A new class of integrable defects

Abstract

An alternative Lagrangian definition of an integrable defect is provided and analyzed. The new approach is sufficiently broad to allow a description of defects within the Tzitzéica model, which was not possible in previous approaches, and may be generalizable. New, two-parameter, sine-Gordon defects are also described, which have characteristics resembling a pair of 'fused' defects of a previously considered type. The relationship between these defects and Bäcklund transformations is described and a Hamiltonian description of integrable defects is proposed.

The classical nonlinear Schrödinger model with a new integrable boundary

“…Not all integrable models can support a type I defect. Amongst the single scalar filed theory that cannot support such a defect it is worth mentioning the Tzitzéica modelalso known as the a (2) 2 affine Toda model -for which a type II defect has been introduced in [9]. The main difference with respect to the type I defect is that the type II requires the presence of extra degrees of freedom at the defect location.…”

“…In fact, it is worth reminding that a complete proof of the defect integrability via the classical r-matrix was missing so far, despite the fact that strong evidence has been collected to support this statement. On this issue, see [4,8,9,[11][12][13] and also [14][15][16].…”

“…Also notice that by applying the same trick to the sine-Gordon model and the corresponding defect (type I [8]), no new boundaries are obtained. The time derivative terms cancel out and in order to find new boundary it is necessary to use the type II defect [9] instead, which contains additional degrees of freedom, as shown in [7]. In order to collect evidence to support the claim of integrability of the new boundary and eventually prove it, the corresponding K matrix is needed and the modified Lax pair technique will be used to find it.…”

Взгляд На Дефекты В Интегрируемой Квантовой Теории Поля

“…Поскольку не во всех теориях поля типа аффинной теории Тоды может присут-ствовать дефект типа I, естественно попытаться добавить дополнительные степени свободы к самому дефекту [7]. Например, рассмотрим две релятивистские теории поля с полями u и v и следующим образом добавим λ(t) в том месте, где расположен дефект:…”

The sinh-Gordon defect matrix generalized for n defects

“…However, there exist integrable models for which such type of Bäcklund transformation are not allowed. This is the case of the Tzitzeica model where additional auxiliary fields are required [2,3,4,5,6]. These are called type II and consist of a new class of Bäcklund transformations.…”