Title: An Example of Catastrophic Self-Focusing in Nonlinear Optics?
Abstract: As the wavelength $\varepsilon$ goes to zero, the slowly varying envelope approximation allows one to replace the fields (solutions to Maxwell equations) with profile solutions to a nonlinear Schrödinger equation (NLS). Depending on the model, this equation may be critical and focusing, and then admits explosive solutions. In this case, the approximation breaks down, and, for $\varepsilon$ fixed, the fields may be globally defined in time, and smooth. This happens in the case of Maxwell--Bloch equations [P. Donnat and J. Rauch, Arch. Ration. Mech. Anal., 136 (1996), pp. 291--303], [E. Dumas, Existence globale pour les systèmes de Maxwell-Bloch, in Séminaire École Polytechnique, 2002--2003, Ecole Polytechnique, Palaiseau, France], of the anharmonic oscillator with saturated nonlinearity [J. L. Joly, G. Métivier, and J. Rauch, SIAM J. Math. Anal., 227 (1996), pp. 903--913], and of propagation in a ferromagnetic medium [J. L. Joly, G. Métivier, and J. Rauch, Ann. Henri Poincaré, 1 (2000), pp. 307--340], [H. Haddar, Modèles asymptotiques en ferromagnétisme: Couches minces et homogénéisation, Ph.D. thesis, thèse INRIA-École Nationale des Ponts et Chaussées, 2000]. We analyze the question of self-focusing for a wave equation in space dimension 2; the same techniques apply to usual models in greater dimensions. We give a new representation of the fields in terms of oscillating profiles, ruled by focusing rays. Furthermore, we prove that the approximation by an explosive solution of NLS is valid up to a time of the order of a negative power of $\ln(1/\varepsilon)$ before explosion; this exhibits an amplification of the fields by a positive power of $\ln(1/\varepsilon)$ between t=0 and that time.
Publication Year: 2003
Publication Date: 2003-01-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 1
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