Institute Of Mathematiques Of Bourgogne

Bourgogne, France

Institute Of Mathematiques Of Bourgogne

Bourgogne, France
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Alexandrov A.,Albert Ludwigs University of Freiburg | Leurent S.,Imperial College London | Leurent S.,Institute Of Mathematiques Of Bourgogne | Tsuboi Z.,Australian National University | And 2 more authors.
Nuclear Physics B | Year: 2014

Following the approach of [1], we construct the master T-operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. We also characterize the class of solutions to the KP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum Gaudin model and the classical Calogero-Moser system of particles. © 2014 The Authors.

Klein C.,Institute Of Mathematiques Of Bourgogne | Sparber C.,University of Illinois at Chicago | Markowich P.,King Abdullah University of Science and Technology
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2014

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. © 2014 The Author(s). All rights reserved.

Nonnenmacher S.,CEA Saclay Nuclear Research Center | Sjostrand J.,Institute Of Mathematiques Of Bourgogne | Zworski M.,University of California at Berkeley
Communications in Mathematical Physics | Year: 2011

For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories. © 2011 The Author(s).

Gaillard P.,Institute Of Mathematiques Of Bourgogne | Gastineau M.,University Pierre and Marie Curie
Communications in Theoretical Physics | Year: 2016

The Peregrine breather of order eleven (P11 breather) solution to the focusing one-dimensional nonlinear Schrödinger equation (NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given: when all parameters are equal to 0 we recover the famous P11 breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the (x; t) plane, in function of the different parameters. © 2016 Chinese Physical Society and IOP Publishing Ltd.

Bonatti C.,Institute Of Mathematiques Of Bourgogne | Diaz L.J.,Pontifical Catholic University of Rio de Janeiro | Kiriki S.,Kyoto University of Education
Nonlinearity | Year: 2012

We consider diffeomorphisms f with heteroclinic cycles associated with saddles P and Q of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to f with a robust cycle associated with hyperbolic sets containing the continuations of P and Q. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized. © 2012 IOP Publishing Ltd & London Mathematical Society.

Bonnard B.,Institute Of Mathematiques Of Bourgogne | Cots O.,French Institute for Research in Computer Science and Automation | Pomet J.-B.,French Institute for Research in Computer Science and Automation | Shcherbakova N.,National Polytechnic Institute of Toulouse
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2014

The Euler-Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret-Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations. © EDP Sciences, SMAI, 2014.

Bonnard B.,Institute Of Mathematiques Of Bourgogne | Charlot G.,Institute Fourier | Ghezzi R.,International School for Advanced Studies | Janin G.,Institute Of Mathematiques Of Bourgogne
Journal of Dynamical and Control Systems | Year: 2011

We study the tangential case in two-dimensional almost-Riemannian geometry and analyze the connection with the Martinet case in sub-Riemannian geometry. We calculate estimates of the exponential map which allow us to describe the conjugate locus and the cut locus at a tangency point. We prove that this tangency point generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set. © 2011 Springer Science+Business Media, LLC.

Ferapontov E.V.,Loughborough University | Khusnutdinova K.R.,Loughborough University | Klein C.,Institute Of Mathematiques Of Bourgogne
Letters in Mathematical Physics | Year: 2011

We investigate (d + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d ≥ 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multidimensional integrable systems in question. In particular, our results imply the nonexistence of 2-component integrable systems of hydrodynamic type for d ≥ 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions. © 2011 Springer.

Bochi J.,University of Santiago de Chile | Bonatti C.,Institute Of Mathematiques Of Bourgogne | Diaz L.J.,Pontifical Catholic University of Rio de Janeiro
Communications in Mathematical Physics | Year: 2016

We give explicit C1-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a C1-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C1-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs. © 2016, Springer-Verlag Berlin Heidelberg.

Bonatti C.,Institute Of Mathematiques Of Bourgogne | Diaz L.J.,Pontifical Catholic University of Rio de Janeiro | Gorodetski A.,University of California at Irvine
Nonlinearity | Year: 2010

We prove that there is a residual subset S in Diff 1(M) such that, for every f ε S, any homoclinic class of f with invariant one-dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f .

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