Dissertations / Theses on the topic 'Integrable'
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Meng, Jinghan. "Bi-Integrable and Tri-Integrable Couplings and Their Hamiltonian Structures." Scholar Commons, 2012. http://scholarcommons.usf.edu/etd/4371.
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An investigation into structures of bi-integrable and tri-integrable couplings is undertaken. Our study is based on semi-direct sums of matrix Lie algebras. By introducing new classes of matrix loop Lie algebras, we form new Lax pairs and generate several new bi-integrable and tri-integrable couplings of soliton hierarchies through zero curvature equations. Moreover, we discuss properties of the resulting bi-integrable couplings, including infinitely many commuting symmetries and conserved densities. Their Hamiltonian structures are furnished by applying the variational identities associated with the presented matrix loop Lie algebras.
The goal of this dissertation is to demonstrate the efficiency of our approach and discover rich structures of bi-integrable and tri-integrable couplings by manipulating matrix Lie algebras.
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Parini, Robert Charles. "Classical integrable field theories with defects and near-integrable boundaries." Thesis, University of York, 2018. http://etheses.whiterose.ac.uk/20428/.
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In the first part of this thesis algebro-geometric solutions for the sine-Gordon and KdV equations in the presence of a type I integrable defect are found, generalising the previously known soliton solutions. Elliptic (genus one) solutions where the defect induces only a phase shift are obtained via ansätze for the fields on each side of the defect. Algebro-geometric solutions for arbitrary genus and involving soliton emission by the defect are constructed using a Darboux transformation, exploiting the fact that the defect equations have the form of a Bäcklund transformation at a point. All the soliton and phase-shifted elliptic solutions to the defect equations are recovered as limits of the algebro-geometric solutions constructed in this way. Certain energy and momentum conserving defects for the Kadomtsev-Petviashvili equation are then presented as a first step towards the construction of integrable defects in higher dimensions. Algebro-geometric solutions to the sine-Gordon equation on the half-line with an integrable two parameter boundary condition are obtained by imposing a corresponding restriction on the Lax pair eigenfunction or, alternatively, as a Darboux transformation of the known algebro-geometric solution for the Dirichlet boundary. Finally, the collision of sine-Gordon solitons with a Robin type boundary is examined. This boundary is typically non-integrable but becomes an integrable Neumann or Dirichlet boundary for certain values of a boundary parameter. Depending on the boundary parameter and initial velocity an antikink may be reflected into various combinations of kinks, antikinks and breathers. The soliton content of the field after the collision is numerically determined by computing the discrete scattering data associated with the inverse scattering method. A highlight of this investigation is the discovery of an intricate structure of resonance windows caused by the production of a breather which can collide multiple times with the boundary before escaping as a lighter breather or antikink.
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Calini, Annalisa Maria. "Integrable curve dynamics." Diss., The University of Arizona, 1994. http://hdl.handle.net/10150/186987.
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The Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³. The role of orthonormal frames of the curve is explored. In this framework a second Poisson structure for the Heisenberg Model is derived and the relation between the Heisenberg Model and the cubic Non-Linear Schrodinger Equation is explained. The Frenet frame of a curve is shown to be a Legendrian curve in the space of orthonormal frames with respect to a natural contact structure. As a consequence, generic singularities of the solution of the Heisenberg Model and topological invariants of the curve are computed. The family of multi-phase solutions of the Heisenberg Model and the corresponding curves are constructed with techniques of algebraic geometry. The relation with the Non-Linear Schrodinger Equation is explained also in this context. A formula for the Backlund transformation for the Heisenberg Model is derived and applied to construct orbits homoclinic to planar circles. As a result singular knots are obtained.
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Saksida, Pavle. "Geometry of integrable systems." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.308545.
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Cheng, Y. "Theory of integrable lattices." Thesis, University of Manchester, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.568779.
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This thesis deals with the theory of integrable lattices in "solitons" throughout. Chapter 1 is a general introduction, which includes an historical survey and a short surrunary of the "solitons" theory and the present work. In Chapter 2, we discuss the equivalence between two kinds of lattice AKNS spectral problems - one includes two potentials, while the other includes four. The two nonlinear lattice systems associated with those two spectral problems, respectively is also proved to be equivalent to each other. In Chapter 3, we derive a class of nonlinear differential-difference equations (NDDEs) and put them into the Hamiltonian systems. Their complete integrability are proved in terms of so called "r-matrix". In the end of this Chapter, we study the symmetry properties and the related topics for lattice systems. In particular, we give detail for the Toda lattice systems. Chapter 4 is concerned with the Backlund transformations (BTs) and nonlinear superposition formulae (NSFs) for a class of NDDEs. A new method is presented to derive the generalized BTs and to prove that these BTs are precisely and really the auto-BTs. The three kinds of NSFs are derived by analysis of so called "elementary BTs". In Chapter 5, we investigate some relations between our lattices and the well-studied continuous systems. The continuum limits of our lattice systems and the discretizations of the continuous systems are discussed. The other study is about how we can consider a BT of continuous systems as a NDDE and then how a BT of such a NDDE can be reduced to the three kinds of NSFs of the continuous systems. The last Chapter is a study of integrable lattices under periodic boundary conditions. It provides a mathematical foundation for the study of integrable models in statistical mechanics. We are particularly interested in the lattice sine-Gordon and sinh-Gordon models. We not only prove the integrability of these models but also derive all kinds of classical phase shifts and some other physically interesting relations.
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Moorhouse, Thomas. "Methods for integrable systems." Thesis, Durham University, 1994. http://etheses.dur.ac.uk/5484/.
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This work concerns the study of certain methods for investigating integrable systems, and the application of these methods to specific problems and examples. After introducing the notion of integrability in chapters 1 and 2, we go on, in chapter 3, to develop a novel type of discrete integrable equation by considering ways of enforcing Leibniz's rule for finite difference operators. We look at several approaches to the problem, derive some solutions and study several examples. Chapter 4 describes a numerical implementation of a method for solving initial value problems for an integrable equation in 2+1 dimensions, exploiting the integrability of the equation. The introduction of twisters enables a powerful scheme to be developed. In chapter 5 Darboux transformations derived from the factorisation of a scattering problem are examined, and a general operator form considered. The topic of chapter 6 is the relationship between the Darboux transform for the sine-Gordon and related equations and certain ansatze established by twistor methods. Finally in chapter 7 a geometric setting for partial differential equations is introduced and used to investigate the structure of Bäcklund transformations and generalised symmetries.
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Berntson, B. K. "Integrable delay-differential equations." Thesis, University College London (University of London), 2017. http://discovery.ucl.ac.uk/1566618/.
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Delay-differential equations are differential-difference equations in which the derivatives and shifts are taken with respect to the same variable. This thesis is concerned with these equations from the perspective of the theory of integrable systems, and more specifically, Painlevé equations. Both the classical Painlevé equations and their discrete analogues can be obtained as deautonomizations of equations solved by two-parameter families of elliptic functions. In analogy with this paradigm, we consider autonomous delay-differential equations solved by elliptic functions, delay-differential extensions of the Painlevé equations, and the interrelations between these classes of equations. We develop a method to identify delay-differential equations that admit families of elliptic solutions with at least two degrees of parametric freedom and apply it to two natural 16-parameter families of delay-differential equations. Some of the resulting equations are related to known models including the differential-difference sine-Gordon equation and the Volterra lattice; the corresponding new solutions to these and other equations are constructed in a number of examples. Other equations we have identified appear to be new. Bäcklund transformations for the classical Painlevé equations provide a source of delay-differential Painlevé equations. These transformations were previously used to derive discrete Painlevé equations. We use similar methods to identify delay-differential equations with continuum limits to the first classical Painlevé equation. The equations we identify are solved by elliptic functions in particular limits corresponding to the autonomous limit of the classical first Painlevé equation.
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Hadad, Yaron. "Integrable Nonlinear Relativistic Equations." Diss., The University of Arizona, 2013. http://hdl.handle.net/10150/293490.
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This work focuses on three nonlinear relativistic equations: the symmetric Chiral field equation, Einstein's field equation for metrics with two commuting Killing vectors and Einstein's field equation for diagonal metrics that depend on three variables. The symmetric Chiral field equation is studied using the Zakharov-Mikhailov transform, with which its infinitely many local conservation laws are derived and its solitons on diagonal backgrounds are studied. It is also proven that it is equivalent to a novel equation that poses a fascinating similarity to the Sinh-Gordon equation. For the 1+1 Einstein equation the Belinski-Zakharov transformation is explored. It is used to derive explicit formula for N gravitational solitons on arbitrary diagonal background. In particular, the method is used to derive gravitational solitons on the Einstein-Rosen background. The similarities and differences between the attributes of the solitons of the symmetric Chiral field equation and those of the 1+1 Einstein equation are emphasized, and their origin is pointed out. For the 1+2 Einstein equation, new equations describing diagonal metrics are derived and their compatibility is proven. Different gravitational waves are studied that naturally extend the class of Bondi-Pirani-Robinson waves. It is further shown that the Bondi-Pirani-Robinson waves are stable with respect to perturbations of the spacetime. Their stability is closely related to the stability of the Schwarzschild black hole and the relation between the two allows to conjecture about the stability of a wide range of gravitational phenomena. Lastly, a new set of equations that describe weak gravitational waves is derived. This new system of equations is closely and fundamentally connected with the nonlinear Schrödinger equation and can be properly called the nonlinear Schrödinger-Einstein equations. A few preliminary solutions are constructed.
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McAnally, Morgan Ashley. "Generalized D-Kaup-Newell integrable systems and their integrable couplings and Darboux transformations." Scholar Commons, 2017. https://scholarcommons.usf.edu/etd/7423.
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We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable.
We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of the original enlarged spectral problem generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original integrable couplings of a generalized D-Kaup-Newell soliton hierarchy to furnish its Hamiltonian structures. Then we apply the variational identity to the reduced integrable couplings. The reduced coupling system has a bi-Hamiltonian structure. Both integrable coupling systems retain the properties of infinitely many commuting high-order symmetries and conserved densities of their original subsystems and, again, are Liouville integrable.
In order to find solutions to a generalized D-Kaup-Newell integrable coupling system, a theory of Darboux transformations on integrable couplings is formulated. The theory pertains to a spectral problem where the spectral matrix is a polynomial in lambda of any order. An application to a generalized D-Kaup-Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Precise one-soliton-like solutions are given for the m-th order generalized D-Kaup-Newell integrable coupling system.
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Jetzer, Frédéric. "Completely integrable systems on supermanifolds." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0020/NQ55399.pdf.
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Rempel, William. "Curve evolution and integrable systems." Thesis, McGill University, 2010. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=95240.
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In this thesis, we explain the connection between completely integrable systems of evolution equations and the evolution of the differential invariants of group actions in the particular case of a curve in Rⁿ acted on by an affine group defined as the semidirect product of G and Rⁿ, where G is semisimple, as it is presented in the work of Marì-Beffa. This connection involves Fels and Olver's work on the theory of moving frames, infinite-dimensional Hamiltonian systems and Poisson brackets, and centrally extended loop groups and algebras. We also summarize the relevant theory both of Lie groups and Lie algebras and of differential invariants and symmetries of differential equations. groups and algebras. We also summarize the relevant theory both of Lie groups and Lie algebras and of differential invariants and symmetries of differential equations.Dans cette thèse, on explique le lien entre les systèmes d'équations d'évolution complètement intégrables et l'évolution des invariants différentiels de l'action d'un groupe dans le cas particulier d'une courbe dans Rⁿ , sous l'action du produit semidirect d'un groupe G avec Rⁿ, où G est semisimple, tel que présenté par Marì-Beffa. Cette connection fait intervenir la théorie de repères mobiles de Fels et Olver, les systèmes hamiltoniens de dimension infinie et les crochets de Poisson, ainsi que les groupes et algèbres de lacets à extension centrale. On présente aussi un ré́sumé de la théorie pertinente des groupes et algèbres Lie et des invariants différentiels ainsi que les symmétries d'équations différentielles.
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Yu, Lei. "Reductions of dispersionless integrable hierarchies." Thesis, Imperial College London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.395540.
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Baraglia, David. "G2 geometry and integrable systems." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:30cf9c7c-157e-4204-b68b-08f6e199ef36.
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We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.
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Löbner, Clemens. "Integrable Approximations for Dynamical Tunneling." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-178216.
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Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. For many applications it is useful to approximate the regular dynamics of such a mixed system H by an integrable approximation Hreg. We present a new, iterative method to construct such integrable approximations. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. In contrast to other known approaches, our method remains applicable to strongly non-integrable systems H. We present its application to 2D maps and 2D billiards. Based on the obtained integrable approximations we finally discuss the theoretical description of dynamical tunneling in mixed systemsTypische Hamiltonsche Systeme haben einen gemischten Phasenraum, in dem disjunkte Bereiche klassisch regulärer und chaotischer Dynamik koexistieren. Für viele Anwendungen ist es zweckmäßig, die reguläre Dynamik eines solchen gemischten Systems H durch eine integrable Näherung Hreg zu beschreiben. Wir stellen eine neue, iterative Methode vor, um solche integrablen Näherungen zu konstruieren. Diese Methode basiert auf der Konstruktion einer integrablen Näherung in Winkel-Wirkungs-Variablen, die im Phasenraum durch iterative Anwendungen kanonischer Transformationen verbessert wird. Im Gegensatz zu bisher bekannten Verfahren bleibt unsere Methode auch auf stark nichtintegrable Systeme H anwendbar. Wir demonstrieren sie anhand von 2D-Abbildungen und 2D-Billards. Mit den gewonnenen integrablen Näherungen diskutieren wir schließlich die theoretische Beschreibung von dynamischem Tunneln in gemischten Systemen
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Mulvey, Joseph Anthony. "Symmetry methods for integrable systems." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5379/.
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This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.
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Sun, Yi Ph D. Massachusetts Institute of Technology. "Quantum intertwiners and integrable systems." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/104579.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 223-229).
We present a collection of results on the relationship between intertwining operators for quantum groups and eigenfunctions for quantum integrable systems. First, we study the Etingof-Kirillov Jr. expression of Macdonald polynomials as traces of intertwiners of quantum groups in the Gelfand-Tsetlin basis. In the quasiclassical limit, we obtain a new Harish-Chandra type integral formula for Heckman- Opdam hypergeometric functions. This formula is related to an integral formula appearing in recent work of Borodin-Gorin by integration over Liouville tori of the Gelfand-Tsetlin integrable system. At the quantum level, we obtain a new proof of the branching rule for Macdonald polynomials which transparently relates branching of Macdonald polynomials to branching of quantum group representations. Second, we study traces of intertwiners for quantum affine algebras. In the sl2 case, we show that, when valued in the three-dimensional evaluation representation, such traces converge in a certain region of parameters and provide a representation-theoretic construction of Felder-Varchenko's hypergeometric solutions to the q-KZB heat equation. This gives the first proof that such a trace function converges and resolves the first case of a conjecture of Etingof-Varchenko. As an application, we prove Felder-Varchenko's conjecture that their elliptic Macdonald polynomials are related to Etingof-Kirillov Jr.'s affine Macdonald polynomials. In the general case, we modify the setting of the work of Etingof-Schiffmann-Varchenko to show that traces of such intertwiners satisfy four commuting systems of q-difference equations - the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations.
by Yi Sun.
Ph. D.
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Mei, Zhongtao. "Wave Functions of Integrable Models." University of Cincinnati / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1530880774625297.
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Ohyama, Yousuke. "Self-duality and Integrable Systems." 京都大学 (Kyoto University), 1990. http://hdl.handle.net/2433/86420.
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Kiesenhofer, Anna. "Integrable systems on b-symplectic manifolds." Doctoral thesis, Universitat Politècnica de Catalunya, 2016. http://hdl.handle.net/10803/457145.
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The study of b-symplectic manifolds was initiated in 2012 by the works of Victor Guillemin, Eva Miranda and Ana Rita Pires (Adv. Math. 264 (2014), 864¿896). These manifolds, which can be understood as symplectic manifolds with singularities, have since then become the object of intense study. In the language of Poisson tensors, a b-symplectic manifold is a manifold $M^{2n}$ with a Poisson tensor $\Pi$ such that $\Pi^n$ vanishes transversally to the zero section of the bundle $¿^{2n}TM$. This thesis contributes important results about the dynamics of b-symplectic manifolds. After reviewing the general theory of Poisson and $b$-symplectic manifolds we present the definitions of integrable systems on these manifolds. The main results of this thesis are the action-angle coordinate theorems for commutative and non-commutative b-integrable systems [KMS, KM2], which state the existence of invariant "Liouville" tori on the singular set of the b-symplectic manifold, in perfect analogy to the symplectic case. We go on to present a cotangent model of this result, identifying a neighborhood of such a Liouville torus with a certain cotangent lift of a torus action [KM1]. These models also allow us to construct examples of b-integrable systems using torus actions as a starting point. The existence of action-angle coordinates motivates us to explore an analogue of the classical stability result for symplectic manifolds known as KAM theory. We prove a result that shows stability of a large number of invariant tori under certain perturbations. Finally, we present several examples of singular symplectic structures that arise naturally as the result of non-canonical transformations used in regularization of singularities in celestial mechanics [DKM].El estudio de variedades b-simplécticas se inició en 2012 con el trabajo de Victor Guillemin, Eva Miranda y Ana Rita Pires (Adv. Math. 264 (2014), 864-896). Estas variedades que pueden ser interpretadas como variedades simplécticas con singularidades se han convertido en un campo de investigación muy activo. Desde el punto de vista de estructuras de Poisson, una variedad b-sympléctica se define como una variedad de dimensión par (orientada) $M^{2n}$ con una estructura de Poisson $\Pi$ tal que $\Pi^n$ se anula transversalmente como sección del fibrado $¿^{2n}TM$. Esta tesis contribuye resultados importantes en geometría y dinámica de estas variedades b-simplécticas. Tras explicar las nociones básicas sobre variedades de Poisson y variedades b-simplécticas introducimos las definiciones de sistemas integrables en estas variedades. En el caso de variedades b-simplécticas los sistemas integrables que investigamos son llamados "sistemas b-integrables". Los resultados principales de esta tesis son teoremas de coordenadas acción-ángulo para sistemas b-integrables en el caso conmutativo y no conmutativo [KMS, KM2]. Este teorema demuestra la existencia de toros invariantes, llamados toros de Liouville, en el conjunto singular de la variedad b-simpléctica, y su estructura b-simpléctica en un entorno del toro. Posteriormente presentamos un modelo cotangente que nos permite de identificar un entorno de un toro Liouville con un tipo de lift cotangente de una acción de un toro [KM1]. Estos modelos también nos permiten la construcción de ejemplos de sistemas b-integrables utilizando acciones de un toro como punto de partida. El teorema de coordenadas acción-ángulo es la motivación para explorar la estibilidad de sistemas b-integrables de manera analoga al resultado clásico de la teoria KAM para variedades simplécticas. Nuestro teorema KAM demuestra la existencia de un "gran número" de toros Liouville que son invariantes bajo pequeñas perturbaciones de cierta forma. Para terminar, presentamos varios ejemplos de estructuras simplécticas singulares que aparecen como resultado de transformaciones non-canónicas en la regularización de singularidades en mecánica celeste [DKM].
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Pehlivan, Yamac. "Matrix Quantum Mechanics And Integrable Systems." Phd thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605065/index.pdf.
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In this thesis we improve and extend an algebraic technique pioneered by M. Gaudin. The technique is based on an infinite dimensional Lie algebra and a related family of mutually commuting Hamiltonians. In order to find energy eigenvalues of such Hamiltonians one has to solve the equations of Bethe ansatz. However, in most cases analytical solutions are not available. In this study we examine a special case for which analytical solutions of Bethe ansatz equations are not needed. Instead, some special properties of these equations are utilized to evaluate the energy eigenvalues. We use this method to find exact expressions for the energy eigenvalues of a class of interacting boson models.
In addition to that, we also introduce a q-deformation of the algebra of Gaudin. This deformation leads us to another family of mutually commuting Hamiltonians which we diagonalize using algebraic Bethe ansatz technique. The motivation for this deformation comes from a relationship between Gaudin algebra and a spin extension of the integrable model of F. Calogero. Observing this relation, we then consider a well known periodic version of Calogero's model which is due to B. Sutherland. The search for a Gaudin-like algebraic structure which is in a similar relationship with the spin extension of Sutherland'
s model naturally leads to the above mentioned q-deformation of Gaudin algebra. The deformation parameter q and the periodicity d of the Sutherland model are related by the formula q=i{pi}/d.
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Hajjar, Ara. "An integrable mixed-signal test system." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape8/PQDD_0027/MQ50616.pdf.
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Hajjar, Ara. "An integrable mixed-signal test system /." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=21298.
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The growing need for integrable test solutions has prompted the creation of various test bus standards. A mixed-signal test core is an ideal complement to these standards. This work presents the design and implementation of an integrable test system. The design consists of two major components: a stimulus generator, and a waveform extractor.A memory-based generator is used to construct the stimulus generation component. Such a circuit repeats a finite portion of an infinite-length PDM sequence in order to produce any arbitrary analog waveform. The circuitry is simple to design---it is comprised of a scan chain, and a 1-bit DAC; it is also area-efficient and robust (mostly digital design). Furthermore, since the analog signal is generated from a digital bit-stream, it is both stable and repeatable.
The extraction component of the test system focuses on the capture of steady-state type responses. A novel A/D algorithm is presented: the Multi-Pass technique. By taking advantage of repetitive waveforms, the Multi-Pass convertor achieves both area-efficiency and high-speed performance. A single on-chip comparator and sample-and-hold circuit is sufficient to extract analog waveforms. In addition, a novel, area-efficient, integrable, and highly-linear voltage reference design is presented.
Experimental results from two prototype boards serve to validate the proposed test system design. The first board implements the system using discrete components; the second makes use of a custom IC fabricated in a 0.5 mum CMOS process. The work presented in this thesis provides the groundwork for obtaining a practical and fully integrable mixed signal test system.
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McCarthy, Oscar Daniel. "Dispersionless integrable systems of KdV type." Thesis, University of Hull, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.395435.
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Sooman, Craig. "Soliton solutions of noncommutative integrable systems." Thesis, University of Glasgow, 2010. http://theses.gla.ac.uk/1449/.
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This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota's direct method for finding multi-soliton solutions of an integrable system. Chapter 2 extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mKP equations. Chapter 2 also covers the concepts of hierarchies and Darboux transformations. The Darboux transformations are iterated to give multi-soliton solutions of the KP and mKP equations. Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian. These determinantal solutions are then verified directly. In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting. We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions. The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants. One family of solutions is termed ``quasiwronskian'' and the other ``quasigrammian'' as both reduce to Wronskian and Grammian determinants when their entries commute. Both families of solutions are then verified directly. The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo. Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices. These solutions have interesting interaction properties not found in the commutative setting. We therefore show various plots of the solutions illustrating these properties. In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mKP equation. The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original. The original material in Chapters 3 and 4 appears in \cite{gilson:nimmo:sooman2008} and in \cite{gilson:nimmo:sooman2009}. Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mKP equations. For both equations, we look at a three-dromion structure from which we then perform a detailed asymptotic analysis. This aymptotic forms show interesting interaction properties which are demonstrated by various plots. This chapter is entirely the author's own work. Chapter 6 presents a summary and conclusions of the thesis.
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Macfarlane, Susan R. "Quasideterminant solutions of noncommutative integrable systems." Thesis, University of Glasgow, 2010. http://theses.gla.ac.uk/1887/.
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Quasideterminants are a relatively new addition to the field of integrable systems. Their simple structure disguises a wealth of interesting and useful properties, enabling solutions of noncommutative integrable equations to be expressed in a straightforward and aesthetically pleasing manner. This thesis investigates the derivation and quasideterminant solutions of two noncommutative integrable equations - the Davey-Stewartson (DS) and Sasa-Satsuma nonlinear Schrodinger (SSNLS) equations. Chapter 1 provides a brief overview of the various concepts to which we will refer during the course of the thesis. We begin by explaining the notion of an integrable system, although no concrete definition has ever been explicitly stated. We then move on to discuss Lax pairs, and also introduce the Hirota bilinear form of an integrable equation, looking at the Kadomtsev-Petviashvili (KP) equation as an example. Wronskian and Grammian determinants will play an important role in later chapters, albeit in a noncommutative setting, and, as such, we give an account of their widespread use in integrable systems. Chapter 2 provides further background information, now focusing on noncommutativity. We explain how noncommutativity can be defined and implemented, both specifically using a star product formalism, and also in a more general manner. It is this general definition to which we will allude in the remainder of the thesis. We then give the definition of a quasideterminant, introduced by Gel'fand and Retakh in 1991, and provide some examples and properties of these noncommutative determinantal analogues. We also explain how to calculate the derivative of a quasideterminant. The chapter concludes by outlining the motivation for studying our particular choice of noncommutative integrable equations and their quasideterminant solutions. We begin with the DS equations in Chapter 3, and derive a noncommutative version of this integrable system using a Lax pair approach. Quasideterminant solutions arise in a natural way by the implementation of Darboux and binary Darboux transformations, and, after describing these transformations in detail, we obtain two types of quasideterminant solution to our system of noncommutative DS equations - a quasi-Wronskian solution from the application of the ordinary Darboux transformation, and a quasi-Grammian solution by applying the binary transformation. After verification of these solutions, in Chapter 4 we select the quasi-Grammian solution to allow us to determine a particular class of solution to our noncommutative DS equations. These solutions, termed dromions, are lump-like objects decaying exponentially in all directions, and are found at the intersection of two perpendicular plane waves. We extend earlier work of Gilson and Nimmo by obtaining plots of these dromion solutions in a noncommutative setting. The work on the noncommutative DS equations and their dromion solutions constitutes our paper published in 2009. Chapter 5 describes how the well-known Darboux and binary Darboux transformations in (2+1)-dimensions discussed in the previous chapter can be dimensionally-reduced to enable their application to (1+1)-dimensional integrable equations. This reduction was discussed briefly by Gilson, Nimmo and Ohta in reference to the self-dual Yang-Mills (SDYM) equations, however we explain these results in more detail, using a reduction from the DS to the nonlinear Schrodinger (NLS) equation as a specific example. Results stated here are utilised in Chapter 6, where we consider higher-order NLS equations in (1+1)-dimension. We choose to focus on one particular equation, the SSNLS equation, and, after deriving a noncommutative version of this equation in a similar manner to the derivation of our noncommutative DS system in Chapter 3, we apply the dimensionally-reduced Darboux transformation to the noncommutative SSNLS equation. We see that this ordinary Darboux transformation does not preserve the properties of the equation and its Lax pair, and we must therefore look to the dimensionally-reduced binary Darboux transformation to obtain a quasi-Grammian solution. After calculating some essential conditions on various terms appearing in our solution, we are then able to determine and obtain plots of soliton solutions in a noncommutative setting. Chapter 7 seeks to bring together the various results obtained in earlier chapters, and also discusses some open questions arising from our work.
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Hassan, M. U. "Aspects of integrable nonlinear sigma models." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603844.
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In the first chapter we introduce the subject and review some existing ideas about conservation laws and dual symmetry in nonlinear sigma models. We also give a brief introduction to supersymmetry in two-dimensional spacetime. In chapter 2 we construct local conserved charges for each PCM (with and without a Wess-Zumino term), which are based on symmetric invariant tensors of the underlying Lie algebra. We then study their Poisson bracket algebra and we find that there exists an infinite family of commuting local charges with spins equal to the exponents modulo the Coxeter number of the algebra. These local charges are shown to be in involution with the nonlocal charges of the PCM (with and without a Wess-Zumino term). Finally we discuss conservation of some of these charges in the quantum PCMs. In chapter 3 we construct two families, each with finitely many members, of commuting local charges with spins equal to the exponents of the algebra (but with no repetition modulo the Coxeter number). We briefly discuss the effect of adding a Wess-Zumino term in the SPCM and its behaviour at critical (WZW) points. The local charges in the PCMs and SPCMs are only indirectly related. At the end of the chapter we comment on quantum conservation of some higher spin superfield currents in the SPCMs. Chapter 4 deals with the extension of dual symmetry to superspace. We explain this for super O(l) sigma models and derive an infinite series of nonlocal conserved charges in terms of both superfields and component fields. We elaborate on the BIZZ procedure in the SPCM and derive Lie algebra valued nonlocal charges in this model. We also show that the nonlocal charges are in involution with the local charges in the SPCM. At the end of the dissertation we give a summary of the main results and briefly discuss some open problems for further research. There follow appendices which collect some background material on Lie algebras, invariant tensors, Yangian algebras, and some details of counting about quantum conservation of higher spin local currents in the PCMs and the SPCMs.
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Adamopoulou, Panagiota-Maria. "Differential equations and quantum integrable systems." Thesis, University of Kent, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655223.
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This thesis explores several aspects of the correspondence between classes of linear ordinary differential equations (ODEs) in the complex plane and certain quantum integrable models (IMs), also known as the ODE/IM correspondence. First, we enlarge the set of ordinary differential equations that enter the correspondence. Differential equations satisfied by Wronskians between solutions of specific ODEs are obtained and are associated to nodes of particular Dynkin diagrams. In the second part of the thesis we generalise the correspondence to encompass massive IMs. Starting from an integrable nonlinear partial differential equation corresponding to the classical A2(l) affine Toda field theory (ATFT), we expand the set of integrable models that enter the correspondence. This establishes an ODE/IM correspondence for a massive IM. We then extend the results to the An-1 (1) ATFTs and the particular example of D3 (l) ATFT.
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Beck, Florian [Verfasser], Emanuel [Akademischer Betreuer] Scheidegger, and Katrin [Akademischer Betreuer] Wendland. "Hitchin and calabi-yau integrable systems." Freiburg : Universität, 2016. http://d-nb.info/1125905840/34.
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Strachan, Ian Alexander Becket. "The twistor description of integrable systems." Thesis, Durham University, 1991. http://etheses.dur.ac.uk/5863/.
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The theory of twistors and the theory of integrable models have, for many years, developed independently of each other. However, in recent years it has been shown that there is considerable overlap between these two apparently disparate areas of mathematical physics. The aim of this thesis is twofold; firstly to show how many known integrable models may be given a natural geometrical/twistorial interpretation, and secondly to show how this leads to new integrable models, and in particular new higher dimensional models. After reviewing those elements of twistor theory that are needed in the thesis, a generalisation of the Yang-Mills self-duality equations is constructed. This is the framework into which many known examples of integrable models may be naturally fitted, and it also provides a simple way to construct higher dimensional generalisations of such models. Having constructed new examples of (2 + l)-dimensional integrable models, one of these is studied in more detail. Embedded within this system are the sine-Gordon and Non-Linear Schrodinger equations. Some solutions of this (2 + l)-dimensional integrable model are found using the 'Riemann Problem with Zeros' method, and these include the sohton solutions of the SG and NLS equations. The relation between this approach and one based the Atiyah-Ward ansatze is dicussed briefly. Scattering of localised structures in integrable models is very different from scattering in non-integrable models, and to illustrate this the scattering of vortices in a modified Abelian-Higgs model is considered. The scattering is studied, for small speeds, using the 'slow motion approximation' which involves the calculation of a moduli space metric. This metric is found for a general TV-lump vortex configuration. Various examples of scattering processes are discussed, and compared with scattering in an integrable model. Finally this geometrical approach is compared with other approaches to the study of integrable systems, such as the Hirota method. The thesis closes with some suggestions for how the KP equation may be fitted into this geometrical/twistorial scheme.
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Dimakos, Michail. "Linear, linearisable and integrable nonlinear PDEs." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.607875.
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Nieri, Fabrizio. "Integrable structures in supersymmetric gauge theories." Thesis, University of Surrey, 2015. http://epubs.surrey.ac.uk/808914/.
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In this thesis we study partition functions of supersymmetric gauge theories on compact backgrounds in various dimensions, with particular focus on infinite dimensional symmetry algebras encoded in these observables. The compact space partition functions of the considered theories can be decomposed into products of holomorphic blocks which are identified with partition functions on elementary geometries. Partition functions on different compact spaces can be obtained by fusing the holomorphic blocks with pairings reflecting the geometric decomposition of the backgrounds. An example of this phenomenon is given by the S4 partition function of 4d N = 2 theories, which can be written as an integral of two copies of the R4 Nekrasov partition function. Remarkably, the AGT correspondence identifies the S4 partition function of class S theories with Liouville CFT correlators. The perturbative integrand is identified with the product of CFT 3-point functions, while each copy of the non-perturbative instanton partition function is identified with conformal blocks of the Virasoro algebra. In this work we define a class of q-deformed CFT correlators, where chiral blocks are controlled by the q-Virasoro algebra and are identified with R4xS1 instanton partition functions. We derive the 3-point functions for two different q-deformed CFTs, and we show that non-chiral correlators can be identified with S5 and S4xS1 partition functions of certain 5d N = 1 theories. Moreover, particular degenerate correlators are mapped to S3 and S2xS1 partition functions of 3d N = 2 theories. This fits the interpretation of the 3d theories as codimension two defects. We also study 4d N = 1 theories on T2 fibrations over S2. We prove that when anomalies are canceled, the compact space partition functions can be expressed through holomorphic blocks associated to R2xT 2. We argue that for particular theories these objects descend from R4xT 2 partition functions, which we identify with the chiral blocks of an elliptically deformed Virasoro algebra.
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Bianchini, D. "Entanglement entropy in integrable quantum systems." Thesis, City, University of London, 2016. http://openaccess.city.ac.uk/17490/.
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In this thesis I present the results I have been developing during my PhD studies at City University London. The original results are based on D Bianchini et al, D Bianchini, O Castro-Alvaredo and B Doyon, D Bianchini and F Ravanini, D Bianchini et al and D Bianchini and O Castro-Alvaredo. In all but one publications, we compute the entanglement of various systems. Using the celebrated “replica trick” we compute the entanglement entropy of non unitary systems using integrable tools in continuum and discrete models. In particular, in the first article we generalise the method described in the seventh article in order to take into account non unitary conformal systems. In the second article we use a form factor expansion to probe a non unitary system outside the critical point. In the fourth article we derive the explicit expressions of one dimensional quantum Hamiltonians which provide a lattice realisation of off critical non unitary minimal models. Using a Corner Transfer Matrix approach we compute the scaling of the entanglement of such spin chains. In the fifth article we study the scaling of various twist field correlation functions in order to compute the entanglement entropy and the logarithmic negativity in free boson massive theories.
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WICKRAMASINGHE, J. M. A. S. P. "MESOSCOPIC FEATURES OF CLASSICALLY INTEGRABLE SYSTEMS." University of Cincinnati / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1140806635.
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Walker, Alan James. "Similarity reductions and integrable lattice equations." Thesis, University of Leeds, 2001. http://etheses.whiterose.ac.uk/7190/.
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In this thesis I extend the theory of integrable partial difference equations (PAEs) and reductions of these systems under scaling symmetries. The main approach used is the direct linearization method which was developed previously and forms a powerful tool for dealing with both continuous and discrete equations. This approach is further developed and applied to several important classes of integrable systems. Whilst the theory of continuous integrable systems is well established, the theory of analogous difference equations is much less advanced. In this context the study of symmetry reductions of integrable (PAEs) which lead to ordinary difference equations (OAEs) of Painleve type, forms a key aspect of a more general theory that is still in its infancy. The first part of the thesis lays down the general framework of the direct linearization scheme and reviews previous results obtained by this method. Most results so far have been obtained for lattice systems of KdV type. One novel result here is a new approach for deriving Lax pairs. New results in this context start with the embedding of the lattice KdV systems into a multi-dimensional lattice, the reduction of which leads to both continuous and discrete Painleve hierarchies associated with the Painleve VI equation. The issue of multidimensional lattice equations also appears, albeit in a different way, in the context of the lattice KP equations, which by dimensional reduction lead to new classes of discrete equations. This brings us in a natural way to a different class of continuous and discrete systems, namely those which can be identified to be of Boussinesq (BSQ) type. The development of this class by means of the direct linearization method forms one of the major parts of the thesis. In particular, within this class we derive new differential-difference equations and exhibit associated linear problems (Lax pairs). The consistency of initial value problems on the multi-dimensional lattice is established. Furthermore, the similarity constraints and their compatibility with the lattice systems guarantee the consistency of the reductions that are considered. As such the resulting systems of lattice equations are conjectured to be of Painleve type. The final part of the thesis contains the general framework for lattice systems of AKNS type for which we establish the basic equations as well as similarity constraints.
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Kameyama, Takashi. "Integrable deformations of the AdS5×S5 superstring." 京都大学 (Kyoto University), 2016. http://hdl.handle.net/2433/215302.
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Takuya Matsumoto and Kentaroh Yoshida 2014 J. Phys.: Conf. Ser. 563 012020Kyoto University (京都大学)
0048
新制・課程博士
博士(理学)
甲第19489号
理博第4149号
新制||理||1596(附属図書館)
32525
京都大学大学院理学研究科物理学・宇宙物理学専攻
(主査)教授 畑 浩之, 教授 田中 貴浩, 教授 川合 光
学位規則第4条第1項該当
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Sorrell, Mark. "Quantum integrable systems and Schrodinger Eigenvalue problems." Thesis, Heriot-Watt University, 2008. http://hdl.handle.net/10399/2176.
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Dunajski, Maciej. "The nonlinear graviton as an integrable system." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298312.
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Wilbourne, Ruth Margaret. "Integrable boundary flows and the g-function." Thesis, Durham University, 2012. http://etheses.dur.ac.uk/4939/.
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This thesis explores renormalisation group flows in integrable quantum field theories with boundaries, as described by the g-function. The main focus is on the g-function in the staircase model, the renormalisation group flow of which passes close to the unitary minimal models. This g-function is used to identify flows between boundary conditions both within and between the minimal models. In certain limits the $\mathcal{M}A_m {(+)}$ theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the $\mathcal{M}A_4 {(+)}$ g-function are discussed, as is initial work on the g-function for the $\mathcal{M}A_4 {(-)}$ theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in $\mathcal{M}A_4 {(+)}$. Expressions are also found for excited state versions of the $\mathcal{M}A_m {(+)}$ g-function, and these allow the unique identification of certain boundary flows.
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Newsham, Samantha. "Linear systems and determinants in integrable systems." Thesis, Lancaster University, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.663238.
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The thesis concerns linear systems and scattering theory. In particular, it presents lineal' systems for some integrable systems and finds discrete analogues for many well known results for continuous variables. It introduces some new tools from linear systems and applies them to standard integrable systems. We begin by expressing the first Painleve equation as the compatibility condition of a certain Lax pair and introduce the Korteweg-de Vries partial differential equation. We introduce the spectral curve for algebraic families and the Toda lattice. The Fredholm determinant of a trace class Hankel integral operator gives rise to a tau function. Dyson used the tau function to solve an inverse spectral problem for Schrodinger operators. When a plane wave is subject to Schrodinger's equation and scattered by a potential u, the output is described at great distances by a scattering function. The spectral problem is to find the spectrum of Schrodinger's operator in L2 and hence the scattering function. The inverse spectral problem is to find the potential given the scattering function. The scattering and inverse scattering problems are linked by the Gelfand- Levitan equation. In this thesis, for a discrete linear system, we introduce a scattering function and Hankel matrix and a version of the Gelfand-Levitan equation for discrete linear systems. We introduce the discrete operator ∑∞/k=n AkBCAk and use it to solve the Gelfand-Levitan equation and compute Fredholm determinants of Hankel operators. We produce a discrete analogue of a calculation of Poppe giving a solution to the Korteweg-de Vries equation and via the methods of linear systems find an analogous solution in terms of Hankel matrices. We then produce a discrete analogue of the Miura transform. Thus the main new contributions of this thesis are the discrete analogues of the R operator, the Gelfand- Levitan equation, the Lyapunov equation and the Miura transform.
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Wacheux, Christophe. "Semi-toric integrable systems and moment polytopes." Phd thesis, Université Rennes 1, 2013. http://tel.archives-ouvertes.fr/tel-00932926.
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Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.
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Rashid, Maher S. "Quasi-integrable models in (2+1) dimensions." Thesis, Durham University, 1992. http://etheses.dur.ac.uk/5780/.
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Recently σ-models have received a lot of attention for many reasons. One interesting aspect of the CP(^n) sigma models is the fact they are the simplest Lorentz invariant models which possess topologically stable (minimum of the action) solutions in (2+0) dimensions. Unfortunately, it appears that Lorentz covariance and integrability are incompatable in (2+1) dimensions. In the literature a few integrable models were constructed in (2+1) dimensions at the expense of Lorentz invariance (e.g. modified chiral model,...). An alternative way to proceed is to retain Lorentz invariance and relax the property of integrability by replacing it with a new property of quasi-integrability. Zakrzewski and others have constructed an example of such quasi-integrable models. Their example is based on the CP(^1) model modified by the addition of two stabilising terms (the first called the "Skyrme-like" term and the second the "potential-like" term) to the basic Lagrangian. In this thesis we have addressed the following relevant questions: How unique is this model? What are the properties of its static structures (skyrmions)? Is it possible to generalise this model? Is quasi-integrabilty, as a property, shared by all CP(^2) models, or it is only restricted to the CP(^1) model? It turns out that the first stabilising term [i.e the Skyrme-like term) is only unique for CP(^1) model and this uniqueness does not survive the generalisations to larger coset spaces, say, CP(^n). The second stabilising term is not unique. By taking advantage of this observation, i.e arbitrariness of the potential term, a generalisation of Zakrzewski's model has become possible. Most important of all is the fact that all the CP" models are quasi-integrable provided one incurs the size instabilities of their soliton solutions.
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Ward, Nicholas Frank Dudley. "Atomic decompositions of integrable or continuous functions." Thesis, University of York, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.306356.
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Arnaudon, Alexis. "Geometric deformations of integrable systems and beyond." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/54827.
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The results of this work were obtained within a single mathematical framework which consists of using the geometrical structure of a given problem not only to study its properties but to implement geometrically consistent deformations. The present interest in such geometric deformations is not only to learn more about the original problem but also to explore new directions, beginning from a common and well-understood starting point and continuing at the interface between several mathematical and physical topics. This approach lays down the foundation of this work upon which various ideas for deformations of several standard systems will be developed and analysed. The mathematical framework that we will be relying on throughout these investigations is the use of symmetries in dynamical systems described by Lie groups which appear in many areas of mathematical physics and in particular in integrable systems, the second theme of this work. Indeed, the integrability of a dynamical system is often related to the existence of symmetries, but usually requires additional structures. The effect of the geometric deformations on the integrability of these systems will be one of the central questions that we will attempt to answer. To undertake such a program, we cannot restrict ourselves to the deformation of a single integrable system, but we must implement various ideas on several and different classic integrable equations in the aim of better characterising their effects and limitations. We will introduce noise, geometric dissipations and Sobolev norms in the classical integrable systems including the rigid body motion, the Toda lattice, the peakon system, partial differential equations such as KdV and NLS as well as the reduced Maxwell-Bloch equation.
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Alsallami, Shami Ali M. "Discrete integrable systems and geometric numerical integration." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/22291/.
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This thesis deals with discrete integrable systems theory and modified Hamiltonian equations in the field of geometric numerical integration. Modified Hamiltonians are used to show that symplectic schemes for Hamiltonian systems are accurate over long times. However, for nonlinear systems the series defining the modified Hamiltonian equation usually diverges. The first part of the thesis demonstrates that there are nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. Specifically, they arise as reductions of a lattice version of the Korteweg-de Vries (KdV) partial differential equation. We present cases of one and two degrees of freedom symplectic mappings, for which the modified Hamiltonian equations can be computed as a closed form expression using techniques of action-angle variables, separation of variables and finite-gap integration. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example is a system of an implicit dependence on the time step, which is obtained by dimensional reduction of a lattice version of the modified KdV equation. The second part of the thesis contains a different class of discrete-time system, namely the Boussinesq type, which can be considered as a higher-order counterpart of the KdV type. The development and analysis of this class by means of the B{\"a}cklund transformation, staircase reductions and Dubrovin equations forms one of the major parts of the thesis. First, we present a new derivation of the main equation, which is a nine-point lattice Boussinesq equation, from the B{\"a}cklund transformation for the continuous Boussinesq equation. Second, we focus on periodic reductions of the lattice equation and derive all necessary ingredients of the corresponding finite-dimensional models. Using the corresponding monodromy matrix and applying techniques from Lax pair and $r$-matrix structure analysis to the Boussinesq mappings, we study the dynamics in terms of the so-called Dubrovin equations for the separated variables.
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Hawkins, Michael Stuart. "Local conservation laws in quantum integrable systems." Thesis, University of Birmingham, 2009. http://etheses.bham.ac.uk//id/eprint/282/.
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One of the phenomena associated with quantum integrable systems is the possibility of persistent currents, i.e. currents which do not decay away entirely, but have some portion that continues to flow undiminished and indefinitely. These residual currents are shown to be the conserved part of the current operator, and calculable from the conservation laws of the system. In a particular system, previous attempts to calculate a known residual current from the conservation laws have failed. A numerical investigation is undertaken, and this disparity with the formal results is resolved by the inclusion of a previously overlooked conservation law. An important corollary to these results is that requiring the mutual commutativity of the conservation laws of a quantum integrable system, previously assumed by analogy with the classical case, is an unnecessary and potentially disastrous restriction. Methods of generating the local conservation laws of a quantum integrable system are investigated, and the current method of using a Boost operator is shown to be subtly flawed. The method is discovered to implicitly require additional knowledge in the form of Hamiltonian identities in order to avoid otherwise unphysical terms. A new method is proposed based on the idea that the logarithm of the Transfer matrix of a system generates these local conservation laws. The method is applicable to a wide class of systems whose Lax operator obeys a certain condition, and the majority of the work required to generate the local conservation laws is entirely general and thus only needs to be done once. This new method is then applied to two quite different spin-chain Hamiltonians, the XXZ and Hubbard models, and shown to successfully generate all of the known local conservation laws of these models and some new ones.
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Hone, Andrew N. W. "Integrable systems and their finite-dimensional reductions." Thesis, University of Edinburgh, 1996. http://hdl.handle.net/1842/15044.
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The first chapter introduces some of the important concepts and structures associated with integrability, and includes a brief overview of some of the applications of integrable systems and their reductions in field theory. Chapter 2 describes the scaling similarity reductions of the Sawada-Kotera, fifth-order KdV, and Kaup-Kupershmidt equations. Similarity solutions of these evolution equations satisfy certain ODEs which are naturally viewed as fourth-order analogues of the Painlevé transcendents; they may also be written as non-autonomous Hamiltonian systems, which are time-dependent generalizations of the integrable Hénon-Heiles systems. The solutions to these systems are encoded into a tau-function, and Bäcklund transformations are presented which allow the construction of rational solutions and some other special solutions. The third chapter is concerned with the motion of the poles of singular solutions (especially rational solutions) of the NLS equation. It is demonstrated that the linear problem for NLS admits an analogue of the well-known Crum transformation for Schrödinger operators, leading to the construction of a sequence of rational solutions. The poles and zeros of these rational solutions are found to satisfy constrained Calogero-Moser equations, and some other singular solutions are also considered. Much use is made of Hirota's bilinear formalism, as well as a trilinear form for NLS related to its reduction from the KP hierarchy. The final chapter deals with soliton solutions of the An(1) affine Toda field theories. By writing the soliton tau-functions as determinants of a particular form, these solutions are related to the hyperbolic spin Ruijsenaars-Schneider system. These results generalize the connection between the ordinary (non-spin) Ruijsenaars-Schneider model and the soliton solutions of the sine-Gordon equation.
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Greenman, Chris. "The spacing distributions of arithmetical integrable systems." Thesis, University of Edinburgh, 1995. http://hdl.handle.net/1842/14949.
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The level spacing distribution of the two dimensional harmonic oscillator is investigated. By obtaining an explicit expression for the spacings, it is observed that the distribution is unstable under the semi-classical limit h → 0. By defining a suitable average, a distribution stable under h → 0 is obtained. Exact expressions are obtained for values of oscillator frequency ratio including the golden mean, 1/2, 1/5 and 1/e. Comparisons are made between these analytic results and the numerical ones in the paper of Berry and Tabor [1]. For a certain class of ratio, including the case 1/e, a delta function is found for the averaged spacing distribution. This is a fractal set shown to have Hausdorff dimension ½ as a subset of possible ratios. The case of generic frequency ratio is also studied for which a closed formula is found. Comments on the distribution follow. For the harmonic oscillator of general dimension n it is shown that the initial value of the level spacing distribution is (n)-1. Reasons for the conjecture that the distribution will be (monotonically) decreasing are also given. By employing a method used in the system above, it is shown for the particle in a two dimensional box, with certain possible box dimensions, that the spacing distribution is distinct from the Poisson one associated with the system.
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Winn, Amanda Elizabeth. "Some classical integrable systems with topological solitons." Thesis, Durham University, 1999. http://etheses.dur.ac.uk/4309/.
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This thesis is concerned with some low dimensional non-linear systems of partial differential equations and their solutions. The systems are all in the classical domain and aside from a version of one model in Appendix D, are continuous. To begin with we examine the field equations of motion derived from Hamiltonian and Lagrangian densities, respectively defining the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models, where the metric on the target manifold is indefinite. The models are integrable in the sense that a suitable Lax pair exists, and admit solitonic solutions classifiable by an integer winding number. Such solutions are explicitly derived in both the static and time dependent cases where physical space X is the circle S(^1). The existence of travelling wave solutions of topological type is discussed for each model with X = S(^1) and X = R; explicit solutions are derived for the X = S(^1) case and it is shown for both the Heisenberg and sigma models, that no such travelling wave solutions exist if X is the real line. Nevertheless, time dependent solutions (not of travelling wave type) are possible in each case for X = R, some examples of which are derived explicitly. A further integrable system; the Hyperbolic 'Pivotal' model is proposed as a special case of a more general model on Hermitian symmetric spaces. Of particular interest is the fact that the Pivotal model interpolates between the previous two models. To begin with the integrability of the model is established via a Lax representation. Solutions analogous to some of those of the previous models are then derived and the interpolative limits examined with respect to the Heisenberg and sigma models. Conserved currents for the model are also briefly discussed. Finally, some conclusions and further possibilities are noted including a brief examination of a discrete version of the sigma model where the target manifold is positive definite. A Bogomol'nyi bound is shown to exist for the systems energy in terms of a well defined winding number.
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Satta, Giovanni. "Algebraic approach to supersymmetric integrable spin chains." Chambéry, 2008. http://www.theses.fr/2008CHAMS001.
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Cette thèse est consacrée à l'étude de la théorie mathématique sous-tendant la construction et la solution d'une classe particulière de systèmes quantiques exactement résolubles : son objectif est d'utiliser les superalgèbres de Lie comme un outil pour construire et résoudre les chaînes de spins intégrales. Nous développons une approche générale et systématique premettant de construire et traiter simultanément une large classe de systèmes intégrables partageant la même supersymétrie, allant du cas bien connu où tous les sites portent la représentation fondamentale (par exemple le modèle t-J) à des situations plus complexes d'intérêt physique comprenant des chaînes de spins alternées, avec impuretés etc. Les deux premiers chapitres sont consacrés à un examen des résultats connus concernant le Yangien du superalgèbre de Lie gl(mIn), nécessaire pour introduire la version graduée de la méthode de diffusion inverse quantique. Nous appliquons notre approche dans le chapitre 3 aux chaînes fermées et dans le chapitre 4 aux chaînes ouvertes. Dans ce chapitre sont étudiées les homologues super-symétriques de l'algèbre de réflexion et du Yangien twisté, qui sont les structures algébriques permettant d'imposer des conditions aux bords qui préservent l'intégralité. Dans le dernier chapitre la méthode de la fusion pour chaînes de spins avec supersymétrie sl(1I2) est traitée en détail. La méthode de solution que nous utilisons, tant dans les cas fermé et ouvert, est la généralisation au cas super-symétrique de l'Ansatz de Bethe analytique, dans laquelle les équations de Bethe paramétrant les nombres quantiques du système sont obtenues comme conditions d'analycité pour les valeurs propres des hamiltoniennesThese thesis deals with the mathematical theory underlying the construction and solution of a particular class of exactly solvable quantum systems : its aim is to use Lie superalgebras as a tool to build and solve integrable quantum spin chains. We develope a general and systematic approach allowing one to build and simultaneously treat a large class of integrable systems sharing the same supersymmetry, ranging from the well known case where all sites carry the fundamental representation (e. G. The t-J model), to more complicated situations of physical interest including alternating spin chains, chains with impurities etc. The two first chapters are devoted to a review of known results about the Yangian of the general Lie super algebra gℓ(m|n), necessary to introduce the graded version of the quantum inverse scattering method. We then apply our approach to periodic spin chains in chapter 3 and to opens spin chains in chapter 4. In this chapter, the supersymmetric counterparts of the reflection algebra and of the twisted Yangian are studied, these being the algebraic structures that allow one to impose boundary conditions that preserve integrability. In the last chapter the fusion method for spin chains with sℓ(1|2) supersymmetry is treated in detail. The solution method we use, both in the closed and open case, is the generalization to the supersymmetric case of the analytical Bethe Ansatz, in which the Bethe equations parametrizing the quantum numbers of the system are obtained as analyticity conditions for the spectrum of the hamiltonians
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HENRIET, GILDAS. "Etude, realisation et caracterisation d'un oscillateur integrable." Paris 6, 1991. http://www.theses.fr/1991PA066156.
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L'experience acquise a l'onera en matiere d'action et de detection capacitives permet la realisation d'oscillateurs utilisant des resonateurs en materiaux non piezoelectriques. Le materiau silicium a ete choisi, pour son faible amortissement mecanique ainsi que pour ses qualites de semiconducteur autorisant dans l'avenir l'integration d'une electronique de commande au voisinage du resonateur. Le mode de vibration est un mode de contour dans une plaque carree appele mode de lame. En l'absence de precautions particulieres, les pertes par les supports constituent la source principale de dissipation d'energie. Il s'est avere necessaire de definir un type de fixation qui minimise le couplage support element vibrant. Des methodes de calcul par elements finis ont ete dans ce but largement utilisees. Apres les etudes theoriques et experimentales du resonateur en silicium (sensibilite a la temperature, a la pression), nous avons realise et caracterise ce nouvel oscillateur