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Global solvability of the Maxwell-Bloch equations from nonlinear optics. (English) Zbl 0873.35093

The authors study the global solvability of the Maxwell-Bloch system of equations relevant to nonlinear optics and especially to self-focusing phenomena. Maxwell-Bloch equations can be written as follows: \[ \partial_tB+ \nabla\times E=0,\tag{1} \]
\[ \partial_tE- \nabla\times B=-\partial_tP,\tag{2} \]
\[ \partial^2_tP+\partial_tP/T_2+ \Omega^2P= c_1NE,\tag{3} \]
\[ \partial_tN+ (N-N_0)/T_1= -c_2\langle\partial_tP, E\rangle.\tag{4} \] Here \(E(t,x)\), \(B(t,x)\), \(P(t,x)\) are the electric field, magnetic field, and polarization of the medium, respectively. A scalar field \(N(t,x)\) stands for the difference between the number of electrons in the excited state and the ground state per unit value. It should be noted that the Maxwell-Bloch system (1)–(4) which is usually used for studying resonance phenomena, here is used to model wave propagation. The electromagnetic field is modelled classically, while the medium is treated quantum mechanically. In (1)–(4) the scalars \(c_1\) and \(c_2\) are assumed to be nonnegative, the scalars \(T_1\) and \(T_2\) are called the inversion population lifetime and the homogeneous dephasing time; \(\Omega\) is a resonance frequency which corresponds to the principal optical spectral line of the medium. Equations (1) and (2) imply that \[ \partial_t\text{div}(E+ P)=\partial_t\text{div}(B)= 0.\tag{5} \] The physically relevant solutions are those which are obeyed to \[ \text{div}(E+ P)= \text{div}(B)=0.\tag{6} \] Denote \(U(t,x):= (E(t,x),B(t,x), P(t,x),\partial_tP(t,x), N(t,x)- N_0)\). Equation (3) can be written as a system for the pair \((P,Q)\) with \(Q:=\partial_tP\), \(\partial_tP= Q\), \(\partial_tQ=- Q/T_2- \Omega^2P+ c_1NE\).
The Maxwell-Bloch system then takes the form of a semilinear symmetric hyperbolic system for \(U\); \(\partial_tU= \sum_{1\leq j\leq 3}A_j\partial_jU+ F(U)\), where \(\partial_j:=\partial/\partial x_j\), the \(A_j\) are symmetric matrices, and \(F:\mathbb{R}^{13}\to\mathbb{R}^{13}\) is a polynomial of degree two which vanishes at the origin. For semilinear hyperbolic equations the following local existence theorem is known: If \(s>3/2\) and \(U(0,\cdot)\in H^s(\mathbb{R}^3)\), then there is a \(T_*\in]0,\infty]\) and a unique \(U\in C([0,T_*[:H^s(\mathbb{R}^3))\) which satisfies the Maxwell-Bloch system and attains these initial values.
The main result of this paper is the proof of the following theorem: If \(s>2\) and the initial data \(U(0,\cdot)\in H^s(\mathbb{R}^3)\) satisfy (6), then \(T_*=\infty\). That is, there is a unique global solution belonging to \(C([0,\infty[:H^s(\mathbb{R}^3))\).
Reviewer: I.E.Tralle (Minsk)

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
35D05 Existence of generalized solutions of PDE (MSC2000)
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