A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their Finite-Gap Solutions

Vladimir P. Kotlyarov

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

The Baker–Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg–de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, Kadomtsev–Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchies. However, extensions of the Baker–Akhiezer function for the Maxwell–Bloch (MB) system or for the Karpman–Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell–Bloch equations. Using different Riemann–Hilbert problems posed on the complex plane with a finite number of cuts we propose such a matrix function that has unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for MB system) and generates a quasi-periodic finite-gap solution to the Maxwell–Bloch equations. The suggested function will be useful in the study of the long time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift–Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell–Bloch equations.

Détail

Keywords: Baker–Akhiezer function; Maxwell–Bloch equations; matrix Riemann–Hilbert problems.

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 Studies in Applied Mathematics, 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981 | MR | Zbl

[3] Belokolos E. D., Bobenko A. I., Enol'skii V. Z., Its A. R., Matveev V. B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994 | Zbl

[4] Bertola M., El G. A., Tovbis A., “Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation”, Proc. A, 472 (2016), 20160340, 12 pp., arXiv: 1605.04713 | DOI | MR | Zbl

[5] Bertola M., Giavedoni P., “A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann–Hilbert problems”, J. Math. Phys., 56 (2015), 061507, 17 pp., arXiv: 1412.2273 | DOI | MR | Zbl

[6] Bertola M., Tovbis A., “Maximal amplitudes of finite-gap solutions for the focusing nonlinear Schrödinger equation”, Comm. Math. Phys., 354 (2017), 525–547, arXiv: 1601.00875 | DOI | MR | Zbl

[7] Boutet de Monvel A., Its A. R., Kotlyarov V. P., “Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition”, C. R. Math. Acad. Sci. Paris, 345 (2007), 615–620 | DOI | MR | Zbl

[8] Boutet de Monvel A., Its A. R., Kotlyarov V. P., “Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition on the half-line”, Comm. Math. Phys., 290 (2009), 479–522 | DOI | MR | Zbl

[9] Boutet de Monvel A., Kotlyarov V. P., “The focusing nonlinear Schrödinger equation on the quarter plane with time-periodic boundary condition: a Riemann–Hilbert approach”, J. Inst. Math. Jussieu, 6 (2007), 579–611 | DOI | MR | Zbl

[10] Boutet de Monvel A., Kotlyarov V. P., Shepelsky D., “Focusing NLS equation: long-time dynamics of step-like initial data”, Int. Math. Res. Not., 2011 (2011), 1613–1653 | DOI | MR | Zbl

[11] Buckingham R., Venakides S., “Long-time asymptotics of the nonlinear Schrödinger equation shock problem”, Comm. Pure Appl. Math., 60 (2007), 1349–1414 | DOI | MR | Zbl

[12] Buckingham R. J., Miller P. D., The sine-{G}ordon equation in the semiclassical limit: dynamics of fluxon condensates, Mem. Amer. Math. Soc., 225, 2013, vi+136 pp., arXiv: 1103.0061 | DOI | MR

[13] Buckingham R. J., Miller P. D., “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour”, Nonlinearity, 27 (2014), 2489–2578, arXiv: 1310.2276 | DOI | MR

[14] Deift P. A., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lecture Notes in Mathematics, 3, New York University, Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999 | MR

[15] Deift P. A., Its A. R., Zhou X., “Long-time asymptotics for integrable nonlinear wave equations”, Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, 181–204 | DOI | MR | Zbl

[16] Deift P. A., Its A. R., Zhou X., “A Riemann–Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics”, Ann. of Math., 146 (1997), 149–235 | DOI | MR | Zbl

[17] Deift P. A., Kriecherbauer T., McLaughlin K. T.-R., Venakides S., Zhou X., “Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory”, Comm. Pure Appl. Math., 52 (1999), 1335–1425 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[18] Deift P. A., Kriecherbauer T., McLaughlin K. T.-R., Venakides S., Zhou X., “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure Appl. Math., 52 (1999), 1491–1552 | 3.3.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[19] Deift P. A., Venakides S., Zhou X., “The collisionless shock region for the long-time behavior of solutions of the KdV equation”, Comm. Pure Appl. Math., 47 (1994), 199–206 | DOI | MR | Zbl

[20] Deift P. A., Venakides S., Zhou X., “New results in small dispersion KdV by an extension of the steepest descent method for Riemann–Hilbert problems”, Int. Math. Res. Not., 1997 (1997), 286–299 | DOI | MR

[21] Deift P. A., Venakides S., Zhou X., “An extension of the steepest descent method for Riemann–Hilbert problems: the small dispersion limit of the Korteweg–de Vries (KdV) equation”, Proc. Natl. Acad. Sci. USA, 95 (1998), 450–454 | DOI | MR | Zbl

[22] Deift P. A., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation”, Ann. of Math., 137 (1993), 295–368, arXiv: math.AP/9201261 | DOI | MR | Zbl

[23] Dubrovin B. A., “Theta functions and non-linear equations”, Russian Math. Surveys, 36:2 (1981), 11–92 | DOI | MR | Zbl

[24] Dubrovin B. A., Krichever I. M., Novikov S. P., “Integrable systems. I”, Current Problems in Mathematics, Fundamental Directions, Itogi Nauki i Tekhniki, 4, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., M., 1985, 179–284 | MR

[25] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nonlinear equations of Korteweg–de Vries type, finite-band linear operators and Abelian varieties”, Russian Math. Surveys, 31:1 (1976), 59–146 | DOI | MR | Zbl

[26] Egorova I., Gladka Z., Kotlyarov V. P., Teschl G., “Long-time asymptotics for the Korteweg–de Vries equation with step-like initial data”, Nonlinearity, 26 (2013), 1839–1864, arXiv: 1210.7434 | DOI | MR | Zbl

[27] 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

[28] Huang L., Chen Y., Localized excitations and interactional solutions for the reduced Maxwell–Bloch equations, arXiv: 1712.02059

[29] Hugot F.-X., Leon J., “Solution of the initial-boundary value problem for the Karpman–Kaup equation”, Inverse Problems, 15 (1999), 701–712 | DOI | MR | Zbl

[30] Its A. R., Kotlyarov V. P., “Explicit formulas for solutions of a nonlinear Schrödinger equation”, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1976, 965–968, arXiv: 1401.4445 | MR | Zbl

[31] Its A. R., Matveev V. B., “Hill's operator with finitely many gaps”, Funct. Anal. Appl., 9 (1975), 65–66 | DOI | Zbl

[32] Its A. R., Matveev V. B., “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, Theoret. and Math. Phys., 23 (1975), 343–355 | DOI | MR

[33] Its A. R., Matveev V. B., “A class of solutions of the Korteweg–de Vries equation”, Problems in Mathematical Physics, 8, Izdat. Leningrad. Univ., L., 1976, 70–92 | MR

[34] Its A. R., Matveev V. B., “Algebrogeometric integration of the MNS equation, finite-gap solutions and their degeneration”, J. Math. Sci., 23 (1983), 2412–2420 | DOI | Zbl

[35] Kamchatnov A. M., Pavlov M. V., “Periodic waves in the theory of self-induced transparency”, JETP, 80 (1995), 22–27

[36] Kamvissis S., McLaughlin K. D.T.-R., Miller P. D., Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation, Annals of Mathematics Studies, 154, Princeton University Press, Princeton, NJ, 2003 | DOI | MR | Zbl

[37] Kamvissis S., Shepelsky D., Zielinski L., “Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation”, J. Nonlinear Math. Phys., 22 (2015), 448–473, arXiv: 1412.7636 | DOI | MR

[38] Kamvissis S., Teschl G., “Long-time asymptotics of the periodic Toda lattice under short-range perturbations”, J. Math. Phys., 53 (2012), 073706, 35 pp., arXiv: 0705.0346 | DOI | MR | Zbl

[39] Karpman V. I., “On the dynamics of sonic-langmuir solitons”, Phys. Scr., 11 (1975), 263–265 | DOI

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

[41] Kotlyarov V. P., “A periodic problem for a nonlinear Schrödinger equation”, Questions on Mathematical Physics and Functional Analysis, Proceedings of the Scientific Seminars of the Institute for Low Temperature Physics and Engineering of the Academy of Sciences of the Ukrainian SSR, Naukova Dumka, Kiev, 1976, 121–131 | MR

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

[43] Kotlyarov V. P., Minakov A. A., “Riemann–Hilbert problem to the modified Korteveg–de Vries equation: long-time dynamics of the steplike initial data”, J. Math. Phys., 51 (2010), 093506, 31 pp. | DOI | MR | Zbl

[44] Kotlyarov V. P., Minakov A. A., “Step-initial function to the MKdV equation: hyper-elliptic long-time asymptotics of the solution”, J. Math. Phys. Anal. Geometry, 8 (2012), 38–62 | MR | Zbl

[45] Kotlyarov V. P., Shepelsky D. G., “Planar unimodular Baker–Akhiezer function for the nonlinear Schrödinger equation”, Ann. Math. Sci. Appl., 2 (2017), 343–384 | DOI | MR | Zbl

[46] Krichever I. M., “An algebraic-geometric construction of the Zaharov–Shabat equations and their periodic solutions”, Dokl. Akad. Nauk SSSR, 227 (1976), 291–294 | MR | Zbl

[47] Krichever I. M., “Integration of nonlinear equations by the methods of algebraic geometry”, Funct. Anal. Appl., 11 (1977), 12–26 | DOI | MR | Zbl

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

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

[50] Lamb Jr. G.L., “Phase variation in coherent-optical-pulse propagation”, Phys. Rev. Lett., 31 (1973), 196–199 | DOI

[51] Lamb Jr. G.L., “Coherent-optical-pulse propagation as an inverse problem”, Phys. Rev. A, 9 (1974), 422–430 | DOI

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

[53] Manakov S. V., Novokshenov V. Yu., “Complete asymptotic representation of an electromagnetic pulse in a long two-level amplifier”, Theoret. and Math. Phys., 69 (1986), 987–997 | DOI | Zbl

[54] Matveev V. B., “30 years of finite-gap integration theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), 837–875 | DOI | MR | Zbl

[55] Matveev V. B., Smirnov A. O., “Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: a unified approach”, Theoret. and Math. Phys., 186 (2016), 156–182 | DOI | MR | Zbl

[56] Moskovchenko E. A., Kotlyarov V. P., “A new Riemann–Hilbert problem in a model of stimulated Raman scattering”, J. Phys. A: Math. Gen., 39 (2006), 14591–14610 | DOI | MR | Zbl

[57] Moskovchenko E. A., Kotlyarov V. P., “Periodic boundary data for an integrable model of stimulated Raman scattering: long-time asymptotic behavior”, J. Phys. A: Math. Theor., 43 (2010), 055205, 31 pp. | DOI | MR | Zbl

[58] Novikov S. P., “The periodic problem for the Korteweg–de Vries equation”, Funct. Anal. Appl., 8 (1974), 236–246 | DOI | MR | Zbl

[59] Tovbis A., Venakides S., Zhou X., “On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation”, Comm. Pure Appl. Math., 57 (2004), 877–985 | DOI | MR | Zbl

[60] Tovbis A., Venakides S., Zhou X., “On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: pure radiation case”, Comm. Pure Appl. Math., 59 (2006), 1379–1432 | DOI | MR | Zbl

[61] Trogdon T., Riemann–Hilbert problems, their numerical solution and the computation of nonlinear special functions, Ph.D. Thesis, University of Washington, 2013 | MR

[62] Trogdon T., Olver S., Riemann–Hilbert problems, their numerical solution, and the computation of nonlinear special functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016 | MR | Zbl

[63] 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

[64] Zverovich E. I., “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces”, Russian Math. Surveys, 26:1 (1971), 117–192 | DOI | MR