Rogue Wave Solutions Of The Types Of Maxwell-Bloch Equations

Rogue waves are famous for its sudden and large amplitude. General speaking, in the relatively calm plane, a big wave, which is of up to dozens of meters, appears sud-denly from nowhere and disappear quickly without a trace. Many researches think that the special waves originated from modulation instability, and some nonlinear modula-tion instability can be explained by the breather solutions of nonlinear partial differential equations. Based on Darboux transformation from soliton and integrable system, the paper mainly studies the rogue wave solutions of integrable nonlinear partial differential equations by the degeneration form of breather solutions. This process mainly depends on zero point of eigenfunction associated with the spectral problem and the coefficients of linear superposition. Generally, rogue wave solutions could be expressed in the form of quasi-rational solutions, and these solutions can be constructed through the degen-eration of breather solutions. Considering the complexity and diversity of higher order rogue wave solutions:the basic patterns; the triangle patterns; the circular patterns, we systematically present the solving method of higher order rogue wave solutions, and analyzes the degeneration process from breather solutions to rogue wave solution-s. We mainly study Maxwell-Bloch equation, which describes coherent-optical-pulse propagation in resonance two-level media associated with self induced transparency phenomena, and considers on the Rotating reduced Maxwell-Bloch equation, which can be given without the classic slowly varying envelope approximation method in physics. Since the Maxwell-Bloch equation and Rotating reduced Maxwell-Bloch e-quation can be written separately integrable Ablowitz-Kaup-Newell-Segur(AKNS) and Kaup-Newell(KN) spectrum problem, so we also give the similarities and differences of quasi-rational solutions in the two systems.