Initial-Boundary Value Problem for the Maxwell–Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function

M. Filipkovska

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

The initial-boundary value problem (IBVP) for the Maxwell–Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann–Hilbert problem for solving the considered IBVP. Using the system of Ablowitz–Kaup–Newell–Segur equations equivalent to the system of the Maxwell–Bloch (MB) equations, we construct the associated matrix Riemann–Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.

Détail

Keywords: integrable nonlinear PDEs, Maxwell–Bloch equations, inverse scattering transform, Riemann–Hilbert problem, singular integral equation, inhomogeneous broadening, periodic boundary function.

Bibliographie

[1] Ablowitz M.J., Kaup D.J., Newell A.C., “Coherent pulse propagation, a dispersive, irreversible phenomenon”, J. Math. Phys., 15 (1974), 1852–1858 | DOI

[2] Ablowitz M.J., Segur H., Solitons and the inverse scattering transform, SIAM Stud. Appl. Math., 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1981 | MR | Zbl

[3] Biondini G., Gabitov I., Kovačič G., Li S., “Inverse scattering transform for two-level systems with nonzero background”, J. Math. Phys., 60 (2019), 073510, 49 pp., arXiv: 1907.06231 | DOI | MR | Zbl

[4] Boutet de Monvel A., Kotlyarov V., “Scattering problem for the Zakharov–Shabat equations on the semi-axis”, Inverse Probl., 16 (2000), 1813–1837 | DOI | MR | Zbl

[5] Deift P.A., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lect. Notes Math., 3, New York University, New York, 1999 | DOI | MR

[6] Filipkovska M.S., Kotlyarov V.P., “Propagation of electric field generated by periodic pumping in a stable medium of two-level atoms of the Maxwell–Bloch model”, J. Math. Phys., 61 (2020), 123502, 31 pp. | DOI | MR | Zbl

[7] Filipkovska M.S., Kotlyarov V.P., Melamedova E.A., “Maxwell–Bloch equations without spectral broadening: gauge equivalence, transformation operators and matrix Riemann–Hilbert problems”, J. Math. Phys. Anal. Geom., 13 (2017), 119–153 | DOI | MR | Zbl

[8] Fokas A.S., “A unified transform method for solving linear and certain nonlinear PDEs”, Proc. Roy. Soc. London Ser. A, 453 (1997), 1411–1443 | DOI | MR | Zbl

[9] Fokas A.S., “Integrable nonlinear evolution equations on the half-line”, Comm. Math. Phys., 230 (2002), 1–39 | DOI | MR | Zbl

[10] Gabitov I.R., Zakharov V.E., Mikhailov A.V., “Superfluorescence pulse shape”, JETP Lett., 37 (1983), 279–282

[11] Gabitov I.R., Zakharov V.E., Mikhailov A.V., “Non-linear theory of superfluorescence”, Sov. Phys. JETP, 59 (1984), 703–709

[12] Gabitov I.R., Zakharov V.E., Mikhailov A.V., “Maxwell–Bloch equation and the inverse scattering method”, Theoret. and Math. Phys., 63 (1985), 328–343 | DOI | MR

[13] Huang L., Chen Y., “Localized excitations and interactional solutions for the reduced Maxwell–Bloch equations”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 237–252, arXiv: 1712.02059 | DOI | MR | Zbl

[14] Kiselev O.M., “Solution of Goursat problem for the Maxwell–Bloch system”, Theoret. and Math. Phys., 98 (1994), 20–26 | DOI | MR | Zbl

[15] Kotlyarov V., “Complete linearization of a mixed problem to the Maxwell–Bloch equations by matrix Riemann–Hilbert problems”, J. Phys. A, 46 (2013), 285206, 24 pp., arXiv: 1301.3649 | DOI | MR | Zbl

[16] Lamb Jr. G.L., “Propagation of ultrashort optical pulses”, Phys. Lett. A, 25 (1967), 181–182 | DOI

[17] Lamb Jr. G.L., “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium”, Rev. Modern Phys., 43 (1971), 99–124 | DOI | MR

[18] Lenells J., “The nonlinear steepest descent method for Riemann–Hilbert problems of low regularity”, Indiana Univ. Math. J., 66 (2017), 1287–1332, arXiv: 1501.05329 | DOI | MR | Zbl

[19] Li S., Miller P.D., “On the Maxwell–Bloch system in the sharp-line limit without solitons”, Comm. Pure Appl. Math., 77 (2024), 457–542, arXiv: 2105.13293 | DOI | MR

[20] Manakov S.V., “Propagation of ultrshort optical pulse in a two-level laser amplifier”, Sov. Phys. JETP, 56 (1982), 37–44

[21] Muskhelishvili N.I., Singular integral equations: Boundary problems of functions theory and their applications to mathematical physics, Springer, Dordrecht, 2011 | DOI | MR

[22] Spitkovskii I.M., “Factorization of measurable matrix-functions in classes $L_{p,\rho}$ with power weight”, Soviet Math., 32:5 (1988), 78–88 | MR | Zbl

[23] Wei J., Wang X., Geng X., “Periodic and rational solutions of the reduced Maxwell–Bloch equations”, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 1–14, arXiv: 1705.09881 | DOI | MR | Zbl

[24] Zakharov V.E., “Propagation of an amplifying pulse in a two-level medium”, JETP Lett., 32 (1980), 589–593

[25] Zhou X., “The Riemann–Hilbert problem and inverse scattering”, SIAM J. Math. Anal., 20 (1989), 966–986 | DOI | MR | Zbl