The two-dimensional Leznov lattice equation and its various solutions

Abstract

In this paper, the two-dimensional Leznov lattice equation is studied by Hirota’s bilinear method and KP-Toda hierarchy reduction approach. We present solutions of the lattice equation in the ratio of τ functions given by the N×N Gram-type determinants. Solitons, breathers and rational solutions on constant and periodic backgrounds are derived. In particular, we find three types of breather solutions that are periodic in x-direction, y-direction and a general oblique line, respectively. Algebraic solitons are obtained by introducing two differential operators applied to the soliton solutions. We investigate the asymptotic behavior and interactions of algebraic soliton. The interaction of two algebraic solitons are analyzed and proved to be completely elastic. We demonstrate that the asymptotic algebraic solitons in the straight lines well approach the exact ones. Multiple algebraic solitons on the constant and periodic backgrounds are derived with higher-order determinant size. We introduce Schur polynomials to generate explicit expression for fundamental and higher-order lump solutions. Dynamics of lump solutions are analyzed with plots.