Discrete nonlinear Schrödinger type equations: Solutions and continuum limits

As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type equations are considered. Through the bilinearization reduction method, we construct double Casoratian solutions of the reduced discrete nonlinear Schr\"odinger type equations, including soliton solutions and Jordan-block solutions.Dynamics of the obtained one-soliton and two-soliton solutions are analyzed and illustrated. Moreover,both semi-continuous limit and full continuous limit, are applied to obtain solutions of the local and nonlocal semi-discrete nonlinear Schr\"odinger type equations, as well as the local and nonlocal continuous nonlinear Schr\"odinger type equations.