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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 same time in analytic and Sobolev class., Comment: 60 pages, 1 figure

We consider the completely resonant non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s>1. We show the existence of solutions of this equation which achieve arbitrarily large growth of Hs Sobolev norms. We also give estimates for the time required to attain this growth.

We study the quintic nonlinear Schrödinger equation on a two-dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one discussed by Colliander et al. for the case of the cubic NLS. This requires an accurate combinatorial analysis.

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 prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.

We provide quasi-periodic solutions with two frequencies $\omega\in \mathbb R^2$ for a class of completely resonant non-linear wave equations in one and two spatial dimensions and with periodic boundary conditions. This is the first existence result for quasi-periodic solutions in the completely resonant case. The main idea is to work in an appropriate invariant subspace, in order to simplify the bifurcation equation. The frequencies, close to that of the linear system, belong to an uncountable Cantor set of measure zero where no small divisor problem arises.

We consider the class of Hamiltonians: [Formula: see text] where [Formula: see text], and the perturbing function f(q) is a rational function of eiq. We prove upper and lower bounds on the splitting for such class of systems, in regions of the phase space characterized by one fast frequency. Finally using an appropriate Normal Form theorem we prove the existence of chains of heteroclinic intersections.

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 an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups., 45 pages

In this paper we consider a class of fully nonlinear forced and reversible Schroedinger equations and prove existence and stability of quasi-periodic solutions. We use a Nash-Moser algorithm together with a reducibility theorem on the linearized operator in a neighborhood of zero. Due to the presence of the highest order derivatives in the non-linearity the classic KAM-reducibility argument fails and one needs to use a wider class of changes of variables such has diffeomorphisms of the torus and pseudo-differential operators. This procedure automtically produces a change of variables, well defined on the phase space of the equation, which diagonalizes the operator linearized at the solution. This gives the linear stability., 47 pages, 1 figure

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, 66 pages

In this note we present a new KAM result which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems., Comment: 13 pages

We discuss the stability of a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus described in a previous paper. The discussion is essentially combinatorial and algebraic in nature. Thus this paper contains the proof of two Theorems of algebraic, combinatorial and geometric nature, which we need in order to prove stability of certain solutions of the non--linear Schr\"odinger (NLS) equation on a torus.

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.

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for quasi-periodic solutions with Diophantine frequency vector converges. If one studies the Lindstedt series, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where KAM theorem is usually formulated, by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. We revisit Moser's theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates. The interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates., Comment: 38 pages, 18 fugures

We consider the one-dimensional nonlinear Schrodinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number.

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichlet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by means of a perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are not analytic but defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem., 29 pages 6 figures

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\f(u)$, with $\f(u)=\F u^{3} + O(u^{5})$ odd and analytic, $\F\neq0$, and we construct small amplitude periodic solutions with frequency $\o$ for a large Lebesgue measure set of $\o$ close to $1$. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and non-resonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = \f(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional.

We discuss normal forms of the completely resonant nonlinear beam equation and nonlinear Schr?dinger equation. We work in n > 1 spatial dimensions and study both periodic and Dirichlet boundary conditions on cubes. We discuss the applications to the problem of finding quasi-periodic solutions. In the case of periodic boundary and the dimension n = 2, we apply KAM theory and prove the existence and stability of quasi-periodic 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 wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation.

We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach., 60 pages 8 figures

We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in [26] , [27] and a generalization of the quasi-Toplitz functions defined in [31] .