Type-II super-Bäcklund transformation and integrable defects for the N = 1 super sinh-Gordon model

A R Aguirre,N I Spano,A H Zimerman,J F Gomes

doi:10.1007/jhep06(2015)125

Abstract

A new super-Bäcklund transformation for the N = 1 supersymmetric sinhGordon equation is constructed. Based on this construction we propose a type-II integrable defect for the supersymmetric sinh-Gordon model consistent with this new transformation through the Lagrangian formalism. Explicit expressions for the modified conserved energy, momentum and supercharges are also computed. In addition, we show for the model that the type-II defect can also been regarded as a pair of fused defects of a previously introduced type. The explicit derivation of the associated defect matrices is also presented as a necessary condition for the integrability of the model.

Highlights

A new super-Backlund transformation for the N = 1 supersymmetric sinhGordon equation is constructed. Based on this construction we propose a type-II integrable defect for the supersymmetric sinh-Gordon model consistent with this new transformation through the Lagrangian formalism
The question of deriving the associated infinite set of conserved quantities in the presence of defects has been initially handled by using the Lax approach for a wide class of models [14]–[17]
This approach was successfully implemented for the nonlinear Schrodinger (NLS) equation and the sine-Gordon model

Summary

Modified conserved quantities

In this subsection we derive explicit expressions for the defect contributions to the modified conserved supercharges, momentum and energy. (3.31) and (3.32) becomes a total time-derivative, and the modified conserved supercharges take the following form,. By computing its time-derivative and using the defect conditions (3.12)–(3.18), it can be shown that the modified energy given by E = E + ED is conserved, where the defect contribution ED is given by, ED =. The conservation of the modified energy does not impose any constraints on the defect boundary potentials This fact was expected since the time-translation invariance has not been broken. (3.5)–(3.8) are particular solutions of the above the PB relations These conditions are valid in general for any model with defect Lagrangian given by (3.4). The computation of the defect matrix associated to the type-II defect for the sshG model is presented in appendix A

Bosonic limit

Fermionic limit

Fusing defects

Backlund solution for the sshG model

Final remarks

A Defect matrix