Group-invariant solutions of a nonlinear acoustics model

Based on a recent classification of subalgebras of the symmetry algebra of the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation into ordinary differential equations are obtained. Large classes of group-invariant solutions of the equation are also determined, and some properties of the reduced equations and exact solutions are discussed.Comment: 14 page

Chemical Reaction Systems, Computer Algebra and Systems Biology

International audienceIn this invited paper, we survey some of the results obtained in the computer algebra team of Lille, in the domain of systems biology. So far, we have mostly focused on models (systems of equations) arising from generalized chemical reaction systems. Eight years ago, our team was involved in a joint project, with physicists and biologists, on the modeling problem of the circadian clock of the green algae Ostreococcus tauri. This cooperation led us to different algorithms dedicated to the reduction problem of the deterministic models of chemical reaction systems. More recently, we have been working more tightly with another team of our lab, the BioComputing group, interested by the stochastic dynamics of chemical reaction systems. This cooperation led us to efficient algorithms for building the ODE systems which define the statistical moments associated to these dynamics. Most of these algorithms were implemented in the MAPLE computer algebra software. We have chosen to present them through the corresponding MAPLE packages

Classification of Simply-Transitive Levi Non-Degenerate Hypersurfaces in C^3

Holomorphically homogeneous Cauchy–Riemann (CR) real hypersurfaces M3⊂C2 were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces M5⊂C3 using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry

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

Laplace transformation of Lie class ω = 1 overdetermined systems

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal theory of PDEs