This master’s thesis investigates the relationship between supersymmetry and integrability in quantum mechanics. This is done by finding a suitable way to systematically add more supersymmetry to the system. Adding more super- symmetry will give constraints on the potential which will lead to an integrable system. A possible way to explore the integrability of supersymmetric quantum mechanics was introduced in a paper by Crombrugghe and Rittenberg in 1983, their method has been used as well as another approach based on expanding a N = 1 system by introducing complex structures. N = 3 or more supersymmetry is shown to give an integrable system.

In this thesis we study anti–self–duality equations in four and eight dimensions on manifolds of special Riemannian holonomy, among these hyper–Kähler, Quaternion–Kähler and Spin(7)–manifolds. We first consider the octonionic anti–self–duality equations on manifolds with holonomy Spin(7). We construct explicit solutions to their symmetry reductions, the non–abelian Seiberg–Witten equations, with gauge group SU(2). These solutions are singular for flat and Eguchi–Hanson backgrounds, however we find a solution on a co–homogeneity one hyper–Kähler metric with a domain wall, and the solution is regular away from the wall. We then turn to Quaternion–Kähler four–manifolds, which are locally determined by one scalar function subject to Przanowski’s equation. Using twistorial methods we construct a Lax Pair for Przanowski’s equation, confirming its integrability. The Lee form of a compatible local complex structure gives rise to a conformally invariant differential operator, special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. Using recursion relations we construct a contour integral formula for perturbations of Przanowski’s function. Finally, we construct an algorithm to retrieve Przanowski’s function from twistor data. At last, we investigate the relationship between anti–self–dual Einstein metrics with non–null symmetry in neutral signature and pseudo–, para– and null–Kähler metrics. We classify real–analytic anti–self–dual null–Kähler metrics with a Killing vector that are conformally Einstein. This allows us to formulate a neutral signature version of Tod’s result, showing that around non-singular points all real–analytic anti–self–dual Einstein metrics with symmetry are conformally pseudo– or para–Kähler.

Esta dissertação investiga uma particular deformação do modelo de Bullough-Dodd. Não se sabe se tais deformações são ou não integráveis, ainda que nossas simulações numéricas apresentem soluções solitônicas. Exploramos o conceito de quase-integrabilidade em mais esse contexto e mostramos um argumento analítico para a quase-conservação das cargas (isto é, as cargas variam com o tempo, mas seus valores inicial e final são os mesmos). Mesmo quando o argumento analítico não pode fazer previsões, nossas simulações mostram que as cargas apresentam o mesmo comportamento. Isso sugere que as deformações consideradas são integráveis e ainda há espaço para explorá-las. / This dissertation investigates a particular deformation of the Bullough-Dodd model. It\'s unknown if such deformations are integrable or not, yet they present solitonic solutions which were obtained through numerical simulations. We further explore the concept of quasi-integrability in this context, showing analiticaly that at least for some sets of parameters, the charges are quasi-conserved (i.e. the charges vary over time, but it\'s initial and final values are the same). Even when the analitical argument can\'t predict what happens, our simulations show the same charge behaviour. This suggests that those deformations are integrable and can be further explored.

Esta dissertação investiga uma particular deformação do modelo de Bullough-Dodd. Não se sabe se tais deformações são ou não integráveis, ainda que nossas simulações numéricas apresentem soluções solitônicas. Exploramos o conceito de quase-integrabilidade em mais esse contexto e mostramos um argumento analítico para a quase-conservação das cargas (isto é, as cargas variam com o tempo, mas seus valores inicial e final são os mesmos). Mesmo quando o argumento analítico não pode fazer previsões, nossas simulações mostram que as cargas apresentam o mesmo comportamento. Isso sugere que as deformações consideradas são integráveis e ainda há espaço para explorá-las. / This dissertation investigates a particular deformation of the Bullough-Dodd model. It\'s unknown if such deformations are integrable or not, yet they present solitonic solutions which were obtained through numerical simulations. We further explore the concept of quasi-integrability in this context, showing analiticaly that at least for some sets of parameters, the charges are quasi-conserved (i.e. the charges vary over time, but it\'s initial and final values are the same). Even when the analitical argument can\'t predict what happens, our simulations show the same charge behaviour. This suggests that those deformations are integrable and can be further explored.

Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourth-order nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the perturbation. Since our family of equations can be written in the Hamiltonian form, the question is whether or not survival of separatrices implies integrability of the corresponding equation. The complete integrability of the system is examined from two viewpoints: 1) the existence of a second first integral in involution (Liouville integrability), and 2) the existence of single-valued, meromorphic solutions (complex analytic integrability). In the latter case, a singular point analysis is done using the technique given by Ablowitz, Ramani, and Segur (the ARS algorithm) to determine whether the system is of Painlevé-type (P-type), lacking movable critical points. The system is shown by the algorithm to fail to be of P-type, a strong indication of nonintegrability.

Μελετούμε το πρόβλημα ολοκληρωσιμότητας διαφορικών 1-μορφών. Αφού αναφερθούμε στις κλασικές περιπτώσεις των ακριβών 1-μορφών και του ολοκληρωτικού παράγοντα, αποδεικνύουμε το περίφημο Λήμμα του Poincaré, καθώς και το Θεώρημα Ολοκληρωσιμότητας του Frobenious. Η εργασία ολοκληρώνεται με παραδείγματα μορφων που εμφανίζουν ιδιομορφίες. / We study the problem of integrability of differential 1-forms. After the classical cases of exact forms and the integrating factor we prove the famous lemma of Poincaré and the Frobenious integrability theorem. Examples of forms with singularities consist the last part of this study.

In this paper we consider the hypo-ellipticity of differential forms on a closed manifold.The main results show that there are some topological obstruct for the existence of the differential forms with hypoellipticity.

No description available.

We report on the first systematic study of correlation functions in N=4 super Yang-Mills theory using integrability techniques. In particular, we show how to compute three- and four- point functions of single-trace gauge-invariant operators at tree level in the SU(2) sector of the theory. Using the AdS/CFT correspondence, the correlation functions that we compute can be thought of as the joining or splitting of strings moving in AdS5 × S5. We show that when one (two) of the operators in the three-(four-)point function are taken to be small BPS operators, our weak coupling results match perfectly with the strong coupling results in the Frolov-Tseytlin limit. We conclude by presenting some results that will be needed to extend the methods presented in this thesis beyond the SU(2) sector of N=4 super Yang-Mills.

The Painleve analysis plays an important role in investigating local structure of the solutions of differential equations, while Lie symmetries provide powerful tools in global solvability of equations. In this research, the method of Painleve analysis is applied to discrete nonlinear Schrodinger equations and to a family of second order nonlinear ordinary differential equations. Lie symmetries are studied together with the Painleve property for second order nonlinear ordinary differential equations.In the study of the local singularity of discrete nonlinear Schrodinger equations, the Painleve method shows the existence of solution blow up at finite time. It also determines the rate of blow-up. For second order nonlinear ordinary differential equations, the Painleve test is introduced and demonstrated in detail using several examples. These examples are used throughout the research. The Painleve property is shown to be significant for the integrability of a differential equation.After introducing one-parameter groups, a family of differential equations is determined for discussing solvability and for drawing more meaningful conclusions. This is the most general family of differential equations invariant under a given one-parameter group. The first part of this research is the classification of the integrals in the general solutions of differential equations obtained by quadratures. The second part is the application of Riemann surfaces and algebraic curves in the projective complex space to the integrands. The theories of Riemann surfaces and algebraic curves lead us to an effective way to understand the nature of the integral defined on a curve. Our theoretical work then concentrates on the blowing-up of algebraic curves at singular points. The calculation of the genus, which essentially determines the shape of a curve, becomes possible after a sequence of blowing-ups.The research shows that when combining both the Painleve property and Lie symmetries possessed by the differential equations studied in the thesis, the general solutions can be represented by either elementary functions or elliptic integrals.