Le jeudi 4 mai 2017 journée scientifique de l’équipe Mathématique-Physique de l’IMB.

Programme

- 13h :
**Kulkarni Giridhar**: Analysis of solutions to the Bethe ansatz equations for low-energy excited states of the 1D isotropic Heisenberg magnet and related quantities.- Diagonalisation of the Hamiltonian of quantum integrable systems using Bethe ansatz technique leads us to a system of coupled algebraic equations known as Bethe ansatz equations (BAE). In the case of isotropic Heisenberg magnet (XXX), it gives us simple rational functions when written in terms of the rapidity parameters. However, the solutions to these equations, known as Bethe roots are far from trivial. For Heisenberg models, this situation is further complicated by the presence of bound-state excitations (solutions with complex rapidity parameters). It is widely assumed that these complex rapidities must eventually condense into ’string’ configurations in the thermodynamic limit, a premise that is known as the ’string hypothesis’. Although there exist notable but extreme examples of solutions which clearly defy this rule, it’s adoption makes it easier for us to analyse quantities such as asymptotic distribution (density function) of the roots of the excited states and the density of shift in the rapidities of any low-energy excitation from that of the vacuum. Quite incidentally, these are the quantities which play an essential role in the computation of the form-factors. In this talk, I will discuss the specific role of these quantities in the computation and derive the functional expressions describing their asymptotic behaviour under the pretext of the string hypothesis.

- 13h30 :
**Jaber Carine**: Siegel’s fundamental domain and Riemann theta functions.- To a Riemann surface one can associate a Riemann matrix and Riemann theta functions. Theta functions appear in the expression of solutions of integrable systems. Such solutions are relevant in the description of surface water waves, non linear optics, etc. Because of these applications, Deconinck and Van Hoeij have developed and implemented algorithms for computing the Riemann matrix and Deconinck et al the corresponding theta functions. An important step in the efficient computation of these functions is the construction of appropriate symplectic transformations for a given Riemann matrix assuring a rapid convergence of the theta functions. Here, the LLL algorithm used by Deconinck et al is replaced by an exact Minkowski reduction for small genus and an exact identification of the shortest lattice vector for larger values of the genus. I present also some new results concerning Siegel’s fundamental domain for genus 3.

- 14H :
**Mignard Michael**: On the classifiaction of twisted quantum doubles.- Fusion categories are semi-simple tensor categories with finitely many isomorphism classes of simple objects, and such that the endomorphism space of a simple

object is of dimension 1. It is also called braided if there exists isomorphism constraints for commutativity of the tensor product. An example of those categories

is the one of representations of twisted quantum double of a finite group, first introduced in the context of orbifold conformal field theory. Using invariants of

braided fusion categories -mostly Frobenius-Schur indicators-, Mason, Goff and Ng showed that one can obtain a classification of twisted doubles for groups of a given

order under gauge equivalences. In this talk, we will recall the indicators in the general case of fusion categories and discuss the case of twisted doubles

for groups of order 16.

- Fusion categories are semi-simple tensor categories with finitely many isomorphism classes of simple objects, and such that the endomorphism space of a simple
- 14h30 : pause
- 15h :
**Assainova Olga**: The inverse scattering method for the Davey-Stewartson equation.- The inverse scattering method is arguably the most powerful method for solving integrable non-linear partial differential equations. Some examples of the implementation of this technique will be presented in this talk. We will discuss the 1+1 dimensional case at the example of the Korteweg-de Vries equation describing waves on shallow water, and a generalization of this method to the 2+1 dimensional case for the Davey-Stewartson equation, which appears in hydrodynamics and nonlinear optics. Whereas the case of one spatial dimension leads to a so-called Riemann-Hilbert problem, in two spatial dimensions a so-called ’dbar problem’ has to be solved.

- 15h30 :
**Stoilov Nikola**: Numerical study of gradient catastrophe stability in Integrable dispersive equations.- Nonlinear dispersive partial differential equations such as the family of non-linear Schrödinger equation posses solutions that develop a singularity in finite time. We numerically study the long time behaviour and potential gradient catastrophe of solutions to the focusing Davey-Stewartson II equation by analysing perturbations of the lump and the Ozawa exact solutions as well as evolution of Gaussian initial data. We demonstrate that the lump is unstable so perturbations either cause it to blow-up or disperse, whereas the blow-up in the Ozawa solution is generic and they posses their respective blow up rates. We will also discuss the numerical implementation on Graphical Processing Units and their use for general high-performance computing.

- 16h30 : colloquium de N. Tzvetkov (Cergy-Pontoise)