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In general, a Hamiltonian system is nonintegrabe if chaotic dynamics occurs. However, chaotic dynamics may not occur even if it is nonintegrable. Here we are interested in the following question: Does chaotic dynamics occur in a Hamiltonian system when it is nonintegrable? We review some previous results related to this question for two-degree-of-freedom Hamiltoniansystems with saddle-centers and homoclinic orbits. We also state some extensions of the results to a higher-order approximation, heteroclinic orbits and more-or infinite-degree-of-freedom systems. In particular, the extended theory shows that Arnold diffusion type motions can occur in three-or more-degree-of-freedom systems.

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic problem can be mapped into a highly resonant Floquet-Schr\"odinger equation, for which, under a diophantine-like assumption on the frequency, we show the existence of a periodic solution. The proof is based on a KAM-type renormalisation. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing also at low frequencies., Comment: 21 pages

In this paper, the nonlinear Schrödinger equations with Legendre potential $\mathbf{i} u_{t} - u_{xx} + V_L(x)u + mu + \sec x \cdot |u|^2 u = 0$ subject to certain boundary conditions is considered, where $V_L(x) = -\frac{1}{2} - \frac{1}{4} \tan^2 x$, $x \in (-\pi/2,\pi/2)$. It is proved that for each given positive constant $m \gt 0$, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.

This article is devoted to the study of a wave equation with a constant potential and an $x$-periodic and $t$-quasi-periodic nonlinear term subject to periodic boundary conditions. It is proved that the equation admits small amplitude, linear stable and $t$-quasi-periodic solutions for any constant potential and most frequency vectors.

We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in $x_j$, $-i \partial_j$ with coefficients which depend quasiperiodically on time., fixed some misprints

In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS (nonlinear Schrödinger equations):\[ \mathrm{i} \partial_{t} u - \Delta u + V * u + \partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad x \in \mathbb{T}^d, \; t \in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an $\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0 = u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm $\|u(t)\|_s$ remains bounded by $C_s \varepsilon$.

We define and describe the class of Quasi-T\"oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr\"odinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment., Comment: 34 pages, 1 figure

We consider the nonlinear Schr\"{o}dinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2\pi$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect., Comment: 32 pages, 1 figure

In this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional ($1d$) beam equations $$ u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m\gt 0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).

We study the solutions of linear Schroedinger equations in which the potential energy is a periodic function of time and is sufficiently localized in space. We consider the potential to be close to one that is time periodic and yet explicitly solvable. A large family of such potentials has been constructed and the corresponding Schroedinger equation solved by Miller and Akhmediev. Exact bound states, or breather modes, exist in the unperturbed problem and are found to be generically metastable in the presence of small periodic perturbations. Thus, these states are long-lived but eventually decay. On a time scale of order $\epsilon^{-2}$, where $\epsilon$ is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermi's golden rule. For times of order $\epsilon^{-1}$ the breather modes are frequency shifted. This behavior is derived first by classical multiple-scale expansions, and then in certain circumstances we are able to apply the rigorous theory developed by Soffer and Weinstein and extended by Kirr and Weinstein to justify the expansions and also provide longer-time asymptotics that indicate eventual dispersive decay of the bound states with behavior that is algebraic in time. As an application, we use our techniques to study the frequency dependence of the guidance properties of certain optical waveguides. We supplement our results with numerical experiments., Comment: 69 pages, 13 figures, to appear in Nonlinearity

In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + g(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in {\mathbb T}^d, \qquad \qquad (*) $$ 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 $$ u_t=-v,\quad v_t=\Delta^2 u+mu, $$, 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 examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in nonlinear physics that small-amplitude solutions of space-multidimensonal hamiltonian PDEs behave in a chaotic way. The proof uses an abstract KAM theorem from another our publication.

This paper deals with one-dimensional (1D) nonlinear Schr o dinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer $b>1$, the equation admits small-amplitude quasi-periodic solutions, whose $b$-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinite-dimensional KAM theory.

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation ¶ i u t = − Δ u + V ⋆ u + ∂ ū g ( u , ū ) , x ∈ T d , ¶ where [math] is a typical smooth Fourier multiplier and [math] is analytic in both variables. More precisely, we prove that if the initial datum is analytic in a strip of width [math] whose norm on this strip is equal to [math] , then if [math] is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width [math] , with norm bounded on this strip by [math] over a very long time interval of order [math] , where [math] is arbitrary and [math] and [math] are positive constants depending on [math] and [math] .

We consider the following degenerate half-wave equation on the one-dimensional torus: ¶ i ∂ t u − | D | u = | u | 2 u , u ( 0 , ⋅ ) = u 0 . ¶ We show that, on a large time interval, the solution may be approximated by the solution of a completely integrable system—the cubic Szegő equation. As a consequence, we prove an instability result for large [math] norms of solutions of this wave equation.

We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schrödinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

We define and describe the class of Quasi-T��plitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr��dinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment., 34 pages, 1 figure

The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is more on the underlying ideas than on the sharpness of the arguments., 33 pages. Small corrections and updated references

We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of $b$ frequencies, $b\leq d+2$, in arbitrary dimension $d$ and for arbitrary non integrable algebraic nonlinearity $p$. This reflects the conservation of $d$ momenta, energy and $L^2$ norm. In 1d, we prove the existence of quasi-periodic solutions with arbitrary $b$ and for arbitrary $p$, solving a problem that started Hamiltonian PDE., Comment: 19 pp, slightly more details on proofs of Theorems 2 and 3

We present a geometric formulation of existence of time quasi-periodic solutions. As an application, we prove the existence of quasi-periodic solutions of $b$ frequencies, $b\leq d+2$, in arbitrary dimension $d$ and for arbitrary non integrable algebraic nonlinearity $p$. This reflects the conservation of $d$ momenta, energy and $L^2$ norm. In 1d, we prove the existence of quasi-periodic solutions with arbitrary $b$ and for arbitrary $p$, solving a problem that started Hamiltonian PDE., 19 pp, slightly more details on proofs of Theorems 2 and 3

We study the dynamics of a chain of coupled particles subjected to a restoring force (Klein-Gordon lattice) in the cases of either periodic or Dirichlet boundary conditions. Precisely, we prove that, when the initial data are of small amplitude and have long wavelength, the main part of the solution is interpolated by a solution of the nonlinear Schr\"odinger equation, which in turn has the property that its Fourier coefficients decay exponentially. The first order correction to the solution has Fourier coefficients that decay exponentially in the periodic case, but only as a power in the Dirichlet case. In particular our result allows one to explain the numerical computations of the paper \cite{BMP07}.

We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrödinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the $d$ -dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions

The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems where the second Melnikov's conditions are eliminated. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schr\"odinger equations and other equations of higher spatial dimension.