Exact Solutions And Their Dynamical Properties For Two Kinds Of Discrete Integrable Equations

The nonlocal integrable systems and discrete integrable systems are the research hotspots in the filed of soliton and integrable system.The study of the nonlinear localized waves such as solitons,breathers,lumps and rough waves have attracted great attention due to their important applications in nonlinear physical areas such as nonlinear optics,oceanography,BoseEinstein condensates,plasma and biophysics.In this paper,we introduce the reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr(?)dinger(DNLS)equations through the nonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation.The muti-soliton solutions of two types of nonlocal discrete DNLS equations are derived by means of the Hirota bilinear method and reduction approach.The dynamics of soliton solutions and rich soliton structures in two discrete nonlocal DNLS models are studied.Our investigation shows that the solitons of these nonlocal equations are bounded for some range of parameters,but breathe and periodically collapse for other range of parameters.Another discrete integrable equation is the(2+1)-dimensional elliptic Toda equation which is a discrete version of the Kadomtsev-Petviashvili-I(KPI)equation.We derive the M-breather solution in determinant form via B(?)cklund transformation and nonlinear superposition formulae and the lump solutions are derived from the breather solutions through the degeneration process for the(2+1)-dimensional elliptic Toda equation.By introducing the velocity resonance mechanism to N-soliton solution,it is found that the(2+1)-dimensional elliptic Toda equation possesses line soliton molecules,breather-soliton molecules and breather molecules.It is interesting to find that the KPI equation doesn't possess line soliton molecule,but its discrete version —the(2+1)-dimensional elliptic Toda equation exhibits line soliton molecules structure.