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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.

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.

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.

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.

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.

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