Thomas Kappeler, Peter Topalov
We prove an infinite dimensional version of the Arnold-Liouville theorem for integrable non-linear PDEs: In a case study we consider the {\em focusing} NLS equation with periodic boundary conditions.
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All the almost periodic solutions for non integrable PDEs found in the literature are very regular (at least $C^\infty$) and, hence, very close to quasi periodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the one dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover many of our solutions are so only in a weak sense. This is the first result on exis…
We discuss a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs.
In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class solutions which are formally asymptotically in $L^2(\mathbb{R})$. A precise rate of convergence is also obtained assuming more regularity.
In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically taylored towards the needs in perturbation theory. They are defined in neighbourhoods of families of finite dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the dNLS Hamiltonian is in normal form up to order three.
In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this…
Uniqueness of normalized ground states for NLS models
本文提出两种证明NLS等PDE归一化基态唯一性的方法，阐明其核心思想并应用于多类方程，且在相应假设下可推广至其他算子、区域与非线性项。
Right and Wrong Ansätze for Nonlinear Waves in Stochastic PDEs
研究通过乘以随机指数寻找随机反应扩散方程显式解的可行性，指出该方法仅在Stratonovich解释下的NLS型方程中有效。
We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of "irregularity" for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well--posedness theory of the equations. We develop two different approaches to the analysis of the effects …
In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schrödinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $\geq 2$. In the high regularity result, an {\it Inviscid - Infinite dimensional (IID) limit} is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an …
We prove a Wong-Zakai theorem for the defocusing mass-critical stochastic nonlinear Schrödinger equation (NLS) on $\mathbb{R}$. The main ingredient are careful mixtures of bootstrapping arguments at both deterministic and stochastic levels. Several subtleties arising from the proof mark the difference between the dispersive case and corresponding situations in SDEs and parabolic stochastic PDEs, as well as the difference between the large-$n$ case and the limiting ($n=\infty$) case.
Almost global existence for some Hamiltonian PDEs on manifolds with globally integrable geodesic flow
该研究针对具有全局可积测地流的黎曼流形上的半线性哈密顿PDE，基于正规形方法证明了小光滑解的几乎全局存在性抽象结果，并将其应用于带卷积势的NLS方程、非线性梁方程，同时证明了NLS基态的$H^s$稳定性。
In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimat…
We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin-Bona-Mahony (BBM) equation with dispersion~$γ>1$ and the periodic one dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS eq…
Remarks on nonlinear dispersive PDEs with rough dispersion management
本研究针对含粗糙时间系数色散管理的色散PDE，证明其在不同假设下的适定性可从无调制方程迁移，还发现粗糙调制的噪声正则化现象，完善拓展了现有结果。
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355:1--23, 2017] derived two $U(1)$-invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schrödinger equation and admit peakons ($non-smooth\ solitons$). In the present paper, these two peakon PDEs are generalized to a family of $U(1)$-invariant peakon PDEs parametrized by the real projective line $ \mathbf{RP} ^1$. All equations in this family are shown to posses $conservative\ peakon \ solutions$ (whose Sobolev $H^1(\mathbf{R})$ norm is time invariant). The Hamiltonian structure …
The inverse scattering theory is a basic tool to solve linear differential equations and some Partial Differential Equations (PDEs). Using this theory the Korteweg-de Vries (KdV), the family of evolutionary Non Linear Schrodinger (NLS) equations, Kadomtzev-Petviashvili and many more completely integrable PDEs of mathematical physics are solved, using Zacharv-Shabath scheme. This last approach includes the use of a Lax pair, and has an advantage to be applied to wider class of equations, like difference equations, but has a disadvantage to be used only for "rapidly decreasing solutions". This t…
We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As …
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…
Asymptotically full measure sets of almost-periodic solutions for the NLS equation
该研究构造了圆周上NLS方程渐近满测度的小振幅几乎周期解，证明极大不变环面为Banach流形，并建立了椭圆不动点的Lyapunov统计稳定性，将经典KAM定理推广至无穷维。
A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS
本文提出了一般统一的抽象框架，证明了一大类PDE基态解的唯一性与非退化性，并研究了归一化解的存在性、非存在性及多重性，成功将其推广至具非自治非线性的分数阶非线性薛定谔方程，进而分析了驻波的轨道稳定性。
It is known that the initial-boundary value problem for certain integrable partial differential equations (PDEs) on the half-line with integrable boundary conditions can be mapped to a special case of the Inverse Scattering Method (ISM) on the full-line. This can also be established within the so-called Unified Transform (UT) for initial-boundary value problems with linearizable boundary conditions. In this paper, we show a converse to this statement within the Ablowitz-Kaup-Newell-Segur (AKNS) scheme: the ISM on the full-line can be mapped to an initial-boundary value problem with linearizabl…
We consider a general class of infinite dimensional reversible differential systems. Assuming a non resonance condition on the linear frequencies, we construct for such systems almost invariant pseudo norms that are closed to Sobolev-like norms. This allows us to prove that if the Sobolev norm of index $s$ of the initial data $z_0$ is sufficiently small (of order $ε$) then the Sobolev norm of the solution is bounded by $2ε$ during very long time (of order $ε^{-r}$ with $r$ arbitrary). It turns out that this theorem applies to a large class of reversible semi linear PDEs including the non linea…
Continuum limit for discrete NLS with memory effect
本文研究带长程相互作用和记忆效应的离散非线性薛定谔方程在网格尺寸趋于零时的连续极限，证明其解在L²中强收敛到连续NLS型方程的解并给出收敛速率，方法推广了Hong-Yang及Kenig-Ponce-Vega的经典估计框架。
We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretiz…
Nonlinear PDEs with modulated dispersion IV: normal form approach and unconditional uniqueness
本文研究了调制KdV方程的无条件唯一性，通过正常形式方法证明了在特定调制条件下解的唯一性，并扩展了非线性Young积分的构造方法。
We consider a system of $N$ bosons where the particles experience a short range two-body interaction given by $N^{-1}v_N(x)=N^{3β-1}v(N^βx)$ where $v \in C^\infty_c(\mathbb{R}^3)$, without a definite sign on $v$. We extend the results of M. Grillakis and M. Machedon, Comm. Math. Phys., $\textbf{324}$, 601(2013) and E. Kuz, Differ. Integral Equ., $\textbf{137}$, 1613(2015) regarding the second-order correction to the mean-field evolution of systems with repulsive interaction to systems with attractive interaction for $0<β<\frac{1}{2}$. Our extension allows for a more general set of initial data…
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
In a 1959 paper by Pitaevskii, a macroscopic model of superfluidity was derived from first principles, to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model couples two of the most fundamental PDEs in mathematics: the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE). In this article, we show the local existence of weak solutions to this system (in a smooth bounded domain in 3D), by deriving the required a priori estimates. (We will also establish an energy inequality obeyed by the weak solutions constructed in Kim's…
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。
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
本文通过理论、数值和机器学习方法研究偏微分方程中的奇点形成问题，提出新的分析框架和数值方法，提升对复杂方程中奇点形成机制的理解与预测能力。
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