Proceedings of the Second International Conference SYMMETRY IN NONLINEAR MATHEMATICAL PHYSICS Memorial Prof. W. FUSHCHYCH Conference July 7–13, 1997, Kyiv, Ukraine Organized by Institute of Mathematics of the National Academy of Sciences of Ukraine Ukrainian Pedagogical University Editors Mykola SHKIL Anatoly NIKITIN Vyacheslav BOYKO Volume 1 Proceedings of the Second International Conference SYMMETRY IN NONLINEAR MATHEMATICAL PHYSICS Memorial Prof. W. FUSHCHYCH Conference July 7–13, 1997, Kyiv, Ukraine Organized by Institute of Mathematics of the National Academy of Sciences of Ukraine Ukrainian Pedagogical University Editors Mykola SHKIL Anatoly NIKITIN Vyacheslav BOYKO Volume 1 Institute of Mathematics, National Academy of Sciences of Ukraine 3 Tereshchenkivs’ka Street, Kyiv 4, Ukraine E-mail: [email protected] Fax: +380 44 225 20 10 Phone: +380 44 224 63 22 ISBN 966–02–0342–X Symmetry in Nonlinear Mathematical Physics, Editors: M. Shkil, A. Nikitin and V. Boyko. ISBN 966–02–0343–8 (Volume 1) and ISBN 966–02–0344–6 (Volume 2) Copyright c 1997 by Institute of Mathematics of the National Academy of Sciences of Ukraine. All rights reserved. No part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Ukraine, Kyiv, 1997. Preface The Second International Conference “Symmetry in Nonlinear Mathematical Physics” was organized mainly due to efforts of Professor Wilhelm Fushchych. It happened, however, that this out- standing scientist and kind person passed away. The Conference was dedicated to his memory. Among 80 participants of the Conference there were representatives of 14 countries from 5 continents. The main part of the lectures was devoted to investigation of symmetries and construction of exact solutions of nonlinear differential equations. Moreover, the latest trends in symmetry analysis, such as conditional symmetry, potential symmetries, discrete symmetries and differential geometry approach to symmetry analysis were represented efficiently. In addition, important branches of symmetry analysis such as repre- sentation theory, quantum groups, complete integrable systems and the corresponding infinite sets of conservation laws were widely dis- cussed. The related papers as well as ones devoted to symmetries in physics and backgrounds of the nonlinear quantum mechanics, are in- cluded into the second volume of the Proceedings. We plan to continue the series of conferences dedicated to the memory of Professor Wilhelm Fushchych and hope that they will make an essential contribution to symmetry approach to nonlinear mathematical physics. Anatoly NIKITIN October, 1997 Contents Volume 1 List of Participants ...........................................................................7 A. NIKITIN, Scientific Heritage of W. Fushchych ............................................11 B.K. HARRISON, Differential Form Symmetry Analysis of Two Equations Cited byFushchych ............................................................................21 P. BASARAB-HORWATH and W. FUSHCHYCH, Implicit and Parabolic Ansatzes: Some New Ansatzes for Old Equations ...................................................34 M.I. SEROV and M.M. SEROVA, The Conditional Symmetry and Connection Between the Equations of Mathematical Physics ..................................................48 R. ZHDANOV, On Conditional Symmetries of Multidimensional Nonlinear Equations of Quantum Field Theory ..................................................................53 L.F. BARANNYK, B. KLOSKOWSKA and V.V. MITYUSHEV, Invariant Solutions of the Multidimensional Boussinesq Equation ...............................................62 M. EULER and N. EULER, Symmetries for a Class of Explicitly Space- and Time-Dependent (1+1)-Dimensional Wave Equations ......................................................70 A.K. LOPATIN, Symmetry in Perturbation Problems ........................................79 (k) Z. JIANG, Lie Symmetries and Preliminary Classification of un (t)=F n(t, un+a,...,un+b).89 A. BARANNYK and I. YURYK, On Some Exact Solutions of Nonlinear Wave Equations ....98 J. BECKERS, On Specific Symmetries of the KdV Equation and on New Representations of Nonlinear sl(2)-algebras ..............................................................108 I.V. KULEMIN and A.G. MESHKOV, To the Classification of Integrable Systems in 1+1 Dimensions ......................................................................115 V. BOYKO, On New Generalizations of the Burgers and Korteweg-de Vries Equations ......122 M.L. GANDARIAS, Nonclassical Potential Symmetries of the Burgers Equation ............130 R. CHERNIHA, New Ans¨atze and Exact Solutions for Nonlinear Reaction-Diffusion Equations Arising in Mathematical Biology .............................................138 O. ROMAN, Nonlinear Conformally Invariant Wave Equations and Their Exact Solutions . .147 L. BERKOVICH, The Generalized Emden-Fowler Equation .................................155 L. BERKOVICH and S. POPOV, Group Analysis of the Ordinary Differential Equations of the Order n>2 ......................................................................164 V. TYCHYNIN, Non-local Symmetry of the 3-Dimensional Burgers-Type Equation .........172 Z. SYMENOH and I. TSYFRA, Equivalence Transformations and Symmetry of the Schr¨odinger Equation with Variable Potential .......................................177 N. EULER, O. LINDBLOM, M. EULER and L.-E. PERSSON, The Higher Dimensional Bateman Equation and Painlev´e Analysis of Nonintegrable Wave Equations .............185 V. FEDORCHUK, Subgroup Structure of the Poincar´e Group P (1, 4) and Symmetry Reduction of Five-Dimensional Equations of Mathematical Physics ......................193 I. YEHORCHENKO, Differential Invariants for a Nonlinear Representation of the Poincar´e Algebra. Invariant Equations ...........................................................200 5 H. LAHNO, Representations of Subalgebras of a Subdirect Sum of the Extended Euclid Algebras and Invariant Equations .......................................................206 A. ANDREITSEV, To Separation of Variables in a (1+2)-Dimensional Fokker-Planck Equation ...............................................................................211 I. REVENKO, On Exact Solutions of the Lorentz-Dirac-Maxwell Equations .................214 M. SEROVA and N. ANDREEVA, Evolution Equations Invariant under the Conformal Algebra .................................................................................217 H. POPOVYCH, Generalization of Translation Flows of an Ideal Incompressible Fluid: a Modification of the ”Ansatz” Method .................................................222 V. POPOVYCH, On Lie Reduction of the MHD Equations to Ordinary Differential Equations ..............................................................................227 M. LUTFULLIN, Symmetry Reduction of Nonlinear Equations of Classical Electrodynamics ........................................................................232 V.V. BUCHYNCHYK, On Symmetries of a Generalized Diffusion Equation .................237 I. FEDORCHUK, Reduction and Some Exact Solutions of the Eikonal Equation ............241 O. LEIBOV, On Reduction and Some Exact Solutions of the Euler-Lagrange-Born-Infeld Equation ...............................................................................245 Volume 2 W. KLINK, Quantum Mechanics in Noninertial Reference Frames and Representations of the Euclidean Line Group .............................................................. 254 G. SVETLICHNY, On Relativistic Non-linear Quantum Mechanics .........................262 P. NATTERMANN, On (Non)Linear Quantum Mechanics ..................................270 Yu.I. SAMOILENKO, A Fiber Bundle Model of the Ice Ih Structure ........................279 R. LEANDRE´ , Spinor Fields over Stochastic Loop Spaces ...................................287 M. SHKIL, S. KOVALENKO and G. ZAVIZION, On a Mixed Problem for a System of Differential Equations of the Hyperbolic Type with Rotation Points .....................297 I. TSYFRA, Conformal Invariance of the Maxwell-Minkowski Equations .....................307 V. LAHNO, Symmetry Reduction and Exact Solutions of the SU(2) Yang-Mills Equations . 313 R. ANDRUSHKIW and A. NIKITIN, Higher Symmetries of the Wave Equation with Scalar and Vector Potentials ...................................................................321 S.P. ONUFRIYCHUK and O.I. PRYLYPKO, Higher Order Symmetry Operators for the Sch¨odinger Equation ................................................................328 A. SERGEYEV, On Parasupersymmetries in a Relativistic Coulomb Problem for the Modified Stueckelberg Equation ....................................................331 S. SAITO, Discrete Integrable Systems and the Moyal Symmetry ...........................336 P. HOLOD, A. KISILEVICH and S. KONDRATYUK, An Orbit Structure for Integrable Equations of Homogeneous and Principal Hierarchies ....................................343 A.M. KOROSTIL and I.A. KOROSTIL, Three-Gap Elliptic Solutions of the KdV Equation .353 R. SMIRNOV, On a Construction Leading to Magri-Morosi-Gel’fand-Dorfman’s Bi-Hamiltonian Systems ................................................................359 A. SVININ, On Some Integrable System of Hyperbolic Type ................................366 Yu.S. SAMOILENKO,˘