Massimiliano Berti, Thomas Kappeler, Riccardo Montalto
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.
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We prove the existence and stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian perturbations of KdV.
In this paper, we prove the long time stability of KAM tori for the nonlinear Schrödinger equation on the torus with arbitrary dimensions.
Eliasson and Kuksin developed a KAM approach to study the persistence of the invariant tori for nonlinear Schrödinger equation on $\mathbb{T}^{d}$. In this note, we improve Eliasson and Kuksin's KAM theorem by using Kolmogorov's iterative scheme and obtain a local normal form for the transformed Hamiltonian. As a consequence, we are able to derive the time $δ^{-1}$ stability of the obtained KAM tori.
We prove the existence of quasi-periodic, small amplitude, solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the solutions. The proofs are based on a combination of different ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a differential operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeo…
The Existence of Full-Dimensional KAM tori for one-dimensional nonlinear Klein-Gordon equation
本文针对周期边界条件下一维非线性克莱因-戈登方程，采用Bourgain2005年提出的方法，建立了其具有次指数衰减的满维KAM环面的存在性与线性稳定性，研究了该方程的概周期解问题。
We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows to reduce to solving only linear forced equations. We remark that our algorithm works at the…
We show that for almost all perturbations in a one-parameter family of KAM Hamiltonians on a smooth compact surface, for almost all KAM Lagrangian tori $Λ_ω$, we can find a semiclassical measure with positive mass on $Λ_ω$.
Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions
该研究证明了一个抽象无穷维KAM定理，可用于证明周期边界下一维受迫Kirchhoff方程小振幅拟周期解的存在性与线性稳定性，将拟线性扰动下的相关结果从单特征值推广到双特征值情形。
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an \emph{a-posteriori} format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functiona…
Infinite dimensional invariant tori for nonlinear Schrödinger equations
本研究证明无外部参数的圆上非线性薛定谔方程存在大量概周期解，在Kuksin--Poschel 1996年构造的多数有限维KAM环面附近存在无穷维非共振克罗内克环面，回答了哈密顿PDE领域长期存在的问题。
On the existence of full dimensional KAM tori for 1D periodic nonlinear Schrödinger equation
本文证明了1维周期非线性薛定谔方程在非线性扰动显式依赖于空间变量的情况下，全维KAM环的存在性。
We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash-Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case, some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations adv…
In this paper, we consider the nonlinear elliptic equations on rectangular tori. Using methods in the study of KAM theory and Anderson localization, we prove that these equations admit many analytic solutions.
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special w…
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schrödinger equation with periodic boundary conditions \begin{equation*}\label{L1} \mathbf{i}u_t-u_{xx}+V*u+ε|u|^4u=0,\hspace{12pt}x\in\mathbb{T}, \end{equation*} where $V*$ is the convolution potential. Here the radius of the invariant torus satisfies a slower decay, i.e. \begin{equation*}\label{031601} I_n\sim e^{- \ln^σ|n|},\qquad \mbox{as}\ |n|\rightarrow\infty, \end{equation*} for any $σ>2$, which improves the result given by Bourgain (J. Funct. Anal. 229 (2005), no.1, 62-94).
In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+Δ^2 u+m u + g(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in {\mathbb T}^d, \qquad \qquad (*) $$ where $g(x,u)=4u^3+ O(u^4)$. Namely, we show that, for generic $m$, most of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{g=0}$, written as the system $$ u_t=-v,\quad v_t=Δ^2 u+mu, $$, persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. If $d\ge2$, then not all the persisted tori are linearly stable, and we construct explicit example…
We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on $x$ in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact…
In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+Δ^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad \qquad (*) $$ where $G(x,u)=u^4+ O(u^5)$. Namely, we show that, for generic $m$, many of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{G=0}$, written as the system $$ u_t=-v,\quad v_t=Δ^2 u+mu, $$ persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. The persisted tori are filled in with time-quasiperiodi…
We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .
We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations
We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash-Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iterat…
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。
In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain \cite{BJFA2005}.
We prove that a system of coupled nonlinear Schr{ö}dinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6]. This is the first example of unstable tori for a 1d PDE.
In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $({\rm gSQG})_α$ in the patch form close to Rankine vortices. We show that invariant tori survive when the order $α$ of the singular operator belongs to a Cantor set contained in $(0,\frac12)$ with almost full Lebesgue measure. The proof is based on several techniques from KAM theory, pseudo-differential calculus together with Nash-Moser scheme in the spirit of the recent works \cite{Baldi-Berti2018,Berti-Bolle15}. One key novelty here is a refined Egorov type theorem es…
Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $\mathbb T^2:= (\mathbb R/2π\mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.
We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary condition…
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.
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