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Nonlinear Pulse Propagation

Jeffrey Rauch — 2001

Journées équations aux dérivées partielles

This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.

Propagation de la régularité locale de solutions d'équations hyperboliques non linéaires

Patrick GérardJeffrey Rauch — 1987

Annales de l'institut Fourier

Pour tout réel positif s , on étudie la propagation de la régularité locale H s pour des solutions d’équations aux dérivées partielles hyperboliques non linéaires, admettant a priori la régularité minimale permettant de définir les expressions non linéaires figurant dans l’équation. En particulier, on démontre le théorème de propagation dans le cas des solutions essentiellement bornées (resp. lipschitziennes) de systèmes du premier ordre semi-linéaires (resp. quasi-linéaires).

Focusing of spherical nonlinear pulses in R. II. Nonlinear caustic.

Rémi CarlesJeffrey Rauch — 2004

Revista Matemática Iberoamericana

We study spherical pulse like families of solutions to semilinear wave equattions in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the L norm.

A Transmission Strategy for Hyperbolic Internal Waves of Small Width

Olivier GuesJeffrey Rauch

Séminaire Équations aux dérivées partielles

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source in the limit of thin layers. The key idea is to use a transmission problem...

Bérenger/Maxwell with Discontinous Absorptions : Existence, Perfection, and No Loss

Laurence HalpernJeffrey Rauch

Séminaire Laurent Schwartz — EDP et applications

We analyse Bérenger’s split algorithm applied to the system version of the two dimensional wave equation with absorptions equal to Heaviside functions of x j , j = 1 , 2 . The methods form the core of the analysis [11] for three dimensional Maxwell equations with absorptions not necessarily piecewise constant. The split problem is well posed, has no loss of derivatives (for divergence free data in the case of Maxwell), and is perfectly matched.

Global in Time Stability of Steady Shocks in Nozzles

Jeffrey RauchChunjing XieZhouping Xin

Séminaire Laurent Schwartz — EDP et applications

We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.

Coherent nonlinear waves and the Wiener algebra

Guy MétivierJean-Luc JolyJeffrey Rauch — 1994

Annales de l'institut Fourier

We study oscillatory solutions of semilinear first order symmetric hyperbolic system L u = f ( t , x , u , u ) , with real analytic f . The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in T , X with only the natural hypothesis of coherence. In the special case where L has constant coefficients and the phases are linear, the solutions have asymptotic description u ϵ = U ( t , x , t / ϵ , x / ϵ ) + o ( 1 ) where the profile U ( t , x , T , X ) is almost periodic in ( T , X ) . ...

Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc JolyGuy MétivierJeffrey Rauch

Séminaire Équations aux dérivées partielles

The nonlinear dissipative wave equation u t t - Δ u + | u t | h - 1 u t = 0 in dimension d > 1 has strong solutions with the following structure. In 0 t < 1 the solutions have a focusing wave of singularity on the incoming light cone | x | = 1 - t . In { t 1 } that is after the focusing time, they are smoother than they were in { 0 t < 1 } . The examples are radial and piecewise smooth in { 0 t < 1 }

Sharp Domains of Determinacy and Hamilton-Jacobi Equations

Jean-Luc JolyGuy MétivierJeffrey Rauch

Séminaire Équations aux dérivées partielles

If L ( t , x , t , x ) is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets Ω 0 { t = 0 } . The frozen constant coefficient operators L ( t ̲ , x ̲ , t , x ) determine local convex propagation cones, Γ + ( t ̲ , x ̲ ) . Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Ω which cannot be reached by influence curves beginning in the exterior of Ω 0 is a domain of...

A Simple Example of Localized Parametric Resonance for the Wave Equation

Colombini, FerruccioRauch, Jeffrey — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40. The problem studied here was suggested to us by V. Petkov. Since the beginning of our careers, we have benefitted from his insights in partial differential equations and mathematical physics. In his writings and many discussions, the conjuction of deep analysis and specially interesting problems has been a source inspiration for us. The research of J. Rauch is partially supported by the U.S. National Science Foundation under...

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