Dynamics of dark multisoliton and rational solutions for three nonlinear differential-difference equations

Abstract

In this paper, three nonlinear differential-difference equations (NDDEs) from the same hierarchy are investigated using the generalised perturbation $$(n,N-n)$$ -fold Darboux transformation (DT) technique. The dark multisoliton solutions in terms of determinants for three equations are obtained by means of the discrete N-fold DT. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied through numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such dark multisolitons without a noise. We find that the solutions of lower-order NDDEs in the same hierarchy are more robust against a small noise than their corresponding higher-order NDDEs. The discrete generalised perturbation $$(1,N-1)$$ -fold DT is used to derive some discrete rational and semirational solutions of the first equation, and a few mathematical features are also discussed. Results in this paper might be helpful for understanding some physical phenomena.