularity properties of nonlinear Schrödinger equations in vector-valued spaces

In this paper, regularity properties of Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained.

The semiclassical modified nonlinear Schrödinger equation II: asymptotic analysis of the Cauchy problem. The elliptic region for transsonic initial data

We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schrödinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation, along with the steepest descent method of Deift and Zhou. The initial conditions are selected both to allow all relevant scattering data to be calculated without approximation and also to place the governing equation in a transsonic state in which the quantum fluid dynamical system formally approximating it is of hyperbolic type for some $x$ and of elliptic typ…

On the Periodic Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated to the system \begin{equation} \{ \begin{array} [c]{l} 2i\partial_{t}u+q\partial_{x}^{2}u+iγ\partial_{x}^{3}u=F_{1}(u,w)\\ 2i\partial_{t}w+q\partial_{x}^{2}w+iγ\partial_{x}^{3}w=F_{2}(u,w), \end{array} . \end{equation} where $F_{1}$ and $F_{2}$ are polynomials of degree 3 involving $u$, $w$ and their derivatives. This system describes the dynamics of two nonlinear short-optical pulses envelopes $u(x,t)$ and $w(x,t)$ in fibers (\cite{31}, \cite{14}). We prove periodic local well-posedness for the IVP with d…

与黎曼zi函数相关的Schrödinger型方程的初值问题

Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function

本文研究了与黎曼zi函数相关的非线性阻尼Schrödinger方程的初值问题，证明了解的唯一性和全局存在性，并指出一维情况下解在有限时间内趋于零。

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrödinger Equations

A classification of large-time and finite-time blow-up asymptotics of solutions of the Cauchy problem for higher-order Schrödinger equations is performed.

几乎临界空间中二次导数非线性薛定谔方程组的适定性

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces

本文研究Colin和Colin提出的二次导数非线性薛定谔方程组的柯西问题，确定了除标度临界情形外光滑流映射存在的几乎最优Sobolev正则性，填补了作者此前论文的空白。

一般区域中非线性薛定谔方程的一个注记

A note on the nonlinear Schrödinger equation in a general domain

本文针对一般区域中的非线性薛定谔方程，提出通过证明近似解构成柯西序列的简单替代方法构造解，并涵盖幂型、对数型及阻尼型三类非线性。

带有漂移的Schrödinger方程类的Strichartz估计

Strichartz estimates for a class of Schrödinger equations with a drift

本文研究了带有漂移的Schrödinger方程类的Strichartz估计，建立了新的内在估计方法。

A remark on inverse problems for nonlinear magnetic Schrödinger equations on complex manifolds

We show that the knowledge of the Dirichlet--to--Neumann map for a nonlinear magnetic Schrödinger operator on the boundary of a compact complex manifold, equipped with a Kähler metric and admitting sufficiently many global holomorphic functions, determines the nonlinear magnetic and electric potentials uniquely.

二次导数分数阶非线性薛定谔方程的范数膨胀

Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

研究二次导数分数阶非线性薛定谔方程在Sobolev空间中的适定性，通过迭代项估计证明范数膨胀和无限正则性损失，从而得出不适定性。

The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization

The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) this "quantum analytic continuation potential" coincid…

The Cauchy problem for a semilinear ordinary differential equation in the homogeneous and isotropic spacetime

A semilinear ordinary differential equation is derived from a semilinear Schrödinger equation in the homogeneous and isotropic spacetime by the Ehrenfest theorem. The Cauchy problem for the equation is considered. Exact solutions and nonexistence of global weak solutions of the equation are also considered in the de Sitter spacetime. The effects of spatial expansion and contraction are studied.

Global well-posedness and instability of an inhomogeneous nonlinear Schrodinger equation with harmonic potential

This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the associated ground state and instability of the standing wave.

Strichartz Inequalities for Lipschitz Metrics on Manifolds and Nonlinear Schrodinger Equation on Domains

We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.