Abstract
Discrete versions for nonlinear Schrödinger equation and modified nonlinear Schrödinger 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.
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