Momentum conserving defects in affine Toda field theories

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 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.

Integrability of generalised type II defects in affine Toda field theory

“…This idea of extra fields at the defect, and JHEP11(2017)067 the fact that one ATFT can be folded to a different ATFT using certain symmetries of the Dynkin diagram [29,30], was used in [7] to fold existing A r ATFT defects to new C r ATFT defects. These type II defects were generalised in [1] and momentum conserving defects were found in the B r and D r ATFTs. Some investigation into defects in non-relativistic theories such as the nonlinear Schrödinger equation and the Kortweg-de Vries equation have also been made [31,32].…”

“…All of the above defects have soliton solutions in which a soliton passes from one bulk theory to the other, experiencing a delay and sometimes a change in topological charge. An interesting consequence of requiring momentum conservation is that the defect equations can always be modified in such a way that they give a Bäcklund transformation (some first order differential equations coupling the solutions to two sets of uncoupled higher order differential equations [33]) for the bulk theories [1][2][3]6].…”

“…We will achieve this using the method of zero curvature and Lax pairs developed for the Kortweg-de Vries equation in [13] and modified to apply to a system with a type I defect in [3]. We first give a recap of momentum conserving defects, giving the generalised type II defects found in [1] in section 2.1, with these momentum conserving defects in ATFTs given in section 2.2. Section 2.3 gives the Tzitzéica defect found in [6] and section 2.4 gives some new, more complete results for a momentum conserving defect in the D 4 ATFT.…”

“…In this paper we consider the integrability of the defects in affine Toda field theories (ATFTs) found in [1]. Since (some of) the interest in integrable systems is due to their ability to model physical phenomena whilst remaining exactly solvable it is important to be able to incorporate common physical occurences without destroying the integrability of the system.…”

“…A defect is some discontinuity in physical media or fields in a mathematical model, and we will check whether incorporating a discontinuity into an integrable model can be achieved without destroying its integrability. We follow the classical Lagrangian picture of defects introduced in [2] and further studied in [1,[3][4][5][6][7][8]. In this approach for a defect at x = 0 there is a field vector u defined in the region x ≤ 0 and a field vector v defined in the region x ≥ 0, with both fields obeying the same bulk theory.…”