# Singularités en optique nonlinéaire: étude mathématique

Abstract : This thesis is aimed at studying two semilinear wave equations with cubic nonlinearity:

(NLCR) \Box u =2\, u^3,

and

(NLCC) \Box u +\alpha \,\frac(\pa u)(\pa z)=2\, u|u|^2+\beta \,u,

with $\alpha\in i\,\R$ et $\beta \in \R$.

The perimeter is a particular kind of space-like HS.
One prooves the existance of solutions blowing-up exactly on each prescribed manifold. To this purpose, one builds solutions using Fuchsian blabla...
The construction of those solutions provides informations about the behaviour of the solutions in the vicinity of the blow up.

First, one prooves, for several space-like hypersurfaces of a particular kind, the existence of solutions blowing-up exactly on each prescribed manifold. To this purpose, one builds solutions using Fuchsian reduction tools developped by S.~Kichenassamy and al. The construction of those solutions provides informations about their behaviour the vicinity of blow-up.

Afterwards, one took advantage of that obtained knowledge in order to partly study three problems:
i) What is the behaviour, near blow-up, of a particular integral built on the model of the so-called energy integral'' associated with the equation (NLCR) ?
ii) In which $L^p$ spaces does a solution of the equation (NLCR) --possibly with a small perturbation-- blow-up ?
iii) To what extent can we run a complete numerical study of the equation (NLCR), which would deal with the blow-up ?
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00008454
Contributor : Gilles Cabart <>
Submitted on : Friday, February 11, 2005 - 4:21:55 PM
Last modification on : Thursday, January 30, 2020 - 3:54:04 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:40:05 PM

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• HAL Id : tel-00008454, version 1

### Citation

Gilles Cabart. Singularités en optique nonlinéaire: étude mathématique. Mathématiques [math]. Université de Reims - Champagne Ardenne, 2005. Français. ⟨tel-00008454⟩

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