Momentum conserving defects in affine Toda field theories

TL;DR

This paper investigates momentum conserving defects in affine Toda field theories, establishing conditions for their existence, linking them to Bäcklund transformations, and analyzing soliton interactions, thereby advancing understanding of integrable defect systems.

Contribution

It introduces a condition on the Lagrangian for momentum conservation in defects and connects these defects to Bäcklund transformations in affine Toda theories.

Findings

Derived conditions for momentum conserving defects.

Constructed Bäcklund transformations for affine Toda theories.

Calculated classical soliton time delays passing through defects.

Abstract

Type II integrable defects with more than one degree of freedom at the defect are investigated. A condition on the form of the Lagrangian for such defects is found which ensures the existence of a conserved momentum in the presence of the defect. In addition it is shown that for any Lagrangian satisfying this condition, the defect equations of motion, when taken to hold everywhere, can be extended to give a B\"{a}cklund transformation between the bulk theories on either side of the defect. This strongly suggests that such systems are integrable. Momentum conserving defects and B\"{a}cklund transformations for affine Toda field theories based on the $A_n$, $B_n$, $C_n$ and $D_n$ series of Lie algebras are found. The defect associated with the $D_4$ affine Toda field theory is examined in more detail. In particular classical time delays for solitons passing through the defect are calculated.