We discuss the integrability of the Bosonic and Grassmannian massive Thirring models in the presence of defects through the inverse scattering approach. We present a general method to compute the generating functions of modified conserved quantities for any integrable field equation associated to the m x m spectral linear problem. We apply the method to derive in particular the defect contributions for the number of occupation, energy and momentum of the massive Thirring models.
There has been a great deal of progress in studying two-dimensional integrable field theories in the presence of defects or internal boundaries, in both classical and quantum context [1,2]. From the classical point of view, the Lagrangian formalism [3,4] has shown to be significantly successful. In this framework, the usual variational principle from a local Lagrangian density located at some fixed point, reveals frozen Bäcklund transformations as the defect conditions for the fields. There are many interesting features of these defects. It turns out that these kind of defect conditions allow for several types of integrable field theories [3,4,5], not only the energy conservation but also the conservation of a modified momentum, which includes a defect contribution. Moreover, their integrability is provided by the existence of a modified Lax pair involving a limit procedure, but in general it was only checked explicitly for a few conserved charges. As a novel feature of most of these models is that only physical fields were present within the formulation and therefore the associated Bäcklund transformations were called type-I [6]. However, it was noticed that not all the possible relativistic integrable models could be accommodate within this framework and then it was proposed a generalization by allowing a defect to have its own degree of freedom, and the associated Bäcklund transformations were named type-II [6]. Many examples were also discussed in [6] like sine/sinh-Gordon, Liouville, massive free field and Tzitzéica-Bullogh-Dodd model. Concerning type-II defects, it is interesting to note that for the supersymmetric extensions of sine-Gordon model [7,8] and for the massive Thirring models [9,10] those auxiliary boundary fields, which correspond to the degree of freedom of the defect itself, had already appeared naturally.
On the other hand, recently it was suggested an alternative and systematic new approach to defects in classical integrable field theories [11]. The inverse scattering method formalism is used and the defect conditions corresponding to frozen Bäcklund transformations are encoded in a well-known defect matrix. This matrix provided an elegant way to compute the modified conserved quantities, ensuring integrability. Using this framework the generating function for the modified conserved charges for any integrable evolution equation of the AKNS scheme were computed, and the type-II Bäcklund transformations for the sine-Gordon and Tzitzéica-Bullough-Dodd models has been also recovered [12]. It is worth noting that this method for constructing integrable initial boundary value problems based on the Bäcklund transformations has already been used in [13,14].
The aim of this paper is to provide an alternative approach in order to establish the integrability of the Bosonic and Grassmannian Thirring models with type-II defects, which were suggested by previous approaches [9,10]. At this point it is worth mentioning that there exists no totally clear definition of what integrability is for classical systems with infinitely many degrees of freedom. However, in this work we adopt the popular point of view where a system is regarded as integrable, if for its describing equations of motion it is possible to determine a constructive way of finding solutions and to show the existence of sufficient number of conserved quantities. Even though this viewpoint is not complete, it is sufficient for our immediate purposes. Since soliton solutions for these models have already been studied in [10], herein we will focus on the explicit construction of their conserved quantities.
The paper is organized as follows. In section 2, for the sake of clearness we present the standard setting of the Lax pair approach for the general m × m spectral linear problem (see for example [20]). One of the most important results of this approach and that is the main point for our purposes is the identification of sets of coupled Riccati equations in order to construct conservation laws by a simple generalization of the method used by Wadatti and Sogo for the particular case m = 2 [15]. Other different generalizations, considering particularly matrix Riccati-type equations, have also been appeared in the literature concerning to multicomponent systems (see for example [16] and references therein).
Within this framework, for the simplest case m = 2 the defect contributions to the infinite set of modified integrals of motion were explicitly derived for several integrable equations associated with the sl(2) Lie algebra-valued Lax pair like: (m)KdV, NLS, Liouville and sine/sinh-Gordon equations for which previous results derived from Lagrangian principles [3,5] were also recovered. More recently, the modified energy and momentum for the Tzitzéica model with type-II defects, which can be described by a
2 Lie algebra-valued Lax pair, were also explicitly computed [12] and showed complete agreement with the results obtained by Lagran
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