Publications

2013

• Hodge theory of the middle convolution
  
  • Dettweiler Michael
  • Sabbah Claude
  Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2013, 49 (4), pp.761-800. We compute the behaviour of Hodge data by tensor product with a unitary rank-one local system and middle convolution by a Kummer unitary rank-one local system for an irreducible variation of polarized complex Hodge structure on a punctured complex affine line. We give applications of these formulas to local systems with G_2-monodromy. (10.4171/PRIMS/119)
  DOI : 10.4171/PRIMS/119
• ESTIMATES FOR SOLUTIONS OF A LOW-VISCOSITY KICK-FORCED GENERALISED BURGERS EQUATION
  
  • Boritchev Alexandre
  Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, Royal Society of Edinburgh, 2013, pp.143(2), 253-268. We consider a non-homogeneous generalised Burgers equation:$$∂u/∂t + f (u) ∂u/∂x − \nu ∂^2u/∂x^2 = η^ω , t ∈ R, x ∈ S^1 .$$Here, $\nu$ is small and positive, $f$ is strongly convex and satisfies a growth assumption, while $η^ω$ is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to $t ≥ 2$. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These estimates are of the form $C \nu^{−β}$ with the same values of $β$ for bounds from above and from below. They depend on $η$ and $f$ , but do not depend on the time $t$ or the initial condition.
• Sur la densite des representations cristallines du groupe de Galois absolu de Q_p
  
  • Chenevier Gaëtan
  Mathematische Annalen, Springer Verlag, 2013, 335, pp.1469-1525. Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of X_d, including the components made of residually irreducible representations. This extends to any dimension d previous results of Colmez and Kisin for d = 2. For this we construct an analogue of the infinite fern of Gouvêa-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline (phi,Gamma)-modules, as well as the density of the crystalline (phi,Gamma)-modules in this family. These results may be viewed as a local analogue of the theory of p-adic families of finite slope automorphic forms, they are new already in dimension 2. The technical heart of the paper is a collection of results about the Fontaine-Herr cohomology of families of trianguline (phi,Gamma)-modules.
• Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension
  
  • Côte Raphaël
  • Zaag Hatem
  Communications on Pure and Applied Mathematics, Wiley, 2013, 66 (10), pp.1541-1581. We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blow-up solution with a characteristic point, we refine the blow-up behavior first derived by Merle and Zaag. We also refine the geometry of the blow-up set near a characteristic point, and show that except may be for one exceptional situation, it is never symmetric with the respect to the characteristic point. Then, we show that all blow-up modalities predicted by those authors do occur. More precisely, given any integer $k\ge 2$ and $\zeta_0 \in \m R$, we construct a blow-up solution with a characteristic point $a$, such that the asymptotic behavior of the solution near $(a,T(a))$ shows a decoupled sum of $k$ solitons with alternate signs, whose centers (in the hyperbolic geometry) have $\zeta_0$ as a center of mass, for all times. (10.1002/cpa.21452)
  DOI : 10.1002/cpa.21452
• From Toda to KdV
  
  • Bambusi Dario
  • Kappeler Thomas
  • Paul Thierry
  Nonlinearity, IOP Publishing, 2013, 28, pp.2461-2496. 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 $L_N$.