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.

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

“…Rogue waves are observed in many fields [54][55][56] other than oceanography. A wide range of authors [57][58][59][60][61][62] have devoted their efforts to finding rogue-wave solutions. This paper is organized as follows.…”

New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System

“…These equations were obtained by coupling the Maxwell equations with the Bloch equation (a linear Schrödinger like equation which describes the evolution of atoms resonantly coupled to the laser field), see [1]. Now in MathSciNet appear 265 articles related with these equations, see for instance [4,5,6,9,10,11].…”

Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

“…Many effective methods, such as the inverse scattering transform (IST) [14,15], the Hirota bilinear method [16][17][18], the Darboux transformation (DT) [19][20][21], the Painlevé analysis [22], etc., have been developed to study the explicit N-soliton solutions of the RMB equations. The RMB equations are one of integrable systems as shown in [23,24] and of course admit other integrable properties including Hamitonian structure and recursion operator [25], the N-degenerate periodic solutions, N-rational solutions and rogue waves [26] as well as interactional solutions [27] by consistent Riccati expansion method. Owing to the practical significance, further study of the characteristics of the RMB equations, especially the interactional solutions, is therefore of great importance.…”