Localized dynamical behavior in the (2+1)-dimensional sine-Gordon equation

In this paper, we investigate the (2+1)-dimensional (D) sine-Gordon (SG) equation, to describe the propagation of localized light waves in nonlinear media. The main purpose is to obtain a simple specific form of exact solutions of the (2+1)-D SG equation by Hirota's bilinear method. The novel dynamical behavior is discussed systematically for some special types of localized excitations, such as multi-dromion, multi-lump, plateau and basin solitons, the quasi-tetragonal breather, and elliptical embedded solitons, which are derived by selecting two arbitrary functions in the solution appropriately. We find that the collision of (2+1)-D solitons could be elastic but also inelastic, depending on the functional form of the solitons.