arXiv subjects

C. Zambon

Publications and source records attributed to C. Zambon.

14 recordsLinked to original sources

arXiv2023

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.

nlin.SI↗

arXiv2020

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.

hep-th↗

arXiv2012

Infinite dimension reflection matrices in the sine-Gordon model with a boundary

Using the sine-Gordon model as the prime example an alternative approach to integrable boundary conditions for a theory restricted to a half-line is proposed. The main idea is to explore the consequences of taking into account the topological charge residing on the boundary and the fact it changes as solitons in the bulk reflect from the boundary. In this context, reflection matrices are intrinsically infinite dimensional, more general than the two-parameter Ghoshal-Zamolodchikov reflection matrix, and related in an intimate manner with defects.

hep-th↗

arXiv2010

Integrable defects in affine Toda field theory and infinite dimensional representations of quantum groups

Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang-Baxter equations, and second by solving a linear intertwining relation between a finite dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite dimensional representation expressed in terms of sets of generalized `quantum' annihilation and creation operators. The principal examples analysed are based on the $a_2^{(2)}$ and $a_n^{(1)}$ affine Toda models but examples of similar infinite dimensional representations for quantum Borel algebras for all other affine Toda theories are also provided.

hep-th↗

arXiv2010

A transmission matrix for a fused pair of integrable defects in the sine-Gordon model

Within the quantum sine-Gordon model a transmission matrix describing the scattering of a soliton with a fused pair of integrable defects is proposed. The result is consistent with the classical picture of scattering and highlights the differences between two defects located at separated points and two defects fused at the same point. Moreover, the analysis reveals how, for certain choices of parameters, both the soliton-soliton and the lightest-breather-soliton S-matrices of the sine-Gordon model are embedded within the transmission matrix, supporting an interpretation in which defects may be regarded as soliton constituents.

hep-th↗

arXiv2009

A new class of integrable defects

An alternative Lagrangian definition of an integrable defect is provided and analyzed. The new approach is sufficiently broad to allow a description of defects within the Tzitzeica model, which was not possible in previous approaches, and may be generalizable. New, two-parameter, sine-Gordon defects are also described, which have characteristics resembling a pair of 'fused' defects of a previously considered type. The relationship between these defects and Backlund transformations is described and a Hamiltonian description of integrable defects is proposed.

hep-th↗

arXiv2009

Comments on defects in the a_r Toda field theories

A simple, basic, argument is given, based solely on energy-momentum considerations to recover conditions under which a_r affine or conformal Toda field theories can support defects of integrable type. Associated triangle relations are solved to provide expressions for transmission matrices that generalize previously known examples calculated for the sine-Gordon model and the a_2 affine Toda model.

hep-th↗

arXiv2007

On the semi-dynamical reflection equation: solutions and structure matrices

Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov-Chekhov-Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are extendible to the non-constant case. In order to simplify future constructions of spin-chain Hamiltonians, a parametrization procedure is applied explicitly to all elements of the semi-dynamical reflection equation available. Interesting expressions for `twists' and R-matrices entering the parametrization procedure are found. In particular, some expressions for the R-matrices seem to appear here for the first time. In addition, a new set of consistent structure matrices for the semi-dynamical reflection equation is obtained.

math.QA↗

arXiv2007

On purely transmitting defects in affine Toda field theory

Affine Toda field theories with a purely transmitting integrable defect are considered and the model based on a_2 is analysed in detail. After providing a complete characterization of the problem in a classical framework, a suitable quantum transmission matrix, able to describe the interaction between an integrable defect and solitons, is found. Two independent paths are taken to reach the result. One is an investigation of the triangle equations using the S-matrix for the imaginary coupling bulk affine Toda field theories proposed by Hollowood, and the other uses a functional integral approach together with a bootstrap procedure. Evidence to support the results is collected in various ways: for instance, through the calculation of the transmission factors for the lightest breathers. While previous discoveries within the sine-Gordon model motivated this study, there are several new phenomena displayed in the a_2 model including intriguing disparities between the classical and the quantum pictures. For example, in the quantum framework, for a specific range of the coupling constant that excludes a neighbourhood of the classical limit, there is an unstable bound state.

hep-th↗

arXiv2006

Jump-defects in the nonlinear Schrodinger model and other non-relativistic field theories

Recent work on purely transmitting 'jump-defects' in the sine-Gordon model and other relativistic field theories is extended to non-relativistic models. In all the cases investigated the defect conditions are provided by 'frozen' Backlund transformations and it is also shown via a Lax pair argument how integrability will be preserved in the presence of this type of defect. Explicit examples of the scattering of solitons by defects are given, and bound states associated with 'jump-defects' in the nonlinear Schrodinger model are described. Although the nonlinear Schrodinger model provides the principal example, some results are also presented for the Korteweg de Vries and modified Korteweg de Vries equations.

nlin.SI↗

arXiv2005

Some aspects of jump-defects in the quantum sine-Gordon model

The classical sine-Gordon model permits integrable discontinuities, or jump-defects, where the conditions relating the fields on either side of a defect are Backlund transformations frozen at the defect location. The purpose of this article is to explore the extent to which this idea may be extended to the quantum sine-Gordon model and how the striking features of the classical model may translate to the quantum version. Assuming a positive defect parameter there are two types of defect. One type, carrying even charge, is stable, but the other type, carrying odd charge, is unstable and may be considered as a resonant bound state of a soliton and a stable defect. The scattering of solitons with defects is considered in detail, as is the scattering of breathers, and in all cases the jump-defect is purely transmitting. One surprising discovery concerns the lightest breather. Its transmission factor is independent of the bulk coupling - a property susceptible to a perturbative check, but not shared with any of the other breathers. It is argued that classical jump-defects can move and some comments are made concerning their quantum scattering matrix.

hep-th↗

arXiv2004

Aspects of sine-Gordon solitons, defects and gates

It was recently noted how the classical sine-Gordon theory can support discontinuities, or `defects', and yet maintain integrability by preserving sufficiently many conservation laws. Since soliton number is not preserved by a defect, a possible application to the construction of logical gates is suggested.

hep-th↗

arXiv2004

Affine Toda field theories with defects

A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on $a_r^{(1)}$ data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is illustrated by noting how defects may preserve a modified form of just one of the spin 2 or spin -2 charges but not both of them.

hep-th↗

arXiv2003

Classically integrable field theories with defects

Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type.

hep-th↗