Curriculum VitaeLong version
Abstract of the thesisStability of solitary waves
This thesis is devoted to the stability of solitary waves, and more precisely to the applications of the Maslov index to the spectral stability problem. We show how the stability problem can be related to a family of linear Hamiltonian ODE. It is then possible to define a Maslov index for periodic waves and solitary waves. We compute the limit, when it exists, of the Maslov index of a sequence of periodic waves which converges to a solitary wave. We describe how exterior algebra can be used to compute the Maslov index, both in the periodic and solitary wave cases. We then use this framework for solitary waves and periodic waves arising in the Kawahara equation and for solitary waves arising in a longwave-shortwave interaction system. Lastly, we deal with the stability of stationary solutions of a model for flows over a non-uniform bottom by using a slightly different method.
CommitteeThomas J. BRIDGES, Professor, Surrey, United Kingdom
Frédéric DIAS, Professor, ENS Cachan
Christopher K.R.T. JONES, Professor, University of North Carolina at Chapel Hill, USA
Juan-Pablo ORTEGA, Chargé de recherches, CNRS
Jean-Claude SAUT, Professor, Paris XI
Nikolay TZVETKOV, Professor, Lille I
F.Chardard, F.Dias, H.-Y. Nguyen, J.-M. Vanden-Broeck. Stability of some stationary
solutions to the forced KdV equation with one or two bumps,
Journal of Engineering Mathematics 70 175-189 (2011).
F.Chardard, F.Dias, T.J.Bridges. Computing the Maslov index of solitary waves. Part 1: Hamiltonian systems on a 4−dimensional phase space,
Physica D 238 1841-1867 (2009).
F.Chardard, F.Dias, T.J.Bridges. Fast computation of the Maslov Index for hyperbolic linear systems with periodic coefficients, Journal of Physics A : Mathematical and General 39 14545-14557 (2006)
F.Chardard, A.Elbert, G.Panasenko. Asymptotic analysis of the thin elastic plate-viscoelastic layer interaction, submitted to Multiscale Modeling and Simulation.
F.Chardard, F.Dias, T.J.Bridges. Computational aspects of the Maslov index of solitary waves, hal-00383888.