The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System

TL;DR

This paper demonstrates how the lattice sine-Gordon equation and its symmetries can generate an integrable NLS-type system, providing superposition formulas and solutions for complex wave interactions.

Contribution

It introduces a novel method to derive an NLS-type system from the lattice sine-Gordon equation using symmetry elimination and constructs superposition formulas for solutions.

Findings

Derived an integrable NLS-type system from the lattice sine-Gordon equation.

Established auto-B"acklund transformation and superposition formulas for the NLS-type system.

Calculated superpositions of multiple elementary solutions using the new formulas.

Abstract

We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An auto-B\"acklund transformation and a superposition formula for the NLS-type system is obtained by elimination of shifts from the lattice sine-Gordon equation and its down-shifted version. We use the obtained formulae to calculate a superposition of two and three elementary solutions.