Preface
This volume represents the 3rd Jairo Charris seminar entitled “Symmetries of
differential and difference equations”, which was held at Universidad Sergio Ar-
boleda, in Bogot´ a, Colombia in August 2009. The aim of this conference was to
discuss recent developments and several approaches to the geometrical and algebraic
aspects of differential and difference equations, such as Lie symmetry groups and
their invariants, differential Galois theory, group theoretical methods in physics,
and geometrization of mechanics.
The contributions by Ibragimov, Jim´ enez, and Olver relate to Lie symmetries,
equivalence transformations, and differential invariants. The paper by Ibragimov is
a survey on integration methods for parabolic equations. The equivalence problem
for parabolic equations is considered, and the equations that can be reduced to the
heat equation by certain equivalence transformation are characterized by terms of
a differential semi-invariant. Also some classical formulas for closed form solutions
of heat equation are revisited. Jim´ enez’s contribution gives some applications of
the theory of Lie correspondences, a geometrical dictionary that translates systems
of partial differential equations of different orders and number of variables. This
theory was proposed by S. Lie at the end of the 19th century but was only recently
developed. As an example, the theory of characteristics can be viewed as a simple
application of Lie correspondences. Olver’s paper surveys the topic of differential
invariants, focusing on explicit computations based on moving frames and Gr¨obner
bases methods. The cases of finite dimensional Lie groups and infinite dimensional
pseudogroups are analyzed, and most recent results on the topic are surveyed.
The papers by Aparicio Monforte and Weil, Mozo, and Sauloy relate to group
theoretical methods in linear equations, namely differential Galois theory and Stokes
phenomenon. The work by Aparicio and Weil is devoted to the study of the struc-
ture of the Galois groups of higher order variational equations as an essential tool
for understanding the integrability of Hamiltonian systems. In particular, they
propose effective tools for proving the nonintegrability of Hamiltonian systems by
means of Morales–Ramis approaches. As a concrete example to test their method
they give a new systematic proof of the nonintegrability of the degenerated case of
the enon-Heiles family. The paper by Mozo surveys the theory and a number of
applications of the summability of solutions of linear differential equations and some
related problems in complex dynamics. Sauloy introduces the Stokes phenomenon
for linear q-difference equations and its connection with q-difference Galois group.
As an interesting illustration he analyzes the q-Euler equation and Tshakaloff series.
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