A Dual Construction of the Isotropic Landau-Lifshitz Model

By interchanging the roles of the space and time coordinates, we describe a dual construction of the isotropic Landau-Lifshitz model, providing equal-space Poisson brackets and dual Hamiltonians conserved with respect to space-evolution. This construction is built in the Lax/zero-curvature formalism, where the duality between the space and time dependencies is evident.Comment: 18 Pages, LaTeX; Version accepted for publication in Physica D: Nonlinear Phenomen

Sonic autoethnographies: personal listening as compositional context

This article discusses a range of self-reflexive tendencies in field recording, soundscape composition and studio production, and explores examples of sonic practices and works in which the personal listening experiences of the composer are a key contextual and compositional element. As broad areas for discussion, particular attention is given to soundscape composition as self-narrative (exploring the representation of the recordist in soundscape works) and to producing the hyperreal and the liminal (considering spatial characteristics of contemporary auditory experience and their consequences for sonic practice). The discussion then focuses on the specific application of autoethnographic research methods to the practice and the understanding of soundscape composition. Compositional strategies employed in two recent pieces by the author are considered in detail. The aim of this discussion is to link autoethnography to specific ideas about sound and listening, and to some tendencies in field recording, soundscape composition and studio production, while also providing context for the discussion of the author’s own practice and works. In drawing together this range of ideas, methods and work, sonic autoethnography is aligned with an emerging discourse around reflexive, embodied sound work

The quantum auxiliary linear problem & Darboux-Backlund transformations

We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole hierarchy of the time components of Lax pairs at the quantum level for both closed and open integrable lattice models. The generic time evolution operator formula is particularly interesting and novel at the quantum level when dealing with systems with open boundary conditions. In the same frame we show that the reflection K-matrix can also be viewed as a particular type of BT, fixed at the boundaries of the system. The q-oscillator (q-boson) model, a variant of the Ablowitz-Ladik model, is then employed as a paradigm to illustrate the method. Particular emphasis is given to the time part of the quantum BT as possible connections and applications to the problem of quantum quenches as well as the time evolution of local quantum impurities are evident. A discussion on the use of Bethe states as well as coherent states and the path integral formulation for the study of the time evolution is also presented.Comment: 20 pages Latex. Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity", 31 August - 25 September 201

Time-like boundary conditions in the NLS model

We focus on the non-linear Schrodinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like `conserved' quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space components of the Lax pairs in the case of time-like boundaries defined by solutions of the reflection equation. Analytical conditions on the boundary Lax pair lead to the time like-boundary conditions. The time-like dressing is also performed for the first time, as an effective means to produce the space components of the Lax pair of the associated hierarchy. This is particularly relevant in the absence of a classical r-matrix, or when considering complicated underlying algebraic structures. The associated time Riccati equations and hence the time-like conserved quantities are also derived. We use as the main paradigm for this purpose the matrix NLS-type hierarchy.Comment: 17 pages, LaTex. A few typos corrected. arXiv admin note: substantial text overlap with arXiv:1810.1093

Non-commutative NLS-type hierarchies: dressing & solutions

We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general $n^{th}$ order matrix-differential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutitave Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.Comment: 29 pages, LaTex. Minor modification

An algebraic approach to discrete time integrability

We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrodinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz-Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two dimensional quantum lattice is thus also examined.Comment: 34 pages, LaTex. References added. Version accepted in J. Phys.

Solitons: conservation laws & dressing methods

We review some of the fundamental notions associated to the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux-Backlund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov-Shabat dressing scheme and the Gelfand-Levitan-Marchenko equation are also introduced. Via this method generic solutions are produced, and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete non-linear Schrodinger model, the Korteweg-de Vries and non-linear Schrodinger equations as well as the sine-Gordon and Liouville models.Comment: 39 pages, LaTex. A few typos correcte

Integrable hierarchies in the Lax/zero-curvature formalism

This thesis focusses on the development of (1+1)-dimensional integrable hierarchies in both the classical and quantum settings via the Lax/zero-curvature picture, where the underlying Poisson structure is found through the use of a classical or quantum R-matrix. After setting the scene by using the non-linear Schrodinger and isotropic Landau-Lifshitz models as examples of the standard approach to constructing hierarchies in this picture, the focus shifts to two more recent developments: equal-space Poisson structures (and the resulting spatially conserved quantities and Lax pairs); and quantum Lax pairs, where previously only the quantum Lax matrix (the spatial component) was considered. The non-linear Schrodinger and isotropic Landau-Lifshitz models (or analogous quantum spin chains) are then used as examples for these recent developments to compare against the familiar results

I played | here for you | like so | now | listen good. On disrupting the exchange in sound composition and reception

The following commentary focuses on the discussion of a developing research practice in sound composition which seeks to apprehend reception context by prioritising acts of ‘reading’ in both the generation and (re)presentation of materials. In particular, this commentary reflects on the development of compositional methods based on: the recording (documentation) of everyday, arbitrary action and events; the editing of recorded materials, and the restaging of materials in contexts which work to destabilise the ‘readability’ and coherence of an emergent ‘work’. By relating this developing practice to examples of minimal, acousmatic and electroacoustic composition, and those of post-modern theory, minimal art and contemporary theatre, this commentary will summarise an cross-disciplinary research practice motivated towards problematising both reception context, and ‘completeness’, as they pertain to ‘music’ - as object, event, and idea

Virtual auditory reality: inhabiting digital pop music as simulated space

This article examines popular music listening in light of recent research in auditory perception and spatial experience, record production, and virtual reality, while considering parallel developments in digital pop music production practice. The discussion begins by considering theories of listening and embodiment by Brandon LaBelle, Eric Clarke, Salomè Voegelin and Linda Salter, examining relations between listening subjects and aural environments, conceptualising listening as a process of environmental ‘inhabiting’, and considering auditory experience as the real-time construction of ‘reality’. These ideas are discussed in relation to recent research on popular music production and perception, with a focus on matters of spatial sound design, the virtual ‘staging’ of music performances and performing bodies, digital editing methods and effects, and on shifting relations between musical spatiality, singer-persona, audio technologies, and listener. Writings on music and virtual space by Martin Knakkergaard, Allan Moore, Ragnhild Brøvig-Hanssen & Anne Danielsen, Denis Smalley, Dale Chapman, Kodwo Eshun and Holger Schulze are discussed, before being related to conceptions of VR sound and user experience by Jaron Lanier, Rolf Nordahl & Niels Nilsson, Mel Slater, Tom Garner and Frances Dyson. This critical framework informs three short aural analyses of digital pop tracks released during the last 10 years - Titanium (Guetta & Sia 2010), Ultralight Beam (West 2016) and 2099 (Charli XCX 2019) - presented in the form of autoethnographic ‘listening notes’. Through this discussion on personal popular music listening and virtual spatiality, a theory of pop listening as embodied inhabiting of simulated narrative space, or virtual story-world, with reference to ‘aural-dominant realities’ (Salter), ‘sonic possible worlds’ (Voegelin), and ‘sonic fictions’ (Eshun), is developed. By examining personal music listening in relation to VR user experience, this study proposes listening to pop music in the 21st century as a mode of immersive, embodied ‘storyliving’, or ‘storydoing’ (Allen & Tucker)