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SYMMETRY in NONLINEAR MATHEMATICAL PHYSICS Memorial Prof ProceeProceedingsdings 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 12 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 2 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. Contents Volume 1 ListofParticipants ...........................................................................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 QuantumFieldTheory ..................................................................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 in1+1Dimensions ......................................................................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,NewAns¨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´eGroupP (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 ...........................................................200 252 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 theEuclideanLineGroup .............................................................. 254 G. SVETLICHNY, On Relativistic Non-linear Quantum Mechanics .........................262 P. NATTERMANN, On(Non)LinearQuantumMechanics ..................................270 Yu.I. SAMOILENKO, A Fiber Bundle Model of the Ice Ih Structure ........................279 R. LEANDRE´ , SpinorFieldsoverStochasticLoopSpaces ...................................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, DiscreteIntegrableSystemsandtheMoyalSymmetry ...........................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,˘ L. TUROWSKA and S. POPOVYCH, Representations of a Cubic Deformation of su(2) and Parasupersymmetric Commutation Relations .................372 253 A. GAVRILIK and N. IORGOV, q-Deformed Inhomogeneous Algebras Uq(ison) and Their Representations .........................................................................384 S. POPOVYCH, Representation of Real Forms of Witten’s First Deformation . ..............393 O. BATSEVYCH, Deformed Oscillators with Interaction ....................................397 Yu. BESPALOV, On Duality for a Braided Cross Product ..................................403 V. SIDORETS and V. VLADIMIROV, On the Peculiarities of Stochastic Invariant Solutions of a Hydrodynamic System Accounting for Non-local Effects ............................409 Y.Z. BOUTROS, M.B. ABD-EL-MALEK, I.A. EL-AWADI and S.M.A. EL-MANSI, Group Method Analysis of the Potential Equation ......................................418 L.F. BARANNYK and P. SULEWSKI , Exact Solutions of the Nonlinear Diffusion − 4 Equation u0 + ∇ u 5 ∇u =0 ......................................................... 429 R. POPOVYCH, On Reduction and Q-conditional (Nonclassical) Symmetry ................437 R. CHERNIHA and M. SEROV, Lie and Non-Lie Symmetries of Nonlinear Diffusion Equations with Convection Term . ....................................................444 S. SPICHAK and V. STOGNII, Conditional
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