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 Bäcklund transformation between the bulk theories on either side of the defect. This strongly suggests that such systems are integrable. Momentum conserving defects and Bäcklund transformations for affine Toda field theories based on the An, Bn, Cn and Dn series of Lie algebras are found. The defect associated with the D4 affine Toda field theory is examined in more detail. In particular classical time delays for solitons passing through the defect are calculated.
Highlights
Fields couple to each other at the defect
This defect, referred to as a type II defect, introduced an additional degree of freedom only at the defect. This type II defect has not been explicitly shown to be integrable there is a strong body of evidence to suggest that it is, namely that momentum and energy are conserved and requiring momentum conservation gave sufficient constraints on the defect to ensure it was integrable in the sine-Gordon and A2 cases, solitons were able to pass through it with no change other than a delay, and that the existence of an infinite number of conserved charges has been shown for the Tzitzeica defect [18]
In [1, 2] it was noted that the defect conditions of any momentum conserving type I defect in an An ATFT were a Backlund transformation if the defect conditions were taken to hold everywhere
Summary
We shall derive conditions for a general class of type II defects to be momentum conserving. The matrix X has a kernel, and we can take this to have a basis consisting of the first r components of u by an orthogonal transformation of u These components of u completely decouple from the auxiliary fields, and so we choose to denote the vector containing only these components of u as u(1), where the superscript indicates that these fields couple like a type I defect. All the coupling matrices apart from A have been set, either to ensure momentum conservation or via field redefinitions Putting this all together we have found that in order for a defect to be momentum conserving its Lagrangian must, up to orthogonal transformations of the bulk fields u and v and field redefinitions of the auxiliary fields μ and ξ, be of the form. It does not affect the defect equations or any of the subsequent working to find the momentum conservation condition in eq (2.48), and so once D and Dsatisfying the condition have been found these field redefinitions can be used to give other D and Dsatisfying the same momentum conservation condition
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