Discrete analogues for two nonlinear Schrödinger type equations

Discrete versions for nonlinear Schrodinger equation and modified nonlinear Schrodinger equation are presented, using the method of generalized Cauchy matrix approach based on a Sylvester equation. By solving the determining equation set, some exact solutions including soliton solutions, Jordan block solutions and mixed solutions for the resulting discrete equations are generated. The associated semi-discrete equations as well as continuous equations are also discussed, applying appropriate continuum limits.