Table Of ContentModem Group Analysis:
Advanced Analytical and Computational Methods
in Mathematical Physics
Modern Group Analysis:
Advanced Analytical and
Computational Methods
in Mathematical Physics
Proceedings ofthe International Workshop Acireale,
Catania, Italy, October 27-31, 1992
Edited by
N. H. Ibragimov
M. Torrisi
and
A. Valenti
Department of Mathematics,
University of Catania,
Catania, Italy
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Modern group analysis, advanced analytical and computational methods
in mathematical phySics : proceedings of the international workshop,
Acireale, Catania, Italy, Octaber 27-31, 1992 I edited by N.H.
Ibragimav, M. Tarrisi, and A. Valenti.
p. cm.
ISBN 978-94-010-4908-5 ISBN 978-94-011-2050-0 (eBook)
DOI 10.1007/978-94-011-2050-0
1. Mathematical phySics--Cangresses. 2. Numerical analysis-
-Congresses. 1. Ibragimov, N. Kh. (Nail' Khalrullavichl
II. Torrisi, M. III. Valenti, A.
OC19.2.M63 1993
530. 1 '5--dc20 93-20973
ISBN 978-94-010-4908-5
Printed on acid-jree paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1s t edition 1993
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic Of mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
The Workshop was dedicated
to the 150th anniversary of
Sophus Lie
PREFACE
On the occasion of the 150th anniversary of Sophus Lie, an International Work-
shop "Modern Group Analysis: advanced analytical and computational methods in
mathematical physics" has been organized in Acireale (Catania, Sicily, October 27-
31, 1992).
The Workshop was aimed to enlighten the present state of this rapidly expanding
branch of applied mathematics.
Main topics of the Conference were:
• classical Lie groups applied for constructing invariant solutions and conservation
laws;
• conditional (partial) symmetries;
• Backlund transformations;
• approximate symmetries;
• group analysis of finite-difference equations;
• problems of group classification;
• software packages in group analysis.
The success of the Workshop was due to the participation of many experts in
Group Analysis from different countries.
This book consists of selected papers presented at the Workshop.
We would like to thank the Scientific Committee for the generous support of
recommending invited lectures and selecting the papers for this volume, as well as
the members of the Organizing Committee for their help.
The Workshop was made possible by the financial support of several sponsors
that are listed below.
It is also a pleasure to thank our colleague Enrico Gregorio for his invaluable help
during the preparation of this volume.
N. H. Ibragimov
M. Torrisi
A. Valenti
Scientific Committee
N. H. Ibragimov (Chairman) M. Torrisi (Co-chairman)
W. F. Ames R. L. Anderson
A. M. Anile G. W. Bluman
V.Boffi C. Cercignani
Y. Choquet-Bruhat A. Donato
D. Fusco G. Grioli
P. G. L. Leach P. J. Olver
L. V. Ovsiannikov
Organizing Committee
M. Lalicata S. Lombardo
A. Grasso A. E. Pompei
M. A. Rigano M. Torrisi (Chairman)
A. Valenti G. Zappala
Sponsors
Mathematical Committee of the Italian National Research Council (C.N.R.)
Gruppo Nazionale per la Fisica Matematica (G.N.F.M.-C.N.R.)
Universita di Catania
Dipartimento di Matematica di Catania
Provincia Regionale di Catania
Assessorato Ecologia ed Ambiente della Provincia Regionale di Catania
A.A.P.I.T. di Catania
Azienda Turismo di Acireale
S.I.M.A.I.
Fondazione IBM Italia
TABLE OF CONTENTS
B. ABRAHAM-SHRAUNER AND A. Guo
Hidden and nonlocal symmetries of nonlinear differential equations 1
I. ANDERSON, N. KAMRAN AND P. J. OLVER
Internal symmetries of differential equations 7
R. L. ANDERSON, P. W. HEBDA AND G. RIDEAU
Examples of completely integrable Bateman pairs 23
N. A. BADRAN AND M. B. ABD-EL-MALEK
Group method analysis of the dispersion of gaseous pollutants
in the presence of a temperature inversion 35
G. BAUMANN
Applications of the generalized symmetry method 43
Yu. Yu. BEREST, N. H. IBRAGIMOV AND A. O. OGANESYAN
Conformal invariance, Huygens principle and fundamental solutions
for scalar second order hyperbolic equations 55
G. BLUMAN
Potential symmetries and equivalent conservation laws 71
S. CARILLO AND B. FUCHSSTEINER
Some remarks on a class of ordinary differential equations:
the Riccati property 85
G. CARRA-FERRO AND S. V. DUZHIN
Differential-algebraic and differential-geometric approach
to the study of involutive symbols 93
P. CASATI, F. MAGRI AND M. PEDRONI
The bihamiltonian approach to integrable systems 101
G. CAVIGLIA AND A. MORRO
Conservation laws in dissipative solids 111
C. CERCIGNANI
Exact solutions of the Boltzmann equation 125
Y. CHOQUET-BRUHAT
Einstein equations with 1 parameter spacelike isometry group 137
G. CICOGNA
Lie Point symmetries and dynamical systems 147
x TABLE OF CONTENTS
P. A. CLARKSON AND E. L. MANSFIELD
Symmetries of the nonlinear heat equation 155
A. DEWISME, S. BOUQUET AND P.G.L. LEACH
Symmetries of time dependent Hamiltonian systems 173
A. DONATO AND F. OLIVERI
Quasilinear hyperbolic systems: reduction to autonomous form
and wave propagation 181
V. A. DORODNITSYN
Finite difference models entirely inheriting symmetry
of original differential equations 191
M. J. ENGLEFIELD
Boundary condition invariance 203
N. EULER AND W.-H STEEB
Nonlinear differential equations, Lie symmetries, and the Painleve test 209
R. FAZIO
Non-iterative transformation methods equivalence 217
D. Fusco AND N. MANGANARO
Reduction procedures for a class of rate-type materials 223
W. FUSHCHYCH
Conditional symmetries of the equations of Mathematical Physics 231
F. GALAS
Pseudopotential symmetries for integrable evolution equations 241
V. P. GERDT AND W. LASSNER
Isomorphism verification for complex and real Lie algebras
by Grobner basis technique 245
P. G. L. LEACH
sl(2, R), Ermakov systems and the magnetic monopole 255
D. LEVI AND P. WINTERNITZ
Symmetries of differential equations on a lattice.
An example: the Toda Lattice 265
F. M. MAHOMED, A. H. KARA AND P. G. L. LEACH
Symmetries of particle Lagrangians 273
L. V. OVSIANNIKOV
The group analysis algorithms 277
E. PUCCI AND G. SACCOMANDI
Potential symmetries of Fokker-Planck equations 291
G. R. W. QUISPEL AND R. SAHADEVAN
Continuous symmetries of difference equations 299
TABLE OF CONTENTS xi
S. RAUCH-WOJCIECHOWSKI
Integrable mechanical systems invariant with respect
to the action of the KdV hierarchy 303
G. J. REID, D. T. WEIH AND A. D. WITTKOPF
A point symmetry group of a differential equation which cannot be found
using infinitesimal methods " 311
C. ROGERS, C. HOENSELAERS AND U. RAMGULAM
Ermakov structure in 2+ I-dimensional systems. Canonical reduction 317
W. SARLET AND E. MARTINEZ
Symmetries of second-order differential equations and decoupling 329
J. SCHU, W. M. SEILER AND J. CALMET
Algorithmic methods for Lie pseudogroups 337
C. SOPHOCLEOUS
A special class of Backlund transformations for certain nonlinear
partial differential equations 345
E. S. SUHUBI
Symmetry groups of balance equations 353
M. TORRISI, R. TRACINA AND A. VALENTI
On equivalence transformations applied to a non-linear wave equation 367
T. WOLF
An efficiency improved program LIEPDE for determining
Lie-symmetries of PDEs 377
S. ZIDOWITZ
Conditional symmetries and the direct reduction
of partial differential equations 387
HIDDEN AND NONLOCAL SYMMETRIES OF NONLINEAR
DIFFERENTIAL EQUATIONS
B. ABRAHAM-SHRAUNER and A. GUO·
Department of Electrical Engineering
Wa3hington Uniller3ity St. Loui3, Miuouri, 63130 U.S.A.
Abstract. New results on hidden and nonlocal symmetries of nonlinear ordinary differential
equations (NLODEs) are presented. Two types of hidden symmetries have been identified. A
type I (II) hidden symmetry of an ODE occurs if a symmetry is lost (gained) when the order of the
ODE is reduced. Both type I and type II hidden symmetries are found in the reduction of a third-
order NLODE invariant under a three-parameter nonsolvable Lie group. Nonlocal group generators
are determined of the exponential form and a new linear form. The ODEs can be reduced by the
nonlocal group generators until first-order ODEs are obtained where the procedure fails because
canonical coordinates cannot be calculated in that case. ODEs cannot be reduced by the linear
nonlocal group generators.
1. Introduction
The pioneering research of Sophus Lie in the latter nineteenth century on the use of
symmetries to solve differential equations has led to widely diverse applications of
Lie groups. The results reported here relate to a further development of his origi-
nal intent, the solution of differential equations by examining the symmetries of the
equations. In the present day the Lie classical method for finding point symmetries
of the differential equations is the most common method used. However, many sym-
metries are not found by the classical method and this has led to the investigation of
contact symmetries [1,2], generalized symmetries [3], nonclassical symmetries [4,5].
All these methods share with the Lie classical method that they are direct methods
which given the differential equations then determine the symmetries of the differen-
tial equations. The term direct method should not be confused with the method of
that name [6]. Nonetheless, Sophus Lie, himself, frequently used an indirect method
where he started with the group and then determined the general form of the differ-
ential equations under which the differential equations were invariant. Tables were
compiled of the general from of ODEs [7].
Not all symmetries are found by the direct methods mentioned above. These
symmetries are called hidden symmetries since for ODEs the type I (II) symmetry is
lost (gained) when the order ofthe ODE is reduced. These symmetries are connected
to nonlocal symmetries since hidden symmetries may be represented by nonlocal
group generators and nonlocal transformations between ODEs of the same order
occur in the presence of hidden symmetries. The significance of the loss or gain of
these symmetries was first stressed by Olver [3] and development of their properties
has been reported in our earlier work [8-11]. In this article the hidden symmetries
• Supported in part by a grant from the Southwestern Bell Corporation.
1
N. H. Ibragimov et al. (eds.),
Modem Group AfUllysis: Advanced AfUllytical and Computational Methods in Mathematical Physics, 1-5.
© 1993 Kluwer Academic Publishers.
