4 research outputs found
Quasi-periodic solutions of the Heisenberg hierarchy
The Heisenberg hierarchy and its Hamiltonian structure are derived
respectively by virtue of the zero curvature equation and the trace identity.
With the help of the Lax matrix we introduce an algebraic curve
of arithmetic genus , from which we define meromorphic
function and straighten out all of the flows associated with the
Heisenberg hierarchy under the Abel-Jacobi coordinates. Finally, we achieve the
explicit theta function representations of solutions for the whole Heisenberg
hierarchy as a result of the asymptotic properties of
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
Lie systems, Lie symmetries and reciprocal transformations
This work represents a PhD thesis concerning three main topics. The first one
deals with the study and applications of Lie systems with compatible geometric
structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie
systems admitting Vessiot--Guldberg Lie algebras of Hamiltonian vector fields
relative to the above mentioned geometric structures are analyzed and their
importance is illustrated by their appearances in physical, biological and
mathematical models. The second part details the study of Lie symmetries and
reductions of relevant hierarchies of differential equations and their
corresponding Lax pairs. For example, the Cammasa-Holm and Qiao hierarchies in
2+1 dimensions. The third and last part is dedicated to the study of reciprocal
transformations and their application in differential equations appearing in
mathematical physics, e.g. Qiao and Camassa-Holm equations equations appearing
in the second part.Comment: PhD thesi
Periodic and almost periodic potentials in the inverse problems
We review basic ideas and basic examples of the theory of the inverse
spectral problems.Comment: 36 pages, 5 figures, LaTe