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This paper has various goals: first, we develop a local and global well-posedness theory for the regularized Benjamin-Ono equation in the periodic setting, second, we show that the Cauchy problem for this equation (in both periodic and non-periodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices, third, a proof of the existence of a smooth curve of periodic travelling wave solutions, for the regularized Benjamin-Ono equation, with fixed minimal period 2L, is given. It is also shown that these solutions are nonlinearly stable in the energy space $H^{1/2}_{per}$ by perturbations of the same wavelength. Finally, an extension of the theory developed for the regularized Benjamin-Ono equation is given and as an example it is proved that the cnoidal wave solutions associated to the Benjamin-Bona-Mahony equation are nonlinearly stable in $H^1_{per}$.

We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.

We continue our study of the well-posedness theory of a one-parameter family of coupled KdV-type systems in the periodic setting. When the value of a coupling parameter \alpha \in (0, 4) \setminus 1, we show that the Gibbs measure is invariant under the flow and the system is globally well-posed almost surely on the statistical ensemble, provided that certain Diophantine conditions are satisfied., Comment: 19 pages. To appear in Diff. Integ. Eqns

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \hat{b}^s_{p, \infty}, sp <-1, contains the support of the white noise. Then, we prove local well-posedness in \hat{b}^s_{p, \infty} for p= 2+, s = -{1/2}+ such that sp <-1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko., Comment: 18 pages. To appear in Comm. Math. Phys

We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter \alpha when \alpha \ne 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local well-posedness of the system in H^s(\mathbb{T}_\lambda), s \geq s^\ast, where s^\ast = s^\ast(\alpha) \in ({1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter \alpha. We also present a sharp local (and global) result in L^2(\mathbb{R}). In the appendix, we briefly discuss the local well-posedness result in H^{-{1/2}}(\mathbb{T}_\lambda) for \alpha= 1 without the mean 0 assumption, by introducing the vector-valued X^{s, b} spaces., Comment: 31 pages. To appear in Internat. Math. Res. Not. "Global Theory" is available in Electron. J. Diff. Eqns

The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation $$u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3,$$ is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces., Comment: 8 pages

We study a family of equations defined on the space of tensor densities of weight $\lambda$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all $\lambda \neq 0,1$, and "shock-peakons" for $\lambda = 3$. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups., Comment: 30 pages

Many "real" inverse spectral data for periodic finite-gap operators (consisting of Riemann Surface with marked "infinite point", local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators cannot be considered as self-adjoint in the ordinary (positive) Hilbert spaces of functions of x. In particular, it is true for the special case of Lame operators with elliptic potential $n(n+1)\wp(x)$ where eigenfunctions were found in XIX Century by Hermit. However, such Baker-Akhiezer (BA) functions present according to the ideas of works by Krichever-Novikov (1989), Grinevich-Novikov (2001) right analog of the Discrete and Continuous Fourier Bases on Riemann Surfaces. It turns out that these operators for the nonzero genus are symmetric in some indefinite inner product, described in this work. The analog of Continuous Fourier Transform is an isometry in this inner product. In the next work with number II we will present exposition of the similar theory for Discrete Fourier Series, Comment: LaTex, 30 pages In the updated version: 3 references added, extensions of the x-space with indefinite metric and the analysis of the Lame potentials are described in more details, relations with Crum transformations are discussed. Discussion of degenerate cases (hyperbolic and trigonometric) and Crum-Darboux transformations is added. Additional reference was added

We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures., Comment: 12 pages; v2: the content isn't changed, submission metadata fixed; v3: a minor correction

In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlev\'e II equation. We prove our results using the Riemann-Hilbert approach., Comment: 27 pages

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq \alpha \leq 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \cite{MST}. Moreover, as a bi-product we prove that if $0<\alpha \leq 1$ the corresponding modified equation (with the nonlinearity $\pm uuu_x$) is locally well-posed in $H^s$ for $s\geq 1/2-\alpha/4$., Comment: 33 pages

Integrable PDEs on the line can be analyzed by the so-called Inverse Scattering Transform (IST) method. A particularly powerful aspect of the IST is its ability to predict the large $t$ behavior of the solution. Namely, starting with initial data $u(x,0)$, IST implies that the solution $u(x,t)$ asymptotes to a collection of solitons as $t \to \infty$, $x/t = O(1)$; moreover the shapes and speeds of these solitons can be computed from $u(x,0)$ using only {\it linear} operations. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value (IBV) problems formulated on the half-line. It can be shown that again $u(x,t)$ asymptotes into a collection of solitons, where now the shapes and the speeds of these solitons depend both on $u(x,0)$ and on the given boundary conditions at $x = 0$. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a {\it nonlinear} Volterra integral equation. However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyze three different types of linearizable boundary conditions. For these cases, the initial conditions are: (a) restrictions of one and two soliton solutions at $t = 0$; (b) profiles of certain exponential type; (c) box-shaped profiles. For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary., Comment: Corrected typos.

We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\R)$. The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at $s=-3/4$ by constructing some special resolution spaces in order to avoid some 'logarithmic divergence' from the high-high interactions. Our local solution has almost the same properties as those for $H^s (s>-3/4)$ solution which enable us to apply the I-method to extend it to a global solution., Comment: 20 pages, 0 figure

In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint works with L.Escauriaza, G.Ponce and L.Vega., Comment: The paper is based on lectures presented at WHAPDE 2008, Merida, Mexico. To appear in Contemp. Math. volume in honor of V.Mazya

Considering the Cauchy problem for the modified finite-depth-fluid equation $\partial_tu-\G_\delta(\partial_x^2u)\mp u^2u_x=0, u(0)=u_0$, where $\G_\delta f=-i \ft ^{-1}[\coth(2\pi \delta \xi)-\frac{1}{2\pi \delta \xi}]\ft f$, $\delta\ges 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 \in H^s (s\geq 1/2)$ with $\norm{u_0}_{L^2}$ sufficiently small for all $\delta \ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $\delta$ tends to $+\infty$., Comment: 29 pages, 0 figures

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u=2(u^{3})_x, u(0)=\phi$, where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\geq1)$ for all $\epsilon \in (0,1]$. Moreover, we prove that for any $s\geq 1$ and $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the MKdV equation if $\epsilon$ tends to 0., Comment: 19 pages

In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality., Comment: 34 pages.A sign error is corrected throught the paper; the statement of Theorem 1.9 is slightly modified since version 3; add a new reference in remark 1.7; some improvement in exposition. To appear Analysis and PDE

We make use of the method of modulus of continuity \cite{K-N-S} and Fourier localization technique \cite{A-H} to prove the global well-posedness of the critical Burgers equation $\partial_{t}u+u\partial_{x}u+\Lambda u=0$ in critical Besov spaces $\dot{B}^{\frac{1}{p}}_{p,1}(\mathbb{R})$ with $p\in[1,\infty)$, where $\Lambda=\sqrt{-\triangle}$., Comment: 21pages

We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a K\"ahler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially., Comment: 23pages

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves)., Comment: submitted for publication. Corrected typo in Theorem 1.1

This note proves the orbital stability in the energy space $H^{1/2}$ of the sum of widely-spaced 1-solitons for the Benjamin-Ono equation, with speeds arranged so as to avoid collisions.

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+\epsilon |\partial_x|^{2\alpha}u+(u^2)_x=0, \ u(0)=\phi, \end{eqnarray*} where $0<\epsilon,\alpha\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\ (s>s_\alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $\epsilon \in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $\epsilon$ tends to 0., Comment: 35 pages, 0 figure

The initial- and boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator $\partial_t +\alpha \partial_x -\gamma\partial_{xxt}$ defined in the space-time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase.

Critical processes of ideal integrable models of Hele-Shaw flows are considered. A regularization method based on multiscaling expansions of solutions of the KdV and Toda hierarchies characterized by string equations is proposed. Examples are exhibited in which the tritronq'ee solution of the Painleve-I equation turns out to provide the leading term of the regularization, Comment: 13 pages, 3 figures

In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-B\"acklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found., Comment: 11 pages

We continue our study of the collision of two solitons for the subcritical generalized KdV equations. In a previous paper, mainly devoted to the case of the quartic gKdV equation, we have introduced a new framework to understand the collision of two solitons in the case where one soliton is small with respect to the other. In this paper, we consider the case of a general nonlinearity $f(u)$ for which the two solitons are nonlinearly stable. We prove that in this situation the two solitons survive and we describe the collision at the main orders.

This paper concerns the problem of collision of two solitons for the quartic generalized Korteweg-de Vries equation. We introduce a new framework to describe the collision in the special case where one soliton is small with respect to the other. We prove that the two soliton survive the collision, we describe the collision phenomenon (computation of the first order of the resulting shifts on the solitons). Moreover, we prove that in this situation, there does not exist pure two-soliton solutions.

One of the difficulties associated with the scattering problems arising in connection with integrable systems is that they are frequently non-self-adjoint, making it difficult to determine where the spectrum lies. In this paper, we consider the problem of locating and counting the discrete eigenvalues associated with the scattering problem for which the sine-Gordon equation is the isospectral flow. In particular, suppose that we take an initially stationary pulse for the sine-Gordon equation, with a profile that has either one extremum point of height less than pi and topological charge 0, or is monotone with topological charge +-1. Then we show that the point spectrum lies on the unit circle and is simple. Furthermore, we give a count of the number of eigenvalues. This result is an analog of that of Klaus and Shaw for the Zakharov-Shabat scattering problem. We also relate our results, as well as those of Klaus and Shaw, to the Krein stability theory for symplectic matrices. In particular we show that the scattering problem associated to the sine-Gordon equation has a symplectic structure, and under the above conditions the point eigenvalues have a definite Krein signature, and are thus simple and lie on the unit circle., Comment: 25 Pages,1 figure

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real s., Comment: 27 pages

The I-method in its first version as developed by Colliander et al. is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H^s, provided s>-1/42., Comment: 6 pages

This paper is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into K\"ahler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method., Comment: 25pages, final version

Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\alpha}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq \alpha\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For $s>-\min\{\frac {3+2\alpha}4, 1\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\frac 12\le\alpha\le 1$ and for $s<-\min\{\frac {3+2\alpha}4, 1\}$ we show some ill-posedness issues., Comment: 20 pages

Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct formally these series by mean of generating functions based on an algebraic framework inspired by the construction of Fock spaces in quantum field theory. Then we build an analytic setting in which all these constructions really make sense and give rise to convergent series., Comment: 38 pages, 1 figure, v2: some sign conventions have been changed, some minor corrections has been added

Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP) hierarchy, we start with a quite general hierarchy of linear ordinary differential equations in a space of matrices and derive from it a matrix Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly nonassociative' (WNA) algebra structure, from which we can conclude, refering to previous work, that any solution of the Riccati system also solves the potential KP hierarchy (in the corresponding matrix algebra). We then turn to the case where the components of the matrices are multiplied using a (generalized) star product. Associated with the deformation parameters, there are additional symmetries (flow equations) which enlarge the respective KP hierarchy. They have a compact formulation in terms of the WNA structure. We also present a formulation of the KP hierarchy equations themselves as deformation flow equations., Comment: 25 pages

We study the Cauchy problem for the modified KdV equation for data u_0 in the space ^H^r_s defined by the norm ||u_0||_{^H^r_s}:=||<\xi>^s u^_0||_{L^r'_\xi}. Local well-posedness of this problem is established in the parameter range 2>=r>1, s>=1/2-1/2r, so the case (s,r)=(0,1), which is critical in view of scaling considerations is almost reached. To show this result, we use an appropriate variant of the Fourier restriction norm method as well as bi- and trilinear estimates for solutions of the Airy equation., Comment: 12 pages

Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of time deformations for each weight, the multi-component KP-hierarchy: these determinants are thus "tau-functions" for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. As an application, the multi-component KP-hierarchy leads to a large set of non-linear PDE's, which are useful in finding partial differential equations for the transition probabilities of certain infinite-dimensional diffusions., Comment: 37 pages, 1 figure

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on $\mathbb T$. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain., Comment: 6 pages

A linear Weingarten surface in Euclidean space ${\bf R}^3$ is a surface whose mean curvature $H$ and Gaussian curvature $K$ satisfy a relation of the form $aH+bK=c$, where $a,b,c\in {\bf R}$. Such a surface is said to be hyperbolic when $a^2+4bc<0$. In this paper we classify all rotational linear Weingarten surfaces of hyperbolic type. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ${\bf R}^3$ that consists into periodic surfaces with self-intersections., Comment: 15 pages, 4 figures

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrodinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg--de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved H^s norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions., Comment: 16 pages. Final version with minor revisions

We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering result for the $L^2$-critical nonlinear Schr\"odinger equation $iu_t + u_{xx} = \mu |u|^4 u$. Secondly, in the defocusing case $\mu > 0$ we present a new dispersion estimate which asserts, roughly speaking, that energy moves to the left faster than the mass, and hence strongly localised soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times., Comment: 16 pages, no figures. A footnote is corrected

Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the $xt$-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schr\"odinger equation. In the reflectionless case such solutions reduce to pure $N$-soliton solutions. An illustrative example is provided., Comment: 17 pages, no figures

We show that the quartic generalised KdV equation $$ u_t + u_{xxx} + (u^4)_x = 0$$ is globally wellposed for data in the critical (scale-invariant) space $\dot H^{-1/6}_x(\R)$ with small norm (and locally wellposed for large norm), improving a result of Gr\"unrock. As an application we obtain scattering results in $H^1_x(\R) \cap \dot H^{-1/6}_x(\R)$ for the radiation component of a perturbed soliton for this equation, improving the asymptotic stability results of Martel and Merle., Comment: 28 pages, no figures, submitted, J. Diff. Eq

It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.

We prove that the flow map of the Kadomtsev-Petviashvili-I (KP-I) equation is not uniformly continuous on bounded sets of the natural energy space., Comment: 14 pages

In this paper we study uniqueness properties of solutions of the k-generalized Korteweg-de Vries equation. Our goal is to obtain sufficient conditions on the behavior of the difference $u_1-u_2$ of two solutions $u_1, u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee that $u_1\equiv u_2$.

We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in $ H^1(\T) $. This leads to a global well-posedness result in $ H^1(\T)$ for the Benjamin-Ono equation., Comment: 27 pages

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on $ H^\infty(\T) $) cannot be of class $ C^{1+\alpha} $, $\alpha>0 $, from $ {\dot H}^s(\T) $ into $ {\dot H}^s(\T) $ as soon as $ s< 0 $., Comment: 47 pages