About nonlinear geometric optics E Dumas

Description

Niveau: Supérieur, Doctorat, Bac+8
About nonlinear geometric optics E. Dumas Institut Fourier, UMR 5582 (CNRS-UJF) 100 rue des Mathematiques Domaine Universitaire BP 74, 38402 Saint Martin d'Heres - France email: Abstract We give an idea of the evolution of mathematical nonlinear geomet- ric optics from its foundation by Lax in 1957, and present applications in various fields of mathematics and physics. 1 Introduction Geometric optics goes back at least to the XVIIth Century, with Fermat, Snell and Descartes, who described the “paths” (rays) followed by the light. Nowadays, Physics tells us that we may reasonably replace the waves from Quantum Mechanics with classical particles, in the semi-classical approx- imation (when considering Planck's constant ~, or the wavelength, as in- finitely small). The mathematical transcription of these problems consists in studying the asymptotic behavior of solutions to partial differential equations where different scales (represented by small parameters) are present, often in a high frequency oscillatory context. We present the first historical results of the field, and then review some extensions and applications of the method. We shall see how geometric optics applies to Maxwell's equations (from optics, ferromagnetism, . . . ), to the wave or Klein-Gordon equation, to fluid dynamics and plasma physics, to general hyperbolic systems and conservation laws, as well as to nonlinear Schrodinger equations, among others.

• phase-amplitude represen- tation


• profile equations


• optics goes


• geometric optics


• weakly nonlinear


• semi-classical approx- imation


• equations takes

Sujets

• Geometrical optics

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 13
• Langue: English

Extrait

1

About

nonlinear geometric

E. Dumas

optics

Institut Fourier, UMR 5582 (CNRS-UJF) 100ruedesMathematiques Domaine Universitaire BP74,38402SaintMartind’Heres-France email: [e-mail]

Abstract We give an idea of the evolution of mathematical nonlinear geomet- ric optics from its foundation by Lax in 1957, and present applications invariouseldsofmathematicsandphysics.

Introduction

Geometric optics goes back at least to the XVIIth Century, with Fermat, Snell and Descartes, who described the “paths” (rays) followed by the light. Nowadays, Physics tells us that we may reasonably replace the waves from Quantum Mechanics with classical particles, in the semi-classical approx- imation (when considering Planck’s constant~, or the wavelength, as in- nitely small). The mathematical transcription of these problems consists in studying the asymptotic behavior of solutions to partial dieren tial equations where dieren t scales (represented by small parameters) are present, often in a high frequency oscillatory context. Wepresentthersthistoricalresultsoftheeld,andthenreviewsome extensions and applications of the method. We shall see how geometric optics applies to Maxwell’s equations (from optics, ferromagnetism, . . . ), to thewaveorKlein-Gordonequation,touiddynamicsandplasmaphysics, to general hyperbolic systems and conservation laws, as well as to nonlinear Schrodingerequations,amongothers.Furthermore,weshallseethatitdoes not apply to oscillatory problems only, but also to boundary layers, shocks and long waves problems.

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Contents

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Introduction

First steps 2.1 Linear geometric optics . . . . . . . . . . . . . . . . . . . . . . 2.2 Weakly nonlinear geometric optics . . . . . . . . . . . . . . . . 2.3 Prole equations . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Rigorous results . . . . . . . . . . . . . . . . . . . . . . . . . .

Other kinds of proles

Caustics

Boundary problems 5.1 Re ection on a boundary . . . . . . . . . . . . . . . . . . . . . 5.2 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Three (and more) scale expansions: diractiv e optics 18 6.1 Some three-scale problems . . . . . . . . . . . . . . . . . . . . 18 6.2 Long-time behavior . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 Transparency and larger amplitudes . . . . . . . . . . . . . . . 24

7 Long waves 27 Quoting some reviews and introductory texts may be useful. A review onnonlineargeometricopticsin1998isduetoJoly,MetivierandRauch [JMR99], and the online book of Rauch [Rau96] is a nice introduction to the subject. Majda [Maj84] made major contributions in the 80ies. Good Physics textbooks on nonlinear optics are Newell and Moloney [NM92] and Boyd [Boy03]. Kalyakin [Kal89] and Hunter [Hun95] review many questions, methods and applications about these multi-scale problems, and Whitham [Whi99] had pioneering contributions in the modeling of nonlinear waves.

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2 First

steps

2.1 Linear geometric optics

The rst rigorous result in mathematical geometric optics is due to Lax [Lax57], who shows that strictly hyperbolic systems admitWKB(for Wentzel, Kramers and Brillouin) solutions. These have thephase-amplituderepresen- tation (t,x)/ε u ε (t, x) =a ε (t, x)e iϕ .

The wavelength corresponds to the small parameterε >0, and the amplitude a ε admits an asymptotic (in general not convergent) expansion, a ε  X ε n a n (t, x),asε→0. n0

In particular, Lax uses such solutions to study the well-posedness of the Cauchy problem. The geometric information is contained in theeikonal equation(of Hamilton- Jacobi type) determining the phaseϕ(plane waves correspond to phases which are linear in (t, x); spherical waves, to phases which are functions oft and|x|only). The amplitudesa n are solutions totransport equationsalong the rays associated with the eikonal equation. The asymptotic expansion is valid for times before rays focus.

2.2 Weakly nonlinear geometric optics

Trying to generalize Lax’s approach to nonlinear systems, one immediately faces several problems. First, a family  a(x)e iϕ(x)/ε  ε>0 is not bounded in any Sobolev spaceH s withs >0, so that the time of existence of the considered exact solutionu ε may shrink to zero asε the Ingoes to zero. same spirit, the size ofu ε must be adapted, large enough to allow nonlinear features at rst order, but sucien tly small to prevent blow-up. Finally, themaininterestanddicultyofnonlinearmodelscomesfrominteractions: one hopes to incorporate in the asymptotic description the self-interaction of a wave (leading for example to generation of harmonics) as well as the possibility of (resonant) interaction of several waves. Thus, the extension to nonlinear systems goes througholesrpU n (t, x, )

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which are periodic or almost periodic w.r.t.∈R q , (1)u ε (t, x) +ε m X ε n U n (~,x,ϕt(t, x)/ε), n0

where is a given groundstate, and~ϕ= (ϕ 1 , . . . , ϕ q ) is a collection of phases. The usual strategy of nonlinear geometric optics consists in: 1-dening a formalsolution,i.e.rgpnhotesle;rsesrntonfutarceihteequatioisesolvabl in(1)denesanapproximatesolution(afunctionapproximatelysolutionto the equation); 2-showing that for any initial data close to the initial value of the approximate solution, an exact solution exists on a time interval indepen- dent ofε; 3(stability)-showing that the exact solution is well approximated by the formal one. In order to observe some nonlinear behavior, the magnitudeε m of oscilla- tionsischosensothatcumulatedeectsofnonlinearitiesbecomeofthesame order as the wave on the typical timeT xed Aof propagation.T(w.r.t. ε) leads tononlinear geometric optics rescaled, the system of partial. Once dieren tial equations takes the form (2)L(t, x, u ε , ε∂)u ε =F(t, x, u ε ),

where the operatorLis in general a rst order symmetric hyperbolic system on some domain inR t 1 ,x +d ,

d (3) L(t, x, u, ε∂)u=ε∂ t u+ j X =1 A j (t, x, u)ε∂ x j u ε +L 0 (t, x)u =:L 1 (t, x, u, ε∂)u+L 0 (t, x)u. The smooth functionsF,BandA j take values inC N , in the space ofN Nmatrices, and in the space ofNNsymmetric matrices, respectively. Furthermore, we will distinguish the hyperbolic case, whenL 0 = 0, and the dispersive case, whenL 0 is skew-symmetric (the dissipative case, when L 0 +L 0 ? 0, will be considered only in Section 7). This coecien tL 0 re ects the interaction between the wave and the material medium, so that, in the dispersive case, the group velocity (see (9)) depends on the frequency. The abovementioned exponentm as follows.is then dened LetJ2 be the order of nonlinearities,  || J 2⇒∂ u A j | u=0 = 0,|| J 1⇒∂  u F | u=0 = 0.

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