Classical integrable defects as quasi B\"acklund transformations

TL;DR

This paper explores classical defects in integrable models, showing their relation to quasi Bäcklund transformations and deriving explicit equations for models like Toda chain and sine-Gordon.

Contribution

It introduces a novel algebraic framework connecting classical defects with quasi Bäcklund transformations in integrable systems.

Findings

Derived equations of motion for defects in Toda and sine-Gordon models

Established structural similarities between defects and Bäcklund transformations

Provided explicit examples demonstrating the theoretical construction

Abstract

We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural similarity of these equations with the discrete and continuous Backlund transformations. And although these equations are similar they are not exactly the same to the Backlund transformations. We also consider specific examples of integrable models to demonstrate our construction, i.e. the Toda chain and the sine-Gordon model. The equations of the time (space) evolution of the defect (discontinuity) degrees of freedom for these models are explicitly derived.