Publication: Sistemas integrables discretos, polinomios matriciales ortogonales y problemas de Riemann-Hilbert
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2016-07-05
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Universidad Complutense de Madrid
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Abstract
El propósito de esta tesis doctoral es el estudio de la conexión, mediante el problema de Riemann-Hilbert, entre sistemas discretos y la teoría de polinomios matriciales ortogonales. La investigación de los modelos integrables se originó en la Mecánica Clásica, en relación a la resolución de las ecuaciones de Newton [2]. Los trabajos de Liouville, Hamilton, Jacobi y otros sentaron las bases de los sistemas integrables como prototipos modelos resolubles por cuadraturas, v.g., por integración directa [7]. Hay una cantidad importante de investigación dedicada a los aspectos geométricos de los sistemas clásicos integrables y superintegrables [66], [82], especialmente en relación a la separación de variables de la ecuación de Hamilton-Jacobi [75]. Fue la aplicación, en la segunda mitad del siglo pasado, de la transformada espectral inversa para la resolución del problema de Cauchy de la ecuación de Korteweg-de Vries [42, 43] la que marcó el inicio de una nueva etapa en este campo, el del estudio de sistemas integrables con un número infinito de grados de libertad, que generalmente se expresan en términos de jerarquías de ecuaciones no lineales en derivadas parciales. Particularmente reseñable, por su aplicación en la hidrodinámica y en la óptica cuántica, es la aparición de las soluciones a un número de solitones arbitrario. En las últimas tres décadas ha habido un importante interés por el estudio de modelos discretos, v.g., sistemas dinámicos de nidos en un retículo de puntos, y expresados en términos de ecuaciones no lineales en diferencia parciales. Muchas de las técnicas encontradas en el mundo continuo se extendieron a este nuevo contexto discreto. Hay dos razones fundamentales para este interés...
The purpose of this Thesis is to relate the notion of integrability for discrete systems with the theory of matrix orthogonal polynomials, by using a Riemann-Hilbert approach. The study of integrable models has originated in Classical Mechanics, in relation with the problem of solving Newton's equations of motion [2]. The work of Liouville, Hamilton, Jacobi and others rmly established integrable systems as prototype modes \solvable by quadratures", i.e. by a direct integration procedure [7]. An impressive amount of research has been devoted to the study of the geometry of classical integrable and superintegrable systems [66], [82], especially in relation with the problem of separation of variables for the associated Hamilton-Jacobi equation [75]. In the second half of twentieth century, the discovery of the Inverse Scattering Method for the Korteweg-de Vries equation [42, 43] signed the beginning of a new eld of research: the study of integrable systems with in nitely many degrees of freedom, expressed in terms of nonlinear eld equations. New classes of integrable models, often encompassed in hierarchies of nonlinear partial di erential equations, were introduced. In particular, equations possessing soliton solutions found interesting applications, for instance, in classical hydrodynamics and quantum optics...
The purpose of this Thesis is to relate the notion of integrability for discrete systems with the theory of matrix orthogonal polynomials, by using a Riemann-Hilbert approach. The study of integrable models has originated in Classical Mechanics, in relation with the problem of solving Newton's equations of motion [2]. The work of Liouville, Hamilton, Jacobi and others rmly established integrable systems as prototype modes \solvable by quadratures", i.e. by a direct integration procedure [7]. An impressive amount of research has been devoted to the study of the geometry of classical integrable and superintegrable systems [66], [82], especially in relation with the problem of separation of variables for the associated Hamilton-Jacobi equation [75]. In the second half of twentieth century, the discovery of the Inverse Scattering Method for the Korteweg-de Vries equation [42, 43] signed the beginning of a new eld of research: the study of integrable systems with in nitely many degrees of freedom, expressed in terms of nonlinear eld equations. New classes of integrable models, often encompassed in hierarchies of nonlinear partial di erential equations, were introduced. In particular, equations possessing soliton solutions found interesting applications, for instance, in classical hydrodynamics and quantum optics...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Teórica II, leída el 02/02/2016. Tesis formato europeo (compendio de artículos)