Monopoles are solutions of an SU(2) gauge theory in R3 satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their charge. Using methods from integrable systems, monopoles can be described in algebraic-geometric terms via their spectral curve, i.e. an algebraic curve, given as a polynomial P in two complex variables, satisfying certain constraints. In this thesis we focus on the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the Ercolani-Sinha constraints, given as relations amongst periods. In this thesis a particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by C3, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve X , subject to the Ercolani-Sinha constraints: the aim of the present work is to establish the existence of such monopoles, which translates into solving the Ercolani-Sinha constraints for X . Exploiting the symmetry of the system,we manage to recast the problem entirely in terms of a genus 2 hyperelliptic curve X, the (unbranched) quotient of X by C3 . A crucial step to this aim involves finding a basis forH1( X; Z), with particular symmetry properties according to a theorem of Fay. This gives a simple formfor the period matrix of X ; moreover, results by Fay and Accola are used to reduce the Ercolani-Sinha constraints to hyperelliptic ones on X. We solve these constraints onX numerically, by iteration using the tetrahedral monopole solution as starting point in the moduli space. We use the Arithmetic-GeometricMean method to find the periods onX: this method iswell understood for a genus 2 curve with real branchpoints; in this work we propose an extension to the situation where the branchpoints appear in complex conjugate pairs, which is the case for X. We are hence able to establish the existence of a curve of solutions corresponding to cyclic 3-monopoles.

This thesis concerns the relationship of submanifold geometry, in both the smooth and discrete sense, to representation theory and the theory of integrable systems. We obtain Lie theoretic generalisations of the transformation theory of projectively and Lie applicable surfaces, and M�obius-flat submanifolds of the conformal n-sphere. In the former case, we propose a discretisation. We develop a projective approach to centro-ane hypersurfaces, analogous to the conformal approach to submanifolds in spaceforms. This yields a characterisation of centro-ane hypersurfaces amongst M�obius-flat projective hypersurfaces using polynomial conserved quantities. We also propose a discretisation of curved flats in symmetric spaces. After developing the transformation theory for this, we see how Darboux pairs of discrete isothermicnets arise as discrete curved flats in the symmetric space of opposite point pairs. We show how discrete curves in the 2-sphere fit into this framework.

No description available.

This dissertation is concerned with analysis of orbital stability of solitary waves and well-posedness of the Cauchy problem in the integrable evolution equations. The analysis is developed by using tools from integrable systems, such as higher-order conserved quantities, B\"{a}cklund transformation, and inverse scattering transform. The main results are obtained for the massive Thirring model, which is an integrable nonlinear Dirac equation, and for the derivative NLS equation. Both equations are related with the same Kaup-Newell spectral problem. Our studies rely on the spectral properties of the Kaup-Newell spectral problem, which convey key information about solution behavior of the nonlinear evolution equations. / Dissertation / Doctor of Philosophy (PhD)

<p>We derive new combinatorail formulas for vector-valued weight functions for the evolution modules over the Yangians Y (gln). We obtain them using the Nested Algebraic Bethe ansatz method.</p> <p>We also describe the asymptotic behavior of the radial solutions of the negative tt∗ equation via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method.</p>

Two Poisson brackets are called compatible if any linear combination of these brackets is a Poisson bracket again. The set of non-zero linear combinations of two compatible Poisson brackets is called a Poisson pencil. A system is called bihamiltonian (with respect to a given pencil) if it is hamiltonian with respect to any bracket of the pencil. The property of being bihamiltonian is closely related to integrability. On the one hand, many integrable systems known from physics and geometry possess a bihamiltonian structure. On the other hand, if we have a bihamiltonian system, then the Casimir functions of the brackets of the pencil are commuting integrals of the system. We consider the situation when these integrals are enough for complete integrability. As it was shown by Bolsinov and Oshemkov, many properties of the system in this case can be deduced from the properties of the Poisson pencil itself, without explicit analysis of the integrals. Developing these ideas, we introduce a notion of linearization of a Poisson pencil. In terms of linearization, we give a criterion for non-degeneracy of a singular point and describe its type. These results are applied to solve the stability problem for a free multidimensional rigid body.

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. We classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions. Furthermore, we develop a theory of integrable dispersive deformations of these Hamiltonian systems following a scheme similar to that proposed by Dubrovin and his collaborators in 1 + 1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.

No description available.

This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally. / text

Στην εργασία αυτή παρουσιάζονται ορισμένοι αλγόριθμοι που συνδέονται με ορθογώνια πολυώνυμα και διακριτά ολοκληρώσιμα συστήματα. Οι κανόνες των αλγορίθμων αυτών είναι ρητού τύπου και συνδέουν τιμές που αφορούν την εξέλιξη των αλγορίθμων στην περίπτωση ιδιομορφιών. Αυτοί οι ιδιάζοντες κανόνες συνιστούν ένα από τα κοινά γνωρίσματα με ορισμένα ολοκληρώσιμα συστήματα στο πλέγμα ΖxZ συγκεκριμένα αυτό του "περιορισμού των ιδιομορφιών". Παρουσιάζονται οι κανόνες των αλγορίθμων ε, ρ και qd όπως και κανόνες που προκύπτουν από τους δύο πρώτους των οποίων η μορφή είναι αναλλοίωτη από μετασχηματισμούς Moebius. Η τελευταία αυτή ιδιότητα βοηθά στην εύρεση ιδιαζόντων κανόνων για τον περιορισμό των ιδιομορφιών. Ο αλγόριθμος qd συνδέεται τόσο με τα ορθογώνια πολυώνυμα στην πραγματική ευθεία όσο και με το διακριτού χρόνου πλέγμα Toda. Παρουσιάζεται η εύρεση του τριδιαγώνιου πίνακα Jacobi από τις σχέσεις που συνδέουν γειτονικές ακολουθίες ορθογωνίων πολυωνύμων. Ο πίνακας Jacobi εκφράζει την γραμμική αναδρομική σχέση τριών διαδοχικών ορθογωνίων πολυωνύμων. Ανάλογη κατασκευή για ορθογώνια πολυώνυμα στον μοναδιαίο κύκλο είναι περισσότερο πολύπλοκη και δεν καταλήγει πάντοτε σε πολυδιαγώνιο πίνακα. Παρουσιάζονται σχετικά πρόσφατα αποτελέσματα για τα ορθογώνια πολυώνυμα στον μοναδιαίο κύκλο και ο πενταδιαγώνιος πίνακας CMV. / In this paper are introduced some algorithms which are connected with orthogonal polynomials and descrete integrable systems. The rules of these algorithms are fraction type and combine the terms which are on the vertex of a rombus. We mainly introduce the rules which relate the evolution of the algorithms in the case of singular rules. These rules introduce one of the common characteristics with some integrable systems in the ZxZ lattice, in particular the "singularity confinement". We introduce the rules of the ε-, ρ- and qd-algorithms as well as the rules which follow from the first two whose type is unchangeable from Moebius transformations. This last property helps in finding proper rules for the singularity confinement. The qd-algorithm is connected not only with the orthogonal polynomials in the real line, but also with the discrete time Toda lattice. We also introduce the finding of the tri-diagonal Jacobi matrix from relations which combine adjacent sequences of orthogonal polynomials. The Jacobi matrix represent the three-term linear reccurence relation of orthogonal polynomials. Correspondent construction for orthogonal polynomials on the unit circle is much more complicated and doesn't conclude always in a poly-diagonal matrix. We introduce some recent results for orthogonal polynomials on the unit circle and the five-diagonal CMV matrix.