15 research outputs found
The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients
The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1)
dimensional soliton equations with variable coefficients. It is well-known that
the gKdV equation is integrable. In this paper a higher-dimensional gKdV
equation, which is integrable in the sense of the Painleve test, is presented.
A transformation that links this equation to the canonical form of the
Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and
similar transformation for the higher-dimensional modified gKdV equation are
also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients
The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained
Discretisations of constrained KP hierarchies
We present a discrete analogue of the so-called symmetry reduced or
`constrained' KP hierarchy. As a result we obtain integrable discretisations,
in both space and time, of some well-known continuous integrable systems such
as the nonlinear Schroedinger equation, the Broer-Kaup equation and the
Yajima-Oikawa system, together with their Lax pairs. It will be shown that
these discretisations also give rise to a discrete description of the entire
hierarchy of associated integrable systems. The discretisations of the
Broer-Kaup equation and of the Yajima-Oikawa system are thought to be new.Comment: Accepted for publication in Journal of Mathematical Sciences, The
University of Toky
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
Solutions de rang k et invariants de Riemann pour les systèmes de type hydrodynamique multidimensionnels
Dans ce travail, nous adaptons la méthode des symétries conditionnelles afin de construire des solutions exprimées en termes des invariants de Riemann. Dans ce contexte, nous considérons des systèmes non elliptiques quasilinéaires homogènes (de type hydrodynamique) du premier ordre d'équations aux dérivées partielles multidimensionnelles. Nous décrivons en détail les conditions nécessaires et suffisantes pour garantir l'existence locale de ce type de solution. Nous étudions les relations entre la structure des éléments intégraux et la possibilité de construire certaines classes de solutions de rang k. Ces classes de solutions incluent les superpositions non linéaires d'ondes de Riemann ainsi que les solutions multisolitoniques. Nous généralisons cette méthode aux systèmes non homogènes quasilinéaires et non elliptiques du premier ordre. Ces méthodes sont appliquées aux équations de la dynamique des fluides en (3+1) dimensions modélisant le flot d'un fluide isentropique. De nouvelles classes de solutions de rang 2 et 3 sont construites et elles incluent des solutions double- et triple-solitoniques. De nouveaux phénomènes non linéaires et linéaires sont établis pour la superposition des ondes de Riemann. Finalement, nous discutons de certains aspects concernant la construction de solutions de rang 2 pour l'équation de Kadomtsev-Petviashvili sans dispersion.In this work, the conditional symmetry method is adapted in order to construct solutions expressed in terms of Riemann invariants. Nonelliptic quasilinear homogeneous systems of multidimensional partial differential equations of hydrodynamic type are considered. A detailed description of the necessary and sufficient conditions for the local existence of these types of solutions is given. The relationship between the structure of integral elements and the possibility of constructing certain classes of rank-k solutions is discussed. These classes of solutions include nonlinear superpositions of Riemann waves and multisolitonic solutions. This approach is generalized to first-order inhomogeneous hyperbolic quasilinear systems. These methods are applied to the equations describing an isentropic fluid flow in (3+1) dimensions. Several new classes of rank-2 and rank-3 solutions are obtained which contain double and triple solitonic solutions. New nonlinear and linear superpositions of Riemann waves are described. Finally, certain aspects of the construction of rank-2 solutions through an application to the dispersionless Kadomtsev-Petviashvili equation are discussed