Nonlinear Maxwell Equations in Inhomogeneous Media

Babin, Anatoli; Figotin, Alexander

doi:10.1007/s00220-003-0939-9

Cited by 12 publications

(31 citation statements)

References 29 publications

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“…We will close this gap by providing a complete local wellposedness theory. We will prove that (1) the system (1.6) has a unique maximal solution u in the function space m j=0 C j ((T − , T + ), H m−j (G)) for all m ∈ N with m ≥ 3 provided the data are sufficiently regular and compatible (in the sense of (2.10) below) with the material law, (2) finite existence time can be characterized by blowup in the Lipschitz-norm, (3) the solution depends continuously on the data.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

141

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In this article we develop the local wellposedness theory for quasilinear Maxwell equations in H m for all m ≥ 3 on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in H 3 is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in H m with m ≥ 3 of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.2010 Mathematics Subject Classification. 35L50, 35L60, 35Q61. Key words and phrases. nonlinear Maxwell equations, perfectly conducting boundary conditions, quasilinear initial boundary value problem, hyperbolic system, local wellposedness, blow-up criterion, continuous dependance. 1 2 MARTIN SPITZThe macroscopic Maxwell equations in a domain G readfor all t ≥ t 0 . In (1.1) we consider the Maxwell system with the boundary conditions of a perfect conductor, where ν denotes the outer normal unit vector of G. System (1.1) has to be complemented by constitutive relations between the electric fields and the magnetic fields, the so called material laws. We choose E and H as state variables and express D and B in terms of E and H. The actual form of these material laws is a question of modelling and different kinds have been considered in the literature. The so called retarded material laws assume that the fields D and B depend also on the past of E and H, see [2] and [33] for instance. In dynamical material laws the material response is modelled by additional evolution equations for the polarization or magnetization, see e.g.[1], [11], [16], or [17]. In this work we concentrate on the instantaneous material laws, see [6] and [14]. Here the fields D and B are given as local functions of E and H, i.e., we assume that there are functions θ 1 , θ 2 : G × R 6 → R 3 such that D(t, x) = θ 1 (x, E(t, x), H(t, x)) and B(t, x) = θ 2 (x, E(t, x), H(t, x)). The most prominent example is the so called Kerr nonlinearity, wherewith ϑ : G → R 3×3 and the vacuum permittivity and permeability set equal to 1 for convenience. We further assume that the current density decomposes as J = J 0 + σ 1 (E, H)E, where J 0 is an external current density and σ 1 denotes the conductivity. If we insert these material laws into (1.1) and formally differentiate, we obtainfor the evolutionary part of (1.1). The resulting equation is a first order quasilinear hyperbolic system. In order to write (1.1) in the standard form of first order systems, we introduce the matrices J 1 =   0 0 0 0 0 −1 0 1 0   , J 2 =   0 0 1 0 0 0 −1 0 0   , J 3 =   0 −1 0 1 0 0 0 0 0   and A co j = 0 −J j J j 0 (1.3)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Spitz

2019

Journal of Differential Equations

141

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“…Observe that if relation (227) does not hold, the FM condition (224) for the phase φ n ( k) does not hold either. Using this fact and integrating the interaction integral (211) by parts m 2 + 1 times as in [17,20] we obtain…”

Section: Non-frequency-matched Interactionsmentioning

confidence: 94%

“…In (36) and (37) the coefficients γ 0 , γ 1 , γ 2 satisfy (20) and h + (x) is a smooth function decaying sufficiently fast as x → ∞. The function h + (x) can be chosen as we please.…”

Section: Relation Between the Nlm And The Nlsmentioning

confidence: 98%

“…Firstly, it should be almost time-harmonic, as in (10) with a deviation from timeharmonicity measured by a small parameter which consequently determines the ratio of the slow time and the fast time scales. Secondly, following the framework described in [17][18][19][20] we choose the excitation current composed of Bloch modes from a single spectral band, described …”

Section: Relation Between the Nlm And The Nlsmentioning

confidence: 99%

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Nonlinear photonic crystals: IV. Nonlinear Schrödinger equation regime

Babin

Figotin

2005

Waves in Random and Complex Media

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We study here the nonlinear Schrödinger (NLS) equation as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell (NLMs) equations. The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current J producing an EM wave. The wave nonlinear evolution is analysed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to three small parameters α, β and . These parameters are introduced through the excitation current J to scale, respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with β −1 scaling its spatial extension; (iii) its frequency bandwidth, with −1 scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current J. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of α, β and , but it also produces a new extended NLS (ENLS) providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compared with the classical NLS equation. Notations and abbreviationsalmost single-mode excitation-see (55) bidirectional quadruplet-see (341) directly excited modes-see (219), (58) doublet-see (339) ENLS extended nonlinear Schrödinger equation-see (100), (101) or (114) and section 1.4 Floquet-Bloch modal decomposition-see (146) FM-frequency matching condition (223), frequency matched, see (227) FNLR first nonlinear response-see (14), (176), (167), (174) Fourier transform-see (425) GVM group velocity matching condition-see (73), (222) indirectly excited modes-see (220), (68), (69) interaction quadruplet-see (72) linear response-see (13) NLM-nonlinear Maxwell equation, see (3) NLS-nonlinear Schrödinger equation, see (36), (37) and (309) NLS regime-a situation when the evolution of an electromagnetic (EM) wave is governed by the NLM equations and it can be approximated by an NLS or, may be, by a slightly more general extended NLS non-FM-non-frequency-matched rectifying coordinates-see (281), (282) susceptibility χ (3) D -see (156) susceptibility χ (3) -see (157) unidirectional excitation-see (188) α π = 3α(2π ) 2d γ (ν) (η)-the Taylor polynomial of ωn (k * + η) of order ν (200) γ (2) (η)-the second-order Taylor polynomial of ωn (k * + η) (201) δ ± ×,ζ -coefficients defined by (349), (345), (346) ζ = ±1 or ζ = ±-band binary number, when used in indices is abbreviated to ζ = ±, . . . , ξ d )-Fourier wavevector variable ωn(k) = ζ ω n (k)-dispersion relation of the band (ζ, n), see (138) ω n 0 (k) = ∇ k ω n 0 (...

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“…5 Below we prove that the superposition principle can hold not only for universal resonance invariant multiwavepackets, but for other cases allowing resonant processes such as second and third harmonic generations, three-wave interaction etc. Here we prove a theorem applicable to such situations, which is more general than Theorem 2.14.…”

Section: Definition 74mentioning

confidence: 76%

Nonlinear Dynamics of a System of Particle-Like Wavepackets

Babin

Figotin

Instability in Models Connected With Fluid Flows I

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This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it is a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

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Nonlinear Maxwell Equations in Inhomogeneous Media

Cited by 12 publications

References 29 publications

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

Nonlinear photonic crystals: IV. Nonlinear Schrödinger equation regime

Nonlinear Dynamics of a System of Particle-Like Wavepackets

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