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

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell–Bloch equations. The nonlocal symmetries of the reduced Maxwell–Bloch equations are obtained by the truncated Painlevé expansion approach and the Möbious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painlevé waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.