Justin Holmer, Maciej Zworski
We clarify the relation between the Hamiltonian and Lagrangian approaches to nonlinear evolution equations, focusing specifically on the nonlinear Schroedinger equation. In particular, we explain the least action principle and the Noether theorem in this context.
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We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrodinger equations with gauge invariant power-law nonlinearities and non-local perturbations. The model includes the Davey--Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variant. Our analysis reveals a new localization phenomenon for non-local perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive ap…
In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-$δ$ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schrödinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.
We apply the GBDT version of the Bäcklund-Darboux transformation to the nonlocal NLS (focusing and defocusing cases). The matrix case is included and solutions in the form of rectangular $m_1 \times m_2$ matrix functions are dealt with. The wave function is also constructed explicitly. Families of explicit examples are considered in detail. Some initial results and representations for the more complicated algebro-geometric solutions are obtained as well.
We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schrödinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved quantities based on the complete integrability of the equation, persistence of regularity, and by separating off the time evolution of finitely many Picard iterates.
Multiple positive bound states for NLS equations on noncompact metric graphs with potential
本文研究了在非紧致度量图上，亚临界NLS方程在足够大的质量下存在多个正束缚态的问题，通过分析吸引势的条件，证明了束缚态数量与图中有界边数的关系。
Three bound states with prescribed angular momentum to the cubic-quintic NLS equations in $\mathbb{R}^{3}$
本文研究三维三次-五次NLS方程具有给定角动量和质量的束缚态，证明了双约束问题存在局部极小值、山隘型和全局极小值三个解，并构造新山隘路径揭示其几何联系与能级比较。
In this work we present an application of a theory of vessels to solution of the evolutionary Non Liner Schrodinger (NLS) equation. The classes of functions for which the initial value problem is solvable relies on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of Zackarov-Shabbath for solving of evolutionary NLS equation, but has an advantage of simpler formulas and new techniques and notions to construct solutions of the evolutionary NLS equation.
The bidirectional NLS approximation for the one-dimensional Euler-Poisson system
本文严格证明一维Euler-Poisson系统的解可由两个相向传播的NLS波包近似，扩展了单向结果，首次给出双向NLS近似的有效证明。
We study the evolution of the one dimensional periodic cubic Schrödinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was first observed by Talbot in an optical experiment performed in 1836. In this paper we prove that a similar phenomenon occurs in the case of the NLS equation.
Orbital stability of kinks in the NLS equation with competing nonlinearities
本文研究了具有竞争非线性的NLS方程中kinks的轨道稳定性，通过严格证明展示了其稳定性，并扩展了分析方法至一般情况。
We revise the classical approach by Brézis-Gallouët to prove global well posedness for nonlinear evolution equations. In particular we prove global well--posedness for the quartic NLS posed on general domains $Ω$ in $\R^2$ with initial data in $H^2(Ω)\cap H^1_0(Ω)$, and for the quartic nonlinear half-wave equation on $\R$ with initial data in $H^1(\R)$.
Energy cascades for NLS on the torus
研究在环面上的非线性薛定谔方程中，通过特殊的小初始数据产生任意大的模式的时间层，采用多相弱非线性几何光学和特定二维离散动力系统进行分析。
For the stationary nonlinear Schrödinger equation $-Δu+ V(x)u- f(u) = λu$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow to match the $L^2$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $L^2$-…
Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$
本文研究波导流形上组合幂次非线性NLS的长时间动力学，通过构造两个重标度变分问题族，建立了孤立子、散射与爆破解的定性定量性质及周期性依赖结果。
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-Δ+V$, which we employ at a nonlinear level. As a first st…
On the formation of singularities for the slightly supercritical NLS equation with nonlinear damping
本文研究了带非线性阻尼的聚焦质量超临界NLS方程，给出了有限时间爆破的充分条件，证明了阻尼项的非正则化效应，并通过调制分析证明了自相似爆破结构的近似保留。
We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schrödinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein-Gordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the …
We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation (NLS), $i \partial_t u+Δu+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Na…
We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is…
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian symmetric spaces $G/SO(N)$. The derivation uses a parallel frame and connection along the curves, involving the Klein geometry of the group $G$. This is shown to yield the two known $U(N-1)$-invariant vector generalizations of the nonlinear Schrodinger (NLS) equation and the complex modified Korteweg-de Vries (mKdV) equation, as well as $U(N-1)$-invariant vector…
We consider a nonlinear Schrödinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of $δ$-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass ($L^2$-norm). When existing, the minimizer is called ground state and it is…
GBDT with nontrivial seeds: explicit solutions of the focusing NLS equations and the corresponding Weyl functions
本文利用 GBDT 方法，针对指数型非平凡种子构造聚焦 NLS 方程的显式解，并给出相应 Baker-Akhiezer 函数与 Weyl 函数演化。
Scattering, random phase and wave turbulence
本文针对实平面上三次二维薛定谔方程，利用色散效应和高斯截断周期函数，在不同时间尺度下研究时间演化算子向动力学算子的两种不同收敛机制，从而揭示共振流形正则性与散射现象的联系。
We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of th…
Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the (1+1)-dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the (3+1)-dimensional case to the variety of solutions of integrable NLS equation of (1+1)-dimensional case. As an example, we illustrated our technique using two lowest order rational solutions of the NLS equation as …
We study Gibbs measures invariant under the flow of the NLS on the unit disc of $\R^2$. For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener measure invariant by the NLS flow. The density of the measure is integrable with respect to the Wiener measure for sub cubic nonlinear interactions. The existence of the dynamics is obtained in Bourgain spaces of low regularity. The key ingredient are bilinear Strichartz estimates for the free evolution. The bilinear effect in our analysis results from simple properties of the Bessel functions and…
We consider the cubic nonlinear Schrödinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrödinger equation from quantum many-body systems. Our geometric constructions are based on a quantized version of the Poisson structure i…
We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in R^3. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key …
Stability of Cnoidal Waves for the Damped Nonlinear Schrödinger Equation
研究一维环面上带线性阻尼的立方非线性薛定谔方程，证明椭圆余弦波族在耗散动力学下轨道稳定，通过一阶近似和指数递减Lyapunov泛函控制扰动。
We study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global …
On scattering for NLS: rigidity properties and numerical simulations via the lens transform
本文研究了非聚焦非线性薛定谔方程的散射算子，通过透镜变换克服了无限时间下的计算挑战，提出高效方法并验证了新理论性质，通过数值实验探讨了长程散射问题。
Global Well-Posedness and Uniform Long-Time Control for a Class of Periodic NLS-Dysthe Equations
本文研究周期NLS-Dysthe方程，证明其在 $s>3/2$ 时的局部适定性、$s\ge2$ 时的全局适定性，还在不对初值加小性假设下得到一致长时间控制，为相关色散方程的调制研究提供了关键理论支撑。
A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space $G/H_+$ where $G$ is an infinite dimensional Lie group and $H_+$ is a subgroup of $G$. It is shown that the HM flows are induced by an action of $\mathbb{R}^2$ on $G/H_+$, and that the HM equation can be integrated by solving a Birkhoff factorization problem for $G$. For the HM flows which are Laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear Schroedinger (NLS) and Heisenberg magnet equations. …
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