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PhD thesisI defended on Friday, May 15th, 2009 at ENS Cachan.PhD dissertationAbstract of the thesisStability of solitary wavesThis 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 KingdomFrédéric 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 PublicationsÉ.Canon, F.Chardard, G.Panasenko, O.Štikonienė, Asymptotics and discretization of a weakly singular
kernel: application to viscous flows in a network of thin
tubes, Journal of Computational Physics 491 112327 (2023).
É.Canon, F.Chardard, G.Panasenko, O.Štikonienė, Numerical solution of the viscous flows in a network of thin tubes: equations on the graph,
Journal of Computational Physics
435 110262 (2021).
F.Chardard, A.Elbert, G.Panasenko. Asymptotic analysis of the thin elastic plate-viscoelastic layer interaction,
Multiscale Modeling and Simulation 16 1258–1282 (2018). T.J.Bridges, F.Chardard. Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function, Nonlinearity 28 77-102 (2015). F.Chardard, F.Dias, T.J.Bridges. Computing the Maslov index of solitary waves. Part 2:Phase space with dimension greater than four,
Physica D 240 1334-1344 (2011). 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. On the Maslov index of multi-pulse orbits,
R. Soc. Lond. Proc. Ser. A 465 2897-2910 (2009).
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. Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves, C. R. Acad. Sci. Paris, Ser. I 345/12 689-694 (2007).
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) Preprints: F.Chardard, F.Dias, T.J.Bridges. Computational aspects of the Maslov index of solitary waves, hal-00383888. Academic databases and search enginesORCID Number: 0000-0001-7147-5001Mathscinet zbMATH Google Scholar ResearchGate |