On a discrete Davey-Stewartson system

We propose a differential difference equation in ${\mathcal R}^1\times {\mathcal Z}^2$ and study it by Hirota's bilinear method. This equation has a singular continuum limit into a system which admits the reduction to the Davey-Stewartson equation. The solutions of this discrete DS system are characterized by Casorati and Grammian determinants. Based on the bilinear form of this discrete DS system, we construct the bilinear B\"{a}cklund transformation which enables us to obtain its Lax pair.Comment: 12 pages, 2 figure

The modified two-dimensional Toda lattice with self-consistent sources

Abstract In this paper, we derive the Grammian determinant solutions to the modified two-dimensional Toda lattice, and then we construct the modified two-dimensional Toda lattice with self-consistent sources via the source generation procedure. We show the integrability of the modified two-dimensional Toda lattice with self-consistent sources by presenting its Casoratian and Grammian structure of the N-soliton solution. It is also demonstrated that the commutativity between the source generation procedure and Bäcklund transformation is valid for the two-dimensional Toda lattice

A difference analogue of the Davey-Stewartson system: discrete Gram-type determinant solution and Lax pair

We consider a difference-difference Davey-Stewartson system together with its bilinear structure. We write some new Gram-type determinantal solutions taking into account a set of Jacobi identities for determinants. A bilinear Backlund transformation is constructed and consequently a Lax pair for the discrete system is derived

On an integrable modified (2+1)-dimensional Lotka-Volterra equation

In this paper, an integrable (2+1)-dimensional modified Lotka–Volterra equation is considered. This is a semi-discrete system, having discrete space and continuous time which has a known (1+1)-dimensional self-dual network equation as a reduction. Its bilinear Bäcklund transformation and Lax pair are also presented and explicit solutions including soliton solutions expressed in terms of pfaffians are obtained. Finally, the reduction of these solutions to (1+1) dimensions is considered