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The modified Volterra lattice equation with nonholonomic constrain has been considered in this paper. The integrability of the deformed model has been demonstrated by providing a Lax pair. Applying the gauge transformation to the Lax pair, we establish Darboux transformation theorem for the nonholonomic deformation equation. Some analytic solutions of the system are obtained via the one-fold and two-fold Darboux transformations. The deformation on explicit solutions exhibits different curvy profiles and propagation trajectories that were not found in modified Volterra lattice equation.

To get more insight into the relation between discrete model and continuous counterpart, a new integrable semi-discrete Kundu–Eckhaus equation is derived from the reduction in an extended Ablowitz–Ladik hierarchy. The integrability of the semi-discrete model is confirmed by showing the existence of Lax pair and infinite number of conservation laws. The dynamic characteristics of the breather and rational solutions have been analyzed in detail for our semi-discrete Kundu–Eckhaus equation to reveal some new interesting phenomena which was not found in continuous one. It is shown that the theory of the discrete system including Lax pair, Darboux transformation and explicit solutions systematically yields their continuous counterparts in the continuous limit.

A three-component integrable differential-difference dynamic system is introduced. It is interesting that the model includes the modified Volterra lattice, the two-component modified Volterra lattice, the relativistic Volterra lattice and the relativistic Toda lattice as special reductions. In this report, we (I) obtain a Lax pair, (II) use the Lax pair to derive the first three conservations laws, (III) present Darboux transformations. and (IV) construct various kinds of exact solutions.

In this paper, a spatial discrete complex modified Korteweg-de Vries equation is investigated. The Lax pair, conservation laws, Darboux transformations, and breather and rational wave solutions to the semi-discrete system are presented. The distinguished feature of the model is that the discrete rational solution can possess new W-shape rational periodic-solitary waves that were not reported before. In addition, the first-order rogue waves reach peak amplitudes which are at least three times of the background amplitude, whereas their continuous counterparts are exactly three times the constant background. Finally, the integrability of the discrete system, including Lax pair, conservation laws, Darboux transformations, and explicit solutions, yields the counterparts of the continuous system in the continuum limit.

In this paper, we propose a new semidiscrete Hirota equation which yields the Hirota equation in the continuum limit. We focus on the topic of how the discrete space step δ affects the simulation for the soliton solution to the Hirota equation. The Darboux transformation and explicit solution for the semidiscrete Hirota equation are constructed. We show that the continuum limit for the semidiscrete Hirota equation, including the Lax pair, the Darboux transformation and the explicit solution, yields the corresponding results for the Hirota equation as δ → 0 .

A new integrable semi-discrete version is proposed for the multi-component coherently coupled nonlinear Schrodinger equation. The integrability of the semi-discrete system is confirmed by existence of Lax pair and infinite number of conservation laws. With the aid of gauge transformations, explicit formulas for N-fold Darboux transformations are derived whereby some physically important solutions of the system are presented. Furthermore, the theory of the semi-discrete system including Lax pair, Darboux transformations, exact solutions and infinite number of conservation laws are shown for their continuous counterparts in the continuous limit.

This paper aims to find new explicit solutions including multisoliton, multipositon, multinegaton, and multiperiodic for a coupled Volterra lattice system. This coupled lattice system is an integrable discrete version of the coupled Korteweg-deVries (KdV) equation which has many physical applications. The dynamical properties of these new solutions are discussed in detail. We also prove that the theory of the coupled Volterra lattice system including the Lax pair, the Darboux transformation, and explicit solutions yield the corresponding theory of the coupled KdV equation in the continuous limit.