Solutions for a System of Difference-Differential Equations Related to the Self-Dual Network Equation

Abstract

We propose a system of difference-differential equations related to the self-dual network equation. By utilizing a Bäcklund transformation equation for the Toda lattice, we derive its N-soliton solution under nonvanishing boundary conditions at infinity. We present explicit expressions of four types of its 1-soliton solutions and examine them. Next, we present new formulas for finding rational solutions, and using them, we determine and analyze four types of rational solutions for these equations. Finally, we derive their elliptic function solution.