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26
Paradifferential calculus and applications to the Cauchy problem for nonlinear systems
, 2008
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Derivation of the Zakharov equations
, 2008
"... This paper continues the study, initiated in [28, 8], of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for wellprepared initial data. We apply this result to the EulerMaxwell equa ..."
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Cited by 18 (2 self)
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This paper continues the study, initiated in [28, 8], of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for wellprepared initial data. We apply this result to the EulerMaxwell equations describing laserplasma interactions, to obtain, in a highfrequency limit, an asymptotic estimate that describes solutions of the EulerMaxwell equations in terms of WKB approximate solutions which leading terms are solutions of the Zakharov equations. Because of transparency properties of the EulerMaxwell equations put in evidence in [28], this study is led in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of Joly, Métivier and Rauch [13, 14], recent results of Lannes on norms of pseudodifferential operators [15], and a semiclassical, paradifferential calculus.
A numerical model for the Raman amplification for laserplasma interaction
 Journal of computational and Applied Mathematics
"... Abstract: In this paper, we continue the study of the Raman amplification initiated in [3]. We use a dispersive, quasilinear system. The quasilinear part is not hyperbolic and this difficulty is overcomed using the dispersion. We give an asymptotic result on a reduced system. We then introduce a sim ..."
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Abstract: In this paper, we continue the study of the Raman amplification initiated in [3]. We use a dispersive, quasilinear system. The quasilinear part is not hyperbolic and this difficulty is overcomed using the dispersion. We give an asymptotic result on a reduced system. We then introduce a simple, robust and efficient numerical scheme on the whole system that takes into account the nonhyperbolicity of the quasilinear part as well as the nonlinear saturation of the Raman growth. The scheme is validated thanks to the asymptotic result. Finally, we present 1D and 2D simulations. 1 Introduction and statement of the result. 1.1 Position of the problem. The aim of this paper is to provide and vaildate a robust numerical method for the study of simulated Raman scattering in a plasma. The starting point is the model introduced for example in [7] that we have modified in [3]. This model
Instabilities in Zakharov equations for laser propagation in a plasma, in Phase space analysis of PDEs
 Progress in Nonlinear Differential equations and their Applications 69, Birkhäuser
, 2006
"... In [LPS], F.Linares, G.Ponce, JC.Saut have proved that a nonfully dispersive Zakharov system arising in the study of Laserplasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields nonvanishin ..."
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Cited by 4 (2 self)
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In [LPS], F.Linares, G.Ponce, JC.Saut have proved that a nonfully dispersive Zakharov system arising in the study of Laserplasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields nonvanishing at infinity, the system develops strong instabilities of Hadamard’s type, implying that the Cauchy problem is strongly illposed. 1
Small data blowup for a system of nonlinear Schrödinger equations
 J. Math. Anal Appl
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From the KleinGordon Zakharov system to a singular nonlinear Schrödinger system
, 2008
"... In this paper, we continue our investigation of the highfrequency and subsonic limits of the KleinGordonZakharov system. Formally, the limit system is the nonlinear Schrödinger equation. However, for some special case of the parameters going to the limits, some new models arise. The main object o ..."
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Cited by 3 (3 self)
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In this paper, we continue our investigation of the highfrequency and subsonic limits of the KleinGordonZakharov system. Formally, the limit system is the nonlinear Schrödinger equation. However, for some special case of the parameters going to the limits, some new models arise. The main object of this paper is the derivation of those new models, together with convergence of the solutions along the limits. Résumé Du système de KleinGordon Zakharov vers un système de Schrödinger nonlinéaire sigulier. Dans cet article, on continue l’investigation des limites haute fréquence et subsonique du système de KleinGordonZakharov. Formellement, le système limite est le système de Schrödinger nonlinéaire. Cependant, pour un cas particulier des paramètres, on trouve un nouveau modèle qui contient un terme sigulier. L’objet de ce papier est de donner une dèrivation rigoureuse de ce moèle et de montrer la
Dispersive Stabilization
"... Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing instability. This idea applies to systems of quasilinear Schröding ..."
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Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing instability. This idea applies to systems of quasilinear Schrödinger equations from nonlinear optics.
Nonlinear models for laserplasma interactions
, 2007
"... In this paper, we present a nonlinear model for laserplasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some wellposedness and illposedness result for some subsystems. ..."
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Cited by 3 (1 self)
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In this paper, we present a nonlinear model for laserplasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some wellposedness and illposedness result for some subsystems.