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.
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References
Avan, Liouville integrable defects: the non-linear Schrödinger paradigm, JHEP 01 (2012) 040 [arXiv:1110.4728] [INSPIRE].
G. Delfino, G. Mussardo and P. Simonetti, Statistical models with a line of defect, Phys. Lett. B 328 (1994) 123 [hep-th/9403049] [INSPIRE].
G. Delfino, G. Mussardo and P. Simonetti, Scattering theory and correlation functions in statistical models with a line of defect, Nucl. Phys. B 432 (1994) 518 [hep-th/9409076] [INSPIRE].
E. Corrigan and C. Zambon, A Transmission matrix for a fused pair of integrable defects in the sine-Gordon model, J. Phys. A 43 (2010) 345201 [arXiv:1006.0939] [INSPIRE].
R. Konik and A. LeClair, Purely transmitting defect field theories, Nucl. Phys. B 538 (1999) 587 [hep-th/9703085].
P. Bowcock, E. Corrigan and C. Zambon, Some aspects of jump-defects in the quantum sine-Gordon model, JHEP 08 (2005) 023 [hep-th/0506169] [INSPIRE].
F. Nemes, Semiclassical analysis of defect sine-Gordon theory, Int. J. Mod. Phys. A 25 (2010) 4493 [arXiv:0909.3268] [INSPIRE].
E. Corrigan and C. Zambon, Comments on defects in the a r Toda field theories, J. Phys. A 42 (2009) 304008 [arXiv:0902.1307] [INSPIRE].
P. Bowcock, E. Corrigan and C. Zambon, Affine Toda field theories with defects, JHEP 01 (2004) 056 [hep-th/0401020] [INSPIRE].
E. Corrigan and C. Zambon, On purely transmitting defects in affine Toda field theory, JHEP 07 (2007) 001 [arXiv:0705.1066] [INSPIRE].
E. Corrigan and C. Zambon, A New class of integrable defects, J. Phys. A 42 (2009) 475203 [arXiv:0908.3126] [INSPIRE].
E. Corrigan and C. Zambon, Jump-defects in the nonlinear Schrodinger model and other non-relativistic field theories, Nonlinearity 19 (2006) 1447 [nlin/0512038].
I. Habibullin and A. Kundu, Quantum and classical integrable sine-Gordon model with defect, Nucl. Phys. B 795 (2008) 549 [arXiv:0709.4611] [INSPIRE].
V. Caudrelier, On a Systematic Approach to Defects in Classical Integrable Field Theories, Int. J. Geom. Methods M. 5 (2008) 1085 [arXiv:0704.2326].
Z. Bajnok and A. George, From defects to boundaries, Int. J. Mod. Phys. A 21 (2006) 1063 [hep-th/0404199] [INSPIRE].
Z. Bajnok, Equivalences between spin models induced by defects, J. Stat. Mech. 0606 (2006) P06010 [hep-th/0601107] [INSPIRE].
Z. Bajnok and Z. Simon, Solving topological defects via fusion, Nucl. Phys. B 802 (2008) 307 [arXiv:0712.4292] [INSPIRE].
R. Weston, An Algebraic Setting for Defects in the XXZ and sine-Gordon Models, arXiv:1006.1555 [INSPIRE].
M. Mintchev, É. Ragoucy and P. Sorba, Scattering in the presence of a reflecting and transmitting impurity, Phys. Lett. B 547 (2002) 313 [hep-th/0209052] [INSPIRE].
M. Mintchev, É. Ragoucy and P. Sorba, Reflection transmission algebras, J. Phys. A 36 (2003) 10407 [hep-th/0303187] [INSPIRE].
V. Caudrelier, M. Mintchev and É. Ragoucy, The Quantum non-linear Schrödinger model with point-like defect, J. Phys. A 37 (2004) L367 [hep-th/0404144] [INSPIRE].
Doikou, Defects in the discrete non-linear Schrödinger model, Nucl. Phys. B 854 (2012) 153 [arXiv:1106.1602] [INSPIRE].
A. Aguirre, T. Araujo, J. Gomes and A. Zimerman, Type-II Bácklund Transformations via Gauge Transformations, JHEP 12 (2011) 056 [arXiv:1110.1589] [INSPIRE].
A.R. Aguirre, Inverse scattering approach for massive Thirring models with integrable type-II defects, J. Phys. A 45 (2012) 205205 [arXiv:1111.5249] [INSPIRE].
J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys. B 269 (1986) 54 [INSPIRE].
F. Magri, P. Casati, G. Falqui and M. Pedroni, Eight Lectures on Integrable Systems, Lect. Notes Phys. 638 (2004) 209.
W. Oevel and O. Ragnisco, R-matrices and higher poisson brackets for integrable systems, Physica A 161 (1989) 181.
J. Avan and E. Ragoucy, Rational Calogero-Moser model: explicit form and R-matrix of the second Poisson structure, SIGMA 8 (2012) 079 [arXiv:1207.5368].
L.D. Faddeev and L.A. Takhtakajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag (1987).
M. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983) 259 [INSPIRE].
M. Jimbo, Quantum R matrix for the generalized Toda system, Commun. Math. Phys. 102 (1986) 537 [http://projecteuclid.org/euclid.cmp/1104114539].
Avan, Boundary Lax pairs for the \( A_n^{(1) } \) Toda field theories, Nucl. Phys. B 821 (2009) 481 [arXiv:0809.2734] [INSPIRE].
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Avan, J., Doikou, A. The sine-Gordon model with integrable defects revisited. J. High Energ. Phys. 2012, 8 (2012). https://doi.org/10.1007/JHEP11(2012)008
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DOI: https://doi.org/10.1007/JHEP11(2012)008