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 H´ 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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