Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration

Sistemas de Lie, simetrías de Lie y transformaciones recíprocas

[ES]En esta tesis, estamos interesados en sistemas de interés físico y matemático, descritos por medio de ecuaciones diferenciales ordinarias y en derivadas parciales. Como es bien sabido, gran parte de los fenómenos naturales pueden modelizarse a través de estas ecuaciones. Por ejemplo, las cuatro ecuaciones de la Electrodinámica de Maxwell, o las ecuaciones de Einstein son ecuaciones diferenciales. Vamos a centrar nuestra investigación en dos tipos de sistemas: los llamados sistemas de Lie, muy recurrentes en la literatura, dadas sus múltiples propiedades geométricas y las ecuaciones diferenciales en derivadas parciales que aparecen en modelos físicos como los pertenecientes a la Mecánica de Fluidos, Física del Plasma o la Neurociencia, entre otros. Dada la importancia de los métodos geométricos en el tratamiento de ecuaciones diferenciales, vamos a formular nuestra investigación desde el punto de vista de la geometría diferencial

Similarity Reductions and Integrable Lattice Equations

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

The Painlevé II hierarchy: geometry and applications

The Painlevé II hierarchy is a sequence of nonlinear ODEs, with the Painlevé II equation as first member. Each member of the hierarchy admits a Lax pair in terms of isomonodromic deformations of a rank 2 system of linear ODEs, with polynomial coefficient for the homogeneous case. It was recently proved that the Tracy-Widom formula for the Hastings-McLeod solution of the homogeneous PII equation can be extended to analogue solutions of the homogeneous PII hierarchy using Fredholm determinants of operators acting through higher order Airy kernels. These integral operators are used in the theory of determinantal point processes with applications in statistical mechanics and random matrix theory. From this starting point, this PhD thesis explored the following directions. We found a formula of Tracy-Widom type connecting the Fredholm determinants of operators acting through matrix-valued analogues of the higher order Airy kernels with particular solution of a matrix-valued PII hierarchy. The result is achieved by using a matrix-valued Riemann-Hilbert problem to study these Fredholm determinants and by deriving a block-matrix Lax pair for the relevant hierarchy. We also found another generalization of the Tracy-Widom formula, this time relating the Fredholm determinants of finite-temperature versions of higher order Airy kernels operators to particular solutions of an integro-differential Painlevé II hierarchy. In this setting, a suitable operator-valued Riemann-Hilbert problem is used to study the relevant Fredholm determinant. The study of its solution produces in the end an operator-valued Lax pair that naturally encodes an integro-differential Painlevé II hierarchy. From a more geometrical point of view, we analyzed the Poisson-symplectic structure of the monodromy manifolds associated to a system of linear ODEs with polynomial coefficient, also known as Stokes manifolds. For the rank 2 case, we found explicit log-canonical coordinates for the symplectic 2-form, forming a cluster algebra of specific type. Moreover, the log-canonical coordinates constructed in this way provide a linearization of the Poisson structure on the Stokes manifolds, first introduced by Flaschka and Newell in their pioneering work of 1981

Un enfoque geométrico a los sistemas de Lie : formalismo de las deformaciones de álgebras de Poisson–Hopf

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 22-01-2021The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, under quantum deformations, Lie systems are extended to generalized systems described by involutive distributions. As a consequence, a quantum deformed Lie system no longer has an underlying Vessiot–Guldberg Lie algebra or a quantum algebra one, but keeps a Poisson–Hopf algebra structure that enables us to obtain, in an explicit way, the t-independent constants of the motion from quantum deformed Casimir invariants, which are potentially useful in a further construction of the generalized notion of superposition rules. We illustrate this approach by considering the non-standard quantum deformation of sl(2) applied to well-known Lie systems, such as the oscillator problem or Milne–Pinney equation, as well as several types of Riccati equations. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass...La noción de álgebras cuánticas se fusiona con la de sistemas de Lie para establecer un nuevo formalismo, las deformaciones del álgebra de Poisson–Hopf de los sistemas de Lie. El procedimiento puede aplicarse a sistemas de Lie dotados de una estructura simpléctica, los denominados sistemas de Lie–Hamilton. Este es un enfoque bastante general, ya que se puede aplicar a cualquier deformación cuántica y a cualquier variedad subyacente. Una de sus principales características es que, bajo deformaciones cuánticas, los sistemas de Lie se extienden a distribuciones involutivas generalizadas. Como consecuencia, un sistema de Lie deformado cuánticamente ya no tiene un álgebra de Vessiot–Guldberg Lie subyacente o un álgebra cuántica, sino que mantiene una estructura de álgebra de Poisson–Hopf que permite obtener, de manera explícita, las constantes del movimientot-independientes a partir de los invariantes de Casimir deformados, que son potencialmente útiles en una construcción adicional de la noción generalizada de reglas de superposición. Ilustramos este enfoque considerando la deformación cuántica no estándar de sl(2) aplicada a sistemas de Lie conocidos, como el problema del oscilador o la ecuación de Milne–Pinney, así como varios tipos de ecuaciones de Riccati. De esta manera, se obtienen sus análogos generalizados (deformados) quedan lugar, en particular, a un nuevo sistema de tipo oscilatorio con una frecuencia dependiente del tiempo y una masa dependiente de la posición...Fac. de Ciencias MatemáticasTRUEunpu