
SIGMA, 2018, Volume 14, 082, 27 pages
(Mi sigma1381)




A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their FiniteGap Solutions
Vladimir P. Kotlyarov^{} ^{} B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103 Kharkiv, Ukraine
Abstract:
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, sineGordon 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 quasiperiodic finitegap 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 initialboundary 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.
Keywords:
Baker–Akhiezer function; Maxwell–Bloch equations; matrix Riemann–Hilbert problems.
DOI:
https://doi.org/10.3842/SIGMA.2018.082
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1802.01622
MSC: 34L25; 34M50; 35F31; 35Q15; 35Q51 Received: February 5, 2018; in final form August 2, 2018; Published online August 10, 2018
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Citation:
Vladimir P. Kotlyarov, “A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations and their FiniteGap Solutions”, SIGMA, 14 (2018), 082, 27 pp.
Citation in format AMSBIB
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\paper A Matrix BakerAkhiezer Function Associated with the MaxwellBloch Equations and their FiniteGap Solutions
\jour SIGMA
\yr 2018
\vol 14
\papernumber 082
\totalpages 27
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