Zone d'oscillations rapides et singularités dans des systèmes dispersives, integrabilité et approches numériques – HOSDINA

The aim of this project is the combination of state of the art analytical and numerical methods to study nonlinear dispersive partial differential equations arising in different fields of mathematics and physics with highly oscillatory solutions. The analytical approach will use Riemann-Hilbert techniques with a steepest descent approach for integrable or almost integrable equations. This will lead to a construction of an asymptotic description of the oscillations in terms of Painlevé transcendents and multidimensional theta functions on the modular space of Riemann surfaces. Both Painlevé equations and modular functions will be studied within this project. The numerical approach will mainly use efficient spectral methods for the spatial coordinates and high order finite difference schemes and exponential integrators for the time integration. The necessary numerical tools are to be developed in this project and will be made generally available in the form of a program library. This will be complemented by an efficient numerical approach for the treatment of moduli spaces associated to compact Riemann surfaces. This code will allow the numerical study of modular functions on the moduli space of the associated Riemann surfaces as determinants of the Laplacian. With the efficient numerical methods of high accuracy a quantitative comparison of the rapid oscillations with the asymptotic solutions in the oscillatory zone will be possible. The numerical results will be used to develop asymptotic descriptions where none exist so far and will help in formulating conjectures which are to be proven in this project. One matrix models in random matrix theory which show oscillatory phenomena in the large N (size of the matrix) limit will be studied with a similar approach.