Dmitry Shepelsky

 a senior researcher, e-mail: shepelsky@ilt.kharkov.ua

Inverse scattering problems for complex electromagnetic media.

We develop rigorous mathematical methods for studying various problems of propagation of electromagnetic waves in complex composite media. The composite microstructure of materials is reflected, at the macroscopic level, in complex constitutive relations, including anisotropic natureof the parameters, coupling of the electric and magnetic fields, resonance frequency dependence, etc. Particularly, we are interesting in inverse problems, which consist in the reconstruction of the electromagnetic parameters of a complex medium by using measurements of the electromagnetic field outside the domain of interest. Our approach is based on the possibility to construct appropriate solutions of the associated differential equations (that are deduced from the Maxwell equations together with the constitutive relations) which are piece-wise analytic in the complex plane of the spectral parameter (frequency). These solutions are related on contours in this plane by jump matrices which can be constructed from the measurements interpreted as the scattering data. Such relations constitute a matrix Riemann-Hilbert problem, the analysis of which provides the uniqueness theorems and the reconstruction algorithms for the electromagnetic parameters.

The approach has been applied for the reconstruction problems for bianisotropic media [1], stratified chiral media [2], inhomogeneous Omega media [3], [4].

The results have been reported at the international conferences: 
Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, Spain, 2000),
European Symposium on Numeric Methods in Electromagnetics (Toulouse, France, 2002),
International Congress of Mathematicians (Beiging, China, 2002).

Initial-boundary value problems for nonlinear equations.

We are developing a method for studying initial-boundary value problems for integrable nonlinear equations, which can be viewed as a generalization of the inverse scattering transform method (developed for the initial-value problems) to the boundary value problems. Our approach is based on the simultaneous spectral analysis of both linear equations of the associated Lax pair.
The solution of the problem is given in terms of the solution of a matrix Riemann-Hilbert problem, formulated in the complex plane of the spectral parameter and having simple, explicit x,t-dependence.
The spectral functions determining the Riemann-Hilbert problem are, in turn, defined by the initial data and the boundary values.
The dependence of the initial and boundary values are described, in the spectral terms, by a simple algebraic relation (''global relation'').
In [5], we have applied this approach to the modified Korteweg-de Vries equation on the half-line. We have shown that this approach is well adapted for studying the large t behavior of solutions, having the same efficiency as the inverse scattering transform method of the whole line. In [6], [7], the approach has been applied to the study of
the modified Korteweg-de Vries equation on a finite interval. In [8], we analyze the global relation in the case of the nonlinear Schroedinger equation on the half-line in order to determine the limiting values of the derivative of the solution on the boundary x=0 in terms of the initial values of the solution (for t=0) and the boundary values on x=0 (generalized Dirichlet-to-Neumann map). 
Using the Gelfand-Levitan-Marchenko representation, we have obtain an explicit description of this map.

[1] D. Sheen and D.Shepelsky, Uniqueness in a frequency-domain inverse problem of a stratified uniaxial bianisotropic medium, Wave Motion, 31, 4 (2000) 371-385.
[2] A. Boutet de Monvel and D.Shepelsky, A frequency-domain inverse problem for a dispersive stratified chiral med-ium, J. Math. Phys. 41, 9 (2000) 6116-6129.
[3] D.Shepelsky, A Riemann—Hilbert problem for propagation of electromagnetic waves in an inhomogeneous, dispersive waveguide, Math. Phys. Anal. Geom. 3, 2 (2000) 179-193.
[4] A. Boutet de Monvel and D.Shepelsky, Reconstruction of a stratified Omega medium and the associated Riemann-Hilbert, Inverse Problems 18, 5 (2002) 1377-1395.
[5] A. Boutet de Monvel, A.Fokas, and D.Shepelsky, The mKdV equation on the half-line, J. Inst. Math. Jussieu 3(2) (2004), 139-164.
[6] A. Boutet de Monvel and D.Shepelsky, The modified KdV equation on a finite interval, C. R. Acad. Sci. Paris Ser. I Math., 337, 8 (2003), 517-522.
[7] A. Boutet de Monvel and D.Shepelsky, Initial boundary value problem for the  mKdV equation on a finite interval, Ann. Inst. Fourier, 54 (2004), no.5, 1477–1495.
[8] A. Boutet de Monvel, A.Fokas, and D.Shepelsky, Analysis of the global relation for the nonlinear SchrЪdinger equation on the half-line Lett. Math. Phys. 65, 3 (2003), 199-212