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

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.

Construction of type-II Bäcklund transformation for the mKdV hierarchy

“…In particular the type II Backlund transformation for the N = 1 super sinh-Gordon models was recently derived in [9] and may be extended to other equations of motion within the hierarchy. For the non linear Schroedinger equation (NLS), the Backlund transformation was derived in terms of a Ricatti equation [10].…”

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

“…The key ingredient in the affinization was the decomposition of the integers and semi integers numbers according to a Z 2 structure assigned to bosonic and fermionic generators defined in (6). For higher rank algebras we expect to systematize such construction decomposing both integers and semi-integers in disjoint subsets compatible with the closure of the algebra, e.g.…”

“…For the supersymmetric mKdV hierarchy, the Bäcklund transformation was derived for the entire hierarchy by an universal gauge transformation that preserves the zero curvature representation and henceforth the equations of motion [1]. The results obtained in [2,5,6] for the super sinh-Gordon were extended to the entire smKdV hierarchy by the construction of a Bäcklundgauge transformation which connects two field configurations of the same equations of motion [1]. Such structure was first introduced in [7,8] for the bosonic sine-Gordon theory in order to describe integrable defects in the sense that two solitons solutions are interpolated by a defect, as a set of internal boundary conditions derived from a Lagrangian density located at certain spatial position connecting two types of solutions.…”

Boundary scattering in the ϕ6 model

“…One object of investigation is to find suitable boundary conditions compatible with integrability [39]. Supersymmetric theories on the half-line keeping integrability were also considered [40][41][42][43]. One interesting aspect that deserves to be more investigated is the consequence of nonintegrability in the interaction of a kink or antikink with a boundary.…”