Integrability, rational solitons and symmetries for nonlinear systems in Biology and Materials Physics

[ES] Los sistemas no lineales constituyen un tema de investigación de creciente interés en las últimas décadas dada su versatilidad en la descripción de fenómenos físicos en diversos campos de estudio. Generalmente, dichos fenómenos vienen modelizados por ecuaciones diferenciales no lineales, cuya estructura matemática ha demostrado ser sumamente rica, aunque de gran complejidad respecto a su análisis. Dentro del conjunto de los sistemas no lineales, cabe destacar un reducido grupo, pero a la vez selecto, que se distingue por las propiedades extraordinarias que presenta: los denominados sistemas integrables. La presente tesis doctoral se centra en el estudio de algunas de las propiedades más relevantes observadas para los sistemas integrables. En esta tesis se pretende proporcionar un marco teórico unificado que permita abordar ecuaciones diferenciales no lineales que potencialmente puedan ser consideradas como integrables. En particular, el análisis de integralidad de dichas ecuaciones se realiza a través de técnicas basadas en la Propiedad de Painlevé, en combinación con la subsiguiente búsqueda de los problemas espectrales asociados y la identificación de soluciones analíticas de naturaleza solitónica. El método de la variedad singular junto con las transformaciones de auto-Bäcklund y de Darboux jugarán un papel fundamental en este estudio. Además, también se lleva a cabo un análisis complementario basado en las simetrías de Lie y reducciones de similaridad, que nos permitirán estudiar desde esta nueva perspectiva los problemas espectrales asociados. Partiendo de la archiconocida ecuación de Schrödinger no lineal, se han investigado diferentes generalizaciones integrables de dicha ecuación con numerosas aplicaciones en diversos campos científicos, como la Física Matemática, Física de Materiales o Biología.[EN] Nonlinear systems emerge as an active research topic of growing interest during the last decades due to their versatility when it comes to describing physical phenomena. Such scenarios are typically modelled by nonlinear differential equations, whose mathematical structure has proved to be incredibly rich, but highly nontrivial to treat. In particular, a narrow but surprisingly special group of this kind stands out: the so-called integrable systems. The present doctoral thesis focuses on the study of some of the extraordinary properties observed for integrable systems. The ultimate purpose of this dissertation lies in providing a unified theoretical framework that allows us to approach nonlinear differential equations that may potentially be considered as integrable. In particular, their integrability characterization is addressed by means of Painlevé analysis, in conjunction with the subsequent quest of the associated spectral problems and the identification of analytical solutions of solitonic nature. The singular manifold method together with auto-Bäckund and Darboux transformations play a critical role in this setting. In addition, a complementary methodology based on Lie symmetries and similarity reductions is proposed so as to analyze integrable systems by studying the symmetry properties of their associated spectral problems. Taking the ubiquitous nonlinear Schrödinger equation as the starting point, we have investigated several integrable generalizations of this equation that possess copious applications in distinct scientific fields, such as Mathematical Physics, Material Sciences and Biology

Studies on the geometry of Painlevé equations

This thesis is a collection of work within the geometric framework for Painlevé equations. This approach was initiated by the Japanese school, and is based on studying Painlevé equations (differential or discrete) via certain rational surfaces associated with affine root systems. Our work is grouped into two main themes: on the one hand making use of tools and techniques from the geometric framework to study problems from applications where Painlevé equations appear, and on the other hand developing and extending the geometric framework itself. Differential and discrete Painlevé equations arise in a wide range of areas of mathematics and physics, and we present a general procedure for solving the identification problem for Painlevé equations. That is, if a differential or discrete system is suspected to be equivalent to one of Painlevé type, we outline a method, based on constructing the associated surfaces explicitly, for identifying the system with a standard example, in which case known results can be used, and demonstrate it in the case of equations appearing in the theory of orthogonal polynomials. Our results on the geometric framework itself begin with an observation of a new class of discrete equations that can described through the geometric theory, beyond those originally defined by Sakai in terms of translation symmetries of families of surfaces. To be precise, we build on previous studies of equations corresponding to non-translation symmetries of infinite order (so-called projective reductions, with fewer parameters than translations of the same surface type) and show that Sakai’s theory allows for integrable discrete equations to be constructed from any element of infinite order in the symmetry group and still have the full parameter freedom for their surface type. We then also make the first steps toward a geometric theory of delay-differential Painlevé equations by giving a description of singularity confinement in this setting in terms of mappings between jet spaces

Inverse Scattering Transform Method for Lattice Equations

The main original contribution of this thesis is the development of a fully discrete inverse scattering transform (IST) method for nonlinear partial difference equations. The equations we solve are nonlinear partial difference equations on a quad-graph, also called lattice equations, which are known to be multidimensionally consistent in N dimensions for arbitrary N. Such equations were discovered by Nijhoff, Quispel and Capel and Adler and later classified by Adler, Bobenko and Suris. The main equation solved by our IST framework is the Q3δ lattice equation. Our approach also solves all of its limiting cases, including H1, known as the lattice potential KdV equation. Our results provide the discrete analogue of the solution of the initial value problem on the real line. We provide a rigorous justification that solves the problem for wide classes of initial data given along initial paths in a multidimensional lattice. Moreover, we show how soliton solutions arise from the IST method and also utilise asymptotics of the eigenfunctions to construct infinitely many conservation laws

Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width

Lagrangian Multiform Structures, Discrete Systems and Quantisation

Lagrangian multiforms are an important recent development in the study of integrable variational problems. In this thesis, we develop two simple examples of the discrete Lagrangian one-form and two-form structures. These linear models still display all the features of the discrete Lagrangian multiform; in particular, the property of Lagrangian closure. That is, the sum of Lagrangians around a closed loop or surface, on solutions, is zero. We study the behaviour of these Lagrangian multiform structures under path integral quantisation and uncover a quantum analogue to the Lagrangian closure property. For the one-form, the quantum mechanical propagator in multiple times is found to be independent of the time-path, depending only on the endpoints. Similarly, for the two-form we define a propagator over a surface in discrete space-time and show that this is independent of the surface geometry, depending only on the boundary. It is not yet clear how to extend these quantised Lagrangian multiforms to non-linear or continuous time models, but by examining two such examples, the generalised McMillan maps and the Degasperis-Ruijsenaars model, we are able to make some steps towards that goal. For the generalised McMillan maps we find a novel formulation of the r-matrix for the dual Lax pair as a normally ordered fraction in elementary shift matrices, which offers a new perspective on the structure. The dual Lax pair may ultimately lead to commuting flows and a one-form structure. We establish the relation between the Degasperis-Ruijsenaars model and the integrable Ruijsenaars-Schneider model, leading to a Lax pair and two particle Lagrangian, as well as finding the quantum mechanical propagator. The link between these results is still needed. A quantum theory of Lagrangian multiforms offers a new paradigm for path integral quantisation of integrable systems; this thesis offers some first steps towards this theory

Quantum Field Theories, Isomonodromic Deformations and Matrix Models

Recent years have seen a proliferation of exact results in quantum field theories, owing mostly to supersymmetric localisation. Coupled with decades of study of dualities, this ensured the development of many novel nontrivial correspondences linking seemingly disparate parts of the mathematical landscape. Among these, the link between supersymmetric gauge theories with 8 supercharges and Painlev{\'e} equations, interpreted as the exact RG flow of their codimension 2 defects and passing through a correspondence with two-dimensional conformal field theory, was highly surprising. Similarly surprising was the realisation that three-dimensional matrix models coming from M-theory compute these solutions, and provide a non-perturbative completion of the topological string. Extending these two results is the focus of my work. After giving a review of the basics, hopefully useful to researchers in the field also for uses besides understanding the thesis, two parts based on published and unpublished results follow. The first is focused on giving Painlev{\'e}-type equations for general groups and linear quivers, and the second on matrix models

Extensions supersymétriques des équations structurelles des supervariétés plongées dans des superespaces

Le but de cette thèse par articles est d’étudier certains aspects géométriques des supervariétés associées aux systèmes supersymétriques intégrables. Ce travail a abouti en quatre articles publiés et un article présentement soumis dans des revues internationales avec des comités de lecture. Dans le premier article, deux extensions supersymétriques des équations de Gauss–Weingarten et de Gauss– Codazzi pour des surfaces plongées dans des superespaces euclidiens ont été construites. Cela a permis de fournir une caractérisation géométrique de telles surfaces avec des vecteurs tangents linéairement indépendants orientés dans la direction des déplacements infinitésimaux des dérivées fermioniques covariantes. De plus, une étude des symétries des versions supersymétriques des équations de Gauss–Codazzi a permis de construire des solutions invariantes au moyen de la méthode de réduction par symétrie impliquant les variables bosoniques et fermioniques, ce qui a mené à des surfaces non triviales, par exemple des surfaces à courbure de Gauss nulle. Dans le second article, l’extension aux cas supersymétriques d’une conjecture énonçant les conditions nécessaires pour qu’un système soit intégrable au sens de la théorie des solitons a été formulée. Cela a été accompli en introduisant un nouvel opérateur de projection et en comparant les symétries du système original avec celles du problème linéaire associé. Cette conjecture a été appliquée à certains exemples et un paramètre « spectral » fermionique a été introduit dans un des systèmes. Dans le troisième article, deux versions supersymétriques de la formule de Fokas–Gel’fand pour l’immersion de surfaces solitoniques dans une superalgèbre de Lie ont été construites. La caractérisation géométrique de la fonction d’immersion, présentée dans cet article, a permis d’investiguer les comportements des surfaces associées. Ces considérations théoriques ont été appliquées à l’équation de sine-Gordon supersymétrique pour laquelle des surfaces à courbure de Gauss constante et de type Weingarten non linéaire ont été obtenues. Le quatrième article est dévoué aux propriétés d’intégrabilité de l’équation de sine-Gordon supersymétrique et à la construction de solutions multisolitoniques explicites. Deux types de problèmes linéaires spectraux, une version supersymétrique d’un ensemble d’équations de Riccati couplées et la transformation d’auto-Bäcklund, tous équivalents à l’équation de sine-Gordon supersymétrique, ont été étudiés. De plus, une analyse détaillée de la énième transformation de Darboux a permis de trouver des solutions multisolitoniques non triviales de l’équation de sine-Gordon supersymétrique. Ces solutions ont été utilisées pour investiguer la version supersymétrique bosonique de la formule d’immersion de Sym–Tafel. Dans le cinquième article, une nouvelle caractérisation géométrique de la formule d’immersion de Fokas–Gel’fand est présentée. Afin d’accomplir cela, trois différents types de problèmes linéaires spectraux sont étudiés, un impliquant les dérivées fermioniques covariantes, un impliquant les dérivées par rapport aux variables bosoniques et un impliquant les dérivées par rapport aux variables fermioniques. Cette caractérisation géométrique implique huit coefficients linéairement indépendants pour les première et deuxième formes fondamentales, contrairement à trois dans le troisième article, ce qui mène à une géométrie plus riche dans le sens où les supervariétés caractérisées de type unidimensionnel (« curve-like ») dans le troisième article sont de type multidimensionnel dans le cinquième article.The goal of this thesis consisting of articles is to study certain geometric aspects of supermanifolds associated with integrable suspersymmetric systems. This work is contained in four published articles and one currently submitted article in international peer-reviewed journals. In the first article, two supersymmetric extensions of the Gauss–Weingarten and Gauss–Codazzi equations for surfaces immersed in Euclidean superspaces were constructed. This allowed us to provide a geometric characterization of such surfaces with linearly independent tangent vectors oriented in the directions of the infinitesimal displacement of the fermionic covariant derivatives. In addition, a study of the symmetries of the supersymmetric versions of the Gauss–Codazzi equations led to the construction of invariant solutions, involving bosonic and fermionic variables, through the symmetry reduction method, which led to nontrivial surfaces, e.g. vanishing Gauss curvature surfaces. In the second article, a conjecture stating the necessary conditions for a system to be integrable in the sense of soliton theory was extended to the supersymmetric cases. This was accomplished by introducing a new projection operator and by comparing the symmetries of the original system to those of the associated linear problem. This conjecture was applied to some examples and a fermionic “spectral” parameter was introduced in one of the systems. In the third article, two supersymmetric versions of the Fokas–Gel’fand formula for the immersion of soliton surfaces in Lie superalgebras were constructed. The geometric characterization of the immersion function presented in this article allowed us to investigate the behavior of the associated surfaces. These theoretical considerations were applied to the supersymmetric sine-Gordon equation, for which constant Gaussian curvature surfaces and nonlinear-type surfaces were obtained. The fourth article was devoted to integrability properties of the supersymmetric sine-Gordon equation and to the construction of explicit multisoliton solutions. Two types of linear spectral problems, a set of coupled super-Riccati equations and the auto-Bäcklund transformation, all equivalent to the supersymmetric sine-Gordon equation, were studied. In addition, a detailed analysis of the nth Darboux transformations allowed us to find nontrivial multisoliton solutions of the supersymmetric sine-Gordon equation. These solutions were used to investigate the bosonic supersymmetric version of the Sym–Tafel immersion formula. In the fifth article, a new geometric characterization of the Fokas–Gel’fand immersion formula was presented. In order to do this, three different types of linear spectral problems were studied, one involving the covariant fermionic derivatives, one involving the bosonic variable derivatives and one involving the fermionic variable derivatives. This geometric characterization involves eight linearly independent coefficients for both the first and second fundamental forms, in constrast with three such coefficients in the third article, which leads to a richer geometry in the sense that curve-like supermanifolds in the third article are of higher dimensions in the fifth article