Periodic and rational solutions of the reduced Maxwell-Bloch equations

Abstract

We investigate the reduced Maxwell-Bloch (RMB) equations which describe the propagation of short optical pulses in dielectric materials with resonant non-degenerate transitions. The general Nth-order periodic solutions are provided by means of the Darboux transformation, and from two different limiting cases of the obtained general periodic solutions, the Nth-order degenerate periodic and Nth-order rational solutions containing several free parameters with compact determinant representations are derived, respectively. Explicit expressions of these solutions from first to second order are presented. Typical nonlinear wave patterns for the four components of the RMB equations such as single-peak, double-peak-double-dip, double-peak and single-dip structures in the second-order rational solutions are shown. This kind of the rational solutions correspond to rogue waves in the reduced Maxwell-Bloch equations.

arXiv comment13 pages

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• Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations
• Several Localized Waves Induced by Linear Interference between a Nonlinear Plane Wave and Bright Solitons
• A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions
• Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function
• The three-level coupled Maxwell-Bloch equations: rogue waves, semirational rogue waves and W-shaped solitons