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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 STABILITY CRITERION FOR HIGHFREQUENCY OSCILLATIONS
, 2014
"... Abstract. We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initialvalue problems issued from highlyoscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase a ..."
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Abstract. We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initialvalue problems issued from highlyoscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition. The analysis relies, in the unstable case, on a Duhamel representation formula for solutions of zerothorder pseudodifferential equations. Our examples include coupled KleinGordon systems, and systems describing Raman and Brillouin instabilities.
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
Semidiscretization in time for Nonlinear Zakharov Waves Equations
 in "Discrete and Continuous Dynamical Systems: Series B
"... In this paper we construct and study discretizations of a nonlinear Zakharovwave system occurring in plasma physics. These systems are generalizations of the Zakharov system that can be recovered by taking a singular limit. We introduce two numerical schemes that take into account this singular li ..."
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In this paper we construct and study discretizations of a nonlinear Zakharovwave system occurring in plasma physics. These systems are generalizations of the Zakharov system that can be recovered by taking a singular limit. We introduce two numerical schemes that take into account this singular limit process. One of the scheme is conservative but sensible to the polarization of the initial data while the other one is able to handle illprepared initial data. We prove some convergence results and we perform some numerical tests.
TWO ASYMPTOTIC PROBLEMS FOR A SINGULAR NONLINEAR SCHRÖDINGER SYSTEM
"... Abstract. In this paper, we continue our investigation of the relation between various systems that can be derived from the KleinGordonZakharov system in the highfrequency and subsonic limits. In this paper we start from the singular nonlinear Schrödinger system which was derived in a previous wo ..."
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Cited by 1 (1 self)
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Abstract. In this paper, we continue our investigation of the relation between various systems that can be derived from the KleinGordonZakharov system in the highfrequency and subsonic limits. In this paper we start from the singular nonlinear Schrödinger system which was derived in a previous work and derive the classical nonlinear Schrödinger system in two different limit cases. These two nonlinear Schrödinger systems have different coefficients in the nonlinearity.
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, 2008
"... Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to timeperiodic galloping solutions ..."
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Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to timeperiodic galloping solutions