Remarks on several 2+1 dimensional lattices

Abstract

In this paper, five 2+1 dimensional lattices considered by several authors are revisited again. First of all we will show that two lattices proposed by Blaszak and Szum [J. Math. Phys. 42, 225 (2001)] become the so-called differential-difference KP equation due to Date, Jimbo, and Miwa [J. Phys. Soc. Jpn. 51, 4116 (1982); 51, 4125 (1982); 52, 388 (1983); 52, 761 (1983); 52, 766 (1983)] by simple variable transformations, while another lattice found by Blaszak and Szum can be viewed as a higher-dimensional generalization of a lattice given by Wu and Hu [J. Phys. A 32, 1515 (1999)]. Some integrable properties on these three lattices are derived. Second, it is shown that a 2+1 dimensional Toda-like lattice studied by Cao, Geng, and Wu [J. Phys. A 32, 8059 (1999)] can be transformed into the bilinear equation given by Hu, Clarkson, and Bullough [J. Phys. A 30, L669 (1997)]. For this bilinear version we also present some rational solutions and Lie symmetries. Finally, a lattice due to Levi, Ragnisco, and Shabat [Can. J. Phys. 72, 439 (1994)] is transformed into coupled bilinear equations. It is shown that these coupled bilinear equations do not have two-soliton solutions. This further confirms that the lattice under consideration is not completely integrable.