Decay of the electromagnetic field in a Maxwell Bloch system

Jochmann, Frank

doi:10.1090/S0033-569X-07-01040-5

Author: Frank Jochmann
Journal: Quart. Appl. Math. 65 (2007), 99-112
MSC (2000): Primary 35Q60; Secondary 35L40, 78A35
DOI: https://doi.org/10.1090/S0033-569X-07-01040-5
Published electronically: January 2, 2007
MathSciNet review: 2313150
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the initial-boundary value problem for the Maxwell-Bloch system which describes the propagation of electromagnetic waves in a polarized quantum-mechanical medium with two energy levels. The main goal is the investigation of the large-time asymptotic behavior of the solutions if there are no relaxation terms in the equations governing the polarization field and the density.

References

A. Adams, Sobolev Spaces, Academic Press, 1980.

R. Boyd, Nonlinear Optics, Academic Press, (1992).

P. Donnat and J. Rauch, Global solvability of the Maxwell-Bloch equations from nonlinear optics, Arch. Rational Mech. Anal. 136 (1996), no. 3, 291–303. MR 1423010, DOI https://doi.org/10.1007/BF02206557

Éric Dumas, Global existence for Maxwell-Bloch systems, J. Differential Equations 219 (2005), no. 2, 484–509. MR 2183269, DOI https://doi.org/10.1016/j.jde.2005.02.001

F. Jochmann, A compactness result for vector fields with divergence and curl in $L^q(\Omega )$ involving mixed boundary conditions, Appl. Anal. 66 (1997), no. 1-2, 189–203. MR 1612136, DOI https://doi.org/10.1080/00036819708840581

Frank Jochmann, Asymptotic behaviour of solutions to a class of semilinear hyperbolic systems in arbitrary domains, J. Differential Equations 160 (2000), no. 2, 439–466. MR 1736995, DOI https://doi.org/10.1006/jdeq.1999.3682

Frank Jochmann, Long time asymptotics of solutions to the anharmonic oscillator model from nonlinear optics, SIAM J. Math. Anal. 32 (2000), no. 4, 887–915. MR 1814743, DOI https://doi.org/10.1137/S0036141099360932

Frank Jochmann, Convergence to stationary states in the Maxwell-Bloch system from nonlinear optics, Quart. Appl. Math. 60 (2002), no. 2, 317–339. MR 1900496, DOI https://doi.org/10.1090/qam/1900496

Frank Jochmann, Decay of the polarization field in a Maxwell Bloch system, Discrete Contin. Dyn. Syst. 9 (2003), no. 3, 663–676. MR 1974532, DOI https://doi.org/10.3934/dcds.2003.9.663

J. L. Joly, G. Metivier, and J. Rauch, Global solvability of the anharmonic oscillator model from nonlinear optics, SIAM J. Math. Anal. 27 (1996), no. 4, 905–913. MR 1393415, DOI https://doi.org/10.1137/S0036141094273672

R. Pantell, H. Puthoff, Fundamental of quantum electronics, Wiley (1969).

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486

Jeffrey Rauch and Michael Taylor, Penetrations into shadow regions and unique continuation properties in hyperbolic mixed problems, Indiana Univ. Math. J. 22 (1972/73), 277–285. MR 303098, DOI https://doi.org/10.1512/iumj.1972.22.22022

Ch. Weber, A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci. 2 (1980), no. 1, 12–25. MR 561375, DOI https://doi.org/10.1002/mma.1670020103