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We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\mathbb{T},\mathbb{R})$, $s > -1/2$, to the scale of weighted $\ell^2-$sequence spaces, $\mathfrak{h}^{s +1/2}_{r,0}(\mathbb{N},\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations., Comment: arXiv admin note: substantial text overlap with arXiv:1812.05391

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform $\Phi$ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev space $H^{s}(\T, \R)$ of elements with mean $0$. As an application we show that for any $-1/2

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE., Comment: The title has been changed. Other minor changes in the text

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail., Comment: 20 pages, 4 figures

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction., Comment: 33 pages. The published version in Potential Analysis (2022) 57, 571--600 has some typos: Theorem 3(i): the first exponent should read $(m-2)/m$; Example 2 Line 2: .... $B(p_{n+1};cn^{-\beta})$.....; Formula (109): $\kappa^{-1}$ missing after the second inequality

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results., Comment: The paper is a greatly extended version of arXiv:1909.07314. In particular, we have included the result on ill-posedness of the Benjamin-Ono equation in $H^{-1/2}$ and added an appendix on the restriction of the Birkhoff map to the scale of Sobolev space $H^s$, $s>0$, and another one on a sharper form of ill-posedness of the Benjamin-Ono equation in $H^{s}_{r}$ with $s<-1/2$

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.

We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torus is globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,in the sense that the solution map, which is known to be defined for smooth data, continuously extends to $H^{-s}(\T,\R)$. The solution map does not extend continuously to $H^{-s}(\T, \R)$ with $s > 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin--Ono equation on the torus are orbitally stable in $H^{-s}(\T,\R)$ for any $0\le s<1/2$., Comment: This is an extended version of the paper submitted on September 16, 2019. It contains additional new results

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., Comment: The new version contains a revised introduction

In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are coordinates which allow to integrate it by quadrature and hence are also referred to as nonlinear Fourier coefficients. As a consequence, all the $L^2(\T)$ solutions of the Benjamin--Ono equation are almost periodic functions of the time variable. The construction of such coordinates relies on the spectral study of the Lax operator in the Lax pair formulation of the Benjamin--Ono equation and on the use of a generating functional, which encodes the entire Benjamin--Ono hierarchy., Comment: 59 pages This new version corrected some typos and updated the references

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.

We prove that there exist potentials so that near them the focusing non-linear Schr\"odinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.

Bounds are obtained for the $L^p$ norm of the torsion function $v_{\Omega}$, i.e. the solution of $-\Delta v=1,\, v\in H_0^1(\Omega),$ in terms of the Lebesgue measure of $\Omega$ and the principal eigenvalue $\lambda_1(\Omega)$ of the Dirichlet Laplacian acting in $L^2(\Omega)$. We show that these bounds are sharp for $1\le p\le 2$., Comment: 14 pages

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$, where $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space $\mathscr{F}\ell^{s,\infty}(\mathbb{T},\mathbb{R})$ with $-1/2 < s \le 0$ and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution $q$ in the Sobolev space $H^{-1}(\mathbb{T},\mathbb{R})$ to be in $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$ in terms of asymptotic behavior of spectral quantities of the Hill operator $-\partial_{x}^{2} + q$. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved., Comment: 45 pages. arXiv admin note: text overlap with arXiv:1502.05857

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.

We prove that the renormalized defocusing mKdV equation on the circle is locally in time $C^{0}$-wellposed on the Fourier Lebesgue space ${\mathcal{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is illposed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the NLS hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing to analytically extend them to ${\mathcal{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$., Comment: 33 pages

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on $H^{s}$ the KdV2 equation is $C^{0}$-wellposed if $s\ge 0$ and illposed (in a strong sense) if $s < 0$., Comment: 76 pages

We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (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 that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to $ 2 \times 2 $ block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.

In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2$\sigma$ + 1 on T^d. We prove that for $\sigma$d$\ge$2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3., Comment: 18 pages

In this paper we prove approximation properties of the solutions of the defoucsing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces $H^s$ with $s\geq 1$ remain bounded for all times.

In this paper we prove that the non-linear Fourier transform of the defocusing NLS equation on the circle is linear up to terms which are one order smoother.

We prove that the nonlinear part $H^{*}$ of the KdV Hamiltonian $H^{kdv}$, when expressed in action variables $I = (I_{n})_{n\ge 1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb N)$ of $\ell^{2}(\mathbb N)$ and is strictly concave near $0$. As a consequence, the differential of $H^{*}$ defines a local diffeomorphism near $0$ of $\ell_{\mathbb C}^{2}(\mathbb N)$., Comment: 48 pages

We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}-$close to the equilibrium and constructed by discretizing any given $C^2-$functions with mesh size $N^{-1}$. For such states we derive asymptotic expansions of the Toda frequencies $(\omega^N_n)_{0 < n < N}$ and the actions $(I^N_n)_{0 < n < N},$ both listed in the standard way, in powers of $N^{-1}$ as $N \to \infty$. %listed in accordance with the ordering of the frequencies at the equilibrium, %$(2 \sin \frac{n\pi} {N})_{0 < n < N}$. At the two edges $n \sim 1$ and $N -n \sim 1$, the expansions of the frequencies are computed up to order $N^{-3}$ with an error term of higher order. Specifically, the coefficients of the expansions of $\omega^N_n$ and $\omega^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials, $q_{-}$ respectively $q_+$, constructed in terms of the states considered. The frequencies $\omega^N_n$ for $n$ away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions $(I^N_n)_{0 < n < N},$ asymptotics of a similar nature are derived.

For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}$-close to the equilibrium and constructed by discretizing arbitrary given $C^2-$functions with mesh size $N^{-1}.$ Our aim is to describe the spectrum of the Jacobi matrices $L\_N$ appearing in the Lax pair formulation of the dynamics of these states as $N \to \infty$. To this end we construct two Hill operators $H\_\pm$ -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as $N \rightarrow \infty $ of the eigenvalues at the edges of the spectrum of $L\_N$ are of the form $\pm (2-(2N)^{-2} \lambda ^\pm \_n + \cdots )$ where $(\lambda ^\pm \_n)\_{n \geq 0}$ are the eigenvalues of $H\_\pm $. In the bulk of the spectrum, the eigenvalues are $o(N^{-2})$-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to $L\_N$.

In this paper we prove new qualitative features of solutions of KdV on the circle. The first result says that the Fourier coefficients of a solution of KdV in Sobolev space $H^N,\, N\geq 0$, admit a WKB type expansion up to first order with strongly oscillating phase factors defined in terms of the KdV frequencies. The second result provides estimates for the approximation of such a solution by trigonometric polynomials of sufficiently large degree.

For $M$ being a closed manifold or the Euclidean space we present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > 1/2\dim M + 1$.

We construct normalized differentials on families of curves of infinite genus. Such curves are used to investigate integrable PDE's such as the focusing Nonlinear Schr{\"o}dinger equation.

We consider the nonlinear Schrodinger equation with a cubic nonlinearity on the circle, which is known to represent an integrable Hamiltonian system. We construct a global coordinate systems, which puts this Hamiltonian into standard normal form, so that it depends only on the actions of the system., Comment: March 2009 version

We consider the large number of particles limit of a periodic Toda lattice for a family of initial data close to the equilibrium state. We show that each of the two edges of the spectra of the corresponding Jacobi matrices is up to an error, determined by the spectra of two Hill operators, associated to this family. We then show that the spectra of the Jacobi matrices remain almost constant when the matrices evolve along the two limiting KdV equations. Finally we prove that the Toda actions, when appropriately renormalized, converge to the ones of KdV .

We prove that the Birkhoff map $\Om$ for KdV constructed on $H^{-1}_0(\T)$ can be interpolated between $H^{-1}_0(\T)$ and $L^2_0(\T)$. In particular, the symplectic phase space $H^{1/2}_0(\T)$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\in H^{-1}(\T)$ in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$.

We study the exponential maps induced by Sobolev type right-invariant (weak) Riemannian metrics of order $k\ge1$ on the Lie group of smooth, orientation preserving diffeomorphisms of the circle. We prove that each of them defines an {\em analytic} Fr\'echet chart of the identity.

In 1974 P. Lax introduced an algebro-analytic mechanism similar to the Lax L-A pair. Using it we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schr{\"o}dinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes., Comment: 35 pages, new results about spectra and eigenfunctions of Schr\"odinger operators added, new references added

The Miura map (introduced by Miura) is a nonlinear map between function spaces which transforms smooth solutions of the modified Korteweg - de Vries equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this paper we study relations between the Miura map and Schroedinger operators with real-valued distributional potentials (possibly not decaying at infinity) from various spaces. We also investigate mapping properties of the Miura map in these spaces. As an application we prove existence of solutions of the Korteweg - de Vries equation in the negative Sobolev space H^{-1} for the initial data in the range of the Miura map., Comment: 33 pages