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.

Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

“…The Darboux transform (DT) of the discrete Ablowitz-Ladik spectral problem could reduce to one of equation (1.1) [59]. A system of difference-differential equations related to equation (1.1) has been proposed, and the N-soliton solutions of equation (1.1) have been derived under the non-vanishing boundary conditions [60]. Other types of solitary wave and periodic solutions of equation (1.1) were obtained [61][62][63][64].…”

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

“…( 1) is derived from a system of lattice equations related to the self-dual network equations. [9] Moreover, Eq. ( 1) is called fractional-type in the sense that the right-hand side is a fraction of the dependent variable.…”