Exact Solutions And Propagation Properties Of (2+1) Dimensional Sine-Gordon Equation And Double Sine-Gordon Equation For The Ultrashort Laser Pulse

(2+1) dimensional sine-Gordon (sG) equation and (2+1) dimensional double sine-Gordon (DsG) equation have been widely applied to describe various phenomena in physic-s fields, such as condensed matter physics, plasma physics, smectic liquid crystal dynam-ics, nonlinear optics and so on. In the nonlinear optics, the properties of ultrashort pulse can be well described by the two equations. Using the multi-scale expansion method, con-sidering the self-induced transparency condition and the short-wave approximation, the sG and DsG equations can be obtained from the Maxwell Bloch equations with the two-level Hamilton’s atomic system.Methods for searching the two equations’exact solutions are very rich. We use two methods to study the (2+1) dimensional sG equation. Firstly, we solve the equation by a generalized tanh function expansion method, which is proposed by professor Lou recent-1y.(2+1) dimensional sG equation satisfy the CRE solvable condition. The consistency condition of (2+1) dimensional sG equation is proved existence after we take some trans-formations, we obtain a type of interaction solutions of solitons and periodic waves. Sec-ondly, constructing the mapping relation between CNKG equation and (2+1) dimensional sG equation, we get abundant multi-interaction solutions by introducing some arbitrary functions. We also investigate the propagation properties of these solutions, and find that the velocities of solitary wave vary with the time and space, indeed, some of velocities have some abrupt changes.For the (2+1) dimensional DsG equation, we construct the mapping relation between the CNKG equation and (2+1) dimensional DsG equation, and obtain various types of solitons by introducing one or two arbitrary functions, such as the fold line type,"snake" type,"V" type, parabolic type and so on.