The sine-Gordon model with integrable defects revisited

Abstract

Application of our algebraic approach to Liouville integrable defects is proposed for the sine-Gordon model. Integrability of the model is ensured by the underlying classical r-matrix algebra. The first local integrals of motion are identified together with the corresponding Lax pairs. Continuity conditions imposed on the time components of the entailed Lax pairs give rise to the sewing conditions on the defect point consistent with Liouville integrability.

Integrability of generalised type II defects in affine Toda field theory

“…A type II defect matrix for the Tzitzéica model is found in [34] and the system is shown to have an infinite number of conserved quantities. A Hamiltonian set-up in which the Lax and r-matrix equations are immediately assumed to be satisfied by some matrix associated with the defect is investigated in [35][36][37][38] for defects in the nonlinear Schrödinger equation, sine-Gordon and ATFTs. While these defects are integrable they do not necessarily describe the same systems as the momentum conserving defects found in the Lagrangian set-up.…”

“…We have also made no attempt to approach these defects from a Hamiltonian perspective, as has been carried out in [35][36][37][38], and have yet to prove that these defects are integrable. It would be interesting to apply the method given in [6] of moving from a Lagrangian to a Hamiltonian picture to these defects.…”

Jumps and twists in affine Toda field theories

“…The presence of impurities in integrable systems has attracted considerable attention over the last years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], especially when dealing with physical applications and confronting experimental data. Integrability offers an elegant framework such that impurities may be naturally incorporated to a physical system in a systematic and controllable manner.…”

Lax pair formulation in the simultaneous presence of boundaries and defects

“…Compatibility conditions of the two differential equations (1.4), (1.5) lead to the zero curvature conditioṅ 6) giving rise to the corresponding classical equations of motion of the system under consideration. Special care should be taken on the defect point; the zero curvature condition on the point reads as [19,20] …”