32 research outputs found
Adding integrable defects to the Boussinesq equation
The purpose of this paper is to extend the store of models able to support
integrable defects by investigating the two-dimensional Boussinesq nonlinear
wave equation. As has been previously noted in many examples, insisting that a
defect contributes to energy and momentum to ensure their conservation, despite
the presence of discontinuities and the explicit breaking of translation
invariance, leads to sewing conditions relating the two fields and their
derivatives on either side of the defect. The manner in which several types of
soliton solutions to the Boussinesq equation are affected by the defect is
explored and reveals new effects that have not been observed in other
integrable systems, such as the possibility of a soliton reflecting from a
defect or of a defect decaying into one or two solitons.Comment: 28 pages, 9 figure
Integrable defects at junctions within a network
The purpose of this article is to explore the properties of integrable, purely transmitting, defects placed at the junctions of several one-dimensional domains within a network. The defect sewing conditions turn out to be quite restrictive—for example, requiring the number of domains meeting at a junction to be even—and there is a clear distinction between the behaviour of conformal and massive integrable models. The ideas are mainly developed within classical field theory and illustrated using a variety of field theory models defined on the branches of the network, including both linear and nonlinear examples
Type II defects revisited
Energy and momentum conservation in the context of a type II, purely transmitting, defect, within a single scalar relativistic two-dimensional field theory, places a severe constraint not only on the nature of the defect but also on the potentials for the scalar fields to either side of it. The constraint is of an unfamiliar type since it requires the Poisson Bracket of the defect contributions to energy and momentum with respect to the defect discontinuity and its conjugate to be balanced by the potential difference across the defect. It is shown that the only solutions to the constraint correspond to the known integrable field theories
Interplay between Zamolodchikov-Faddeev and Reflection-Transmission algebras
We show that a suitable coset algebra, constructed in terms of an extension
of the Zamolodchikov-Faddeev algebra, is homomorphic to the
Reflection-Transmission algebra, as it appears in the study of integrable
systems with impurity.Comment: 8 pages; a misprint in eq. (2.14) and (2.15) has been correcte
The quantum non-linear Schrodinger model with point-like defect
We establish a family of point-like impurities which preserve the quantum
integrability of the non-linear Schrodinger model in 1+1 space-time dimensions.
We briefly describe the construction of the exact second quantized solution of
this model in terms of an appropriate reflection-transmission algebra. The
basic physical properties of the solution, including the space-time symmetry of
the bulk scattering matrix, are also discussed.Comment: Comments on the integrability and the impurity free limit adde
A multisymplectic approach to defects in integrable classical field theory
We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1+1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schr\"odinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen B\"acklund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions
Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect
We introduce and solve the one-dimensional quantum non-linear Schrodinger
(NLS) equation for an N-component field defined on the real line with a defect
sitting at the origin. The quantum solution is constructed using the quantum
inverse scattering method based on the concept of Reflection-Transmission (RT)
algebras recently introduced. The symmetry of the model is generated by the
reflection and transmission defect generators defining a defect subalgebra. We
classify all the corresponding reflection and transmission matrices. This
provides the possible boundary conditions obeyed by the canonical field and we
compute these boundary conditions explicitly. Finally, we exhibit a phenomenon
of spontaneous symmetry breaking induced by the defect and identify the
unbroken generators as well as the exact remaining symmetry.Comment: discussion on symmetry breaking has been improved and examples adde
Generalized q-Onsager Algebras and Dynamical K-matrices
A procedure to construct -matrices from the generalized -Onsager
algebra \cO_{q}(\hat{g}) is proposed. This procedure extends the intertwiner
techniques used to obtain scalar (c-number) solutions of the reflection
equation to dynamical (non-c-number) solutions. It shows the relation between
soliton non-preserving reflection equations or twisted reflection equations and
the generalized -Onsager algebras. These dynamical -matrices are
important to quantum integrable models with extra degrees of freedom located at
the boundaries: for instance, in the quantum affine Toda field theories on the
half-line they yield the boundary amplitudes. As examples, the cases of
\cO_{q}(a^{(2)}_{2}) and \cO_{q}(a^{(1)}_{2}) are treated in details