volume | PIER Journals

Abstract

The quasi-linear Maxwell equations describing electromagnetic wave propagation in nonlinear media permit several weak solutions, which may be discontinuous (shock waves). It is often conjectured that the solutions are unique if they satisfy an additional entropy condition. The entropy condition states that the energy contained in the electromagnetic fields is irreversibly dissipated to other energy forms, which are not described by the Maxwell equations. We use the method employed by Krûzkov to scalar conservation laws to analyze the implications of this additional condition in the electromagnetic case, i.e., systems of equations in three dimensions. It is shown that if a cubic term can be ignored, the solutions are unique and depend continuously on given data.

1. Bloom, F., Mathematical Problems of Classical Nonlinear Electromagnetic Theory, Longman Scientific & Technical, 1993.

2. Coleman, B. D., "B. D. and E. H. Dill. Thermodynamic restrictions on the constitutive equations of electromagnetic theory," Z. Angew. Math. Phys., Vol. 22, 691-702, 1971.
doi:10.1007/BF01587765

3. Courant, R. and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, 1948.

4. Dafermos, C. M., "The entropy rate admissibility criteria for solutions of hyperbolic conservation laws," Journal of Differential Equations, Vol. 14, 202-212, 1973.
doi:10.1016/0022-0396(73)90043-0

5. Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics, 325, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 2000.

6. Evans, L. C., Partial Differential Equations, American Mathematical Society, 1998.

7. Godlewski, E. and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, 1996.

8. Gustafsson, M., "Wave splitting in direct and inverse scattering problems," Ph.D. thesis, 2000.

9. Hadamard, J., Lectures on the Cauchy Problem in Linear Partial Differential Equations, Yale University Press, 1923.

10. Hopf, E., "The partial differential equation u_t + uu_x = μu_xx," Comm. Pure Appl. Math., Vol. 3, 201-230, 1950.
doi:10.1002/cpa.3160030302

11. Hörmander, L., The Analysis of Linear Partial Differential Operators I, Grundlehren der mathematischen Wissenschaften 256, Springer-Verlag, 1983.

12. Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations, Number 26 in Mathemathiques & Applications, Springer-Verlag, 1997.

13. Jackson, J. D., Classical Electrodynamics, 3rd Ed., John Wiley & Sons, 1999.

14. Jouguet, E., "Sur la propagation des discontinuites dans les fluides," C. R. Acad. Sci., Vol. 132, 673-676, 1901.

15. Kreiss, H.-O. and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, 1989.

16. Kristensson, G. and D. J. N. Wall, "Direct and inverse scattering for transient electromagnetic waves in nonlinear media," Inverse Problems, Vol. 14, 113-137, 1998.
doi:10.1088/0266-5611/14/1/011

17. Kržkov, S., "First order quasilinear equations with several space variables," Math. USSR Sbornik, Vol. 10, 217-273, 1970.
doi:10.1070/SM1970v010n02ABEH002156

18. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskiǐ, Electrodynamics of Continuous Media, 2nd Ed., Pergamon, 1984.

19. Lax, P. D., "Shock waves and entropy," Contributions to Nonlinear Functional Analysis, 603-634, 1971.

20. Lax, P. D., "Hyperbolic systems of conservation laws and the mathematical theory of shock waves," Conf. Board. Math. Sci. Regional Conference Series in Applied Mathematics 11, 1973.

21. Lindell, I. V., A. H. Sihvola, and K. Suchy, "Six-vector formalism in electromagnetics of bi-anisotropic media," J. Electro. Waves Applic., Vol. 9, No. 7/8, 887-903, 1995.

22. Liu, T.-P., "The entropy condition and the admissibility of shocks," J. Math. Anal. Appl., Vol. 53, 78-88, 1976.
doi:10.1016/0022-247X(76)90146-3

23. Maugin, G. A., "On shock waves and phase-transition fronts in continua," ARI, Vol. 50, 141-150, 1998.

24. Maugin, G. A., "On the universality of the thermomechanics of forces driving singular sets," Archive of Applied Mechanics, Vol. 70, 31-45, 2000.

25. Oleǐnik, O. A., "Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation," Amer. Math. Soc. Transl., Vol. 14, No. 2(86), 87-158, 1959.

26. Serre, D., "Systems of conservation laws, A challenge for the XXIst century," Mathematics Unlimited - 2001 and Beyond, 1061-1080, 2001.

27. Sjöberg, D., "Reconstruction of nonlinear material properties for homogeneous, isotropic slabs using electromagnetic waves," Inverse Problems, Vol. 15, No. 2, 431-444, 1999.
doi:10.1088/0266-5611/15/2/006

28. Sjöberg, D., "Simple wave solutions for the Maxwell equations in bianisotropic, nonlinear media, with application to oblique incidence," Wave Motion, Vol. 32, No. 3, 217-232, 2000.
doi:10.1016/S0165-2125(00)00039-1

29. Styer, D. F., "Insight into entropy," Am. J. Phys, Vol. 68, No. 12, 1090-1096, 2000.
doi:10.1119/1.1287353

30. Taylor, M., Partial Differential Equations III, Nonlinear Equations, 1996.

31. Åberg, I., "High-frequency switching and Kerr effect â€” Nonlinear problems solved with nonstationary time domain techniques," Progress In Electromagnetics Research, Vol. 17, 185-235, 1997.
doi:10.2528/PIER97021200

32. Kung, F. and H. T. Chuah, "Stability of classical finite-difference time-domain (FDTD) formulation with nonlinear elements â€” A new perspective," Progress In Electromagnetics Research, Vol. 42, 49-89, 2003.
doi:10.2528/PIER03010901

33. Makeeva, G. S., O. A. Golovanov, and M. Pardavi-Horvath, "Mathematical modeling of nonlinear waves and oscillations in gyromagnetic structures by bifurcation theory methods," J. of Electromagn. Waves and Appl., Vol. 20, No. 11, 1503-1510, 2006.
doi:10.1163/156939306779274363

34. Norgren, M. and S. He, "Effective boundary conditions for a 2D inhomogeneous nonlinear thin layer coated on a metallic surface," Progress In Electromagnetics Research, Vol. 23, 301-314, 1999.
doi:10.2528/PIER99020206

Dr. Daniel Sjöberg