DOCSLIB.ORG

Symmetry Groups of Linear Partial Differential Equations and Representation Theory: the Laplace and Axially Symmetric Wave Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetry Groups of Linear Partial Differential Equations and Representation Theory: the Laplace and Axially Symmetric Wave Equations Journal of Differential Equations 166, 107131 (2000) doi:10.1006Âjdeq.2000.3786, available online at http:ÂÂwww.idealibrary.com on Symmetry Groups of Linear Partial Differential Equations and Representation Theory: The Laplace and Axially Symmetric Wave Equations Mark Craddock1 School of Finance and Economics, University of Technology Sydney, P.O. Box 123, Broadway, New South Wales 2007, Australia Received June 28, 1999; revised October 25, 1999 View metadata,We examinecitation and the similar Lie point papers symmetry at core.ac.uk groups of two important equations of brought to you by CORE mathematics and mathematical physics. We establish that the action of the provided by Elsevier - Publisher Connector symmetry groups are in fact equivalent to principal series representations of the underlying group. Some applications are given. 2000 Academic Press Key Words: symmetry groups; PDEs; harmonic analysis. 1. INTRODUCTION The origins of the theory of Lie groups are to be found in Lie's attempt to develop a method of integrating differential equations using group properties. Lie's analysis led him to consider one parameter groups of transformations of PDEs which preserve solutions. These groups usually turn out to be what are now known as Lie groups. A symmetry of a differential equation is an automorphism of the solution space. In other words a map which takes a solution of the differential equa- tion and transforms it to another solution. Most work on symmetries of differential equations involves the study of groups of symmetries, though one can also consider symmetries which do not form groups. For example, if we have a solution u(x, t) of the heat equation uxx=ut , then the map Ä {a : u(x, t) u(x+a, t) is a symmetry, since u(x+a, t) is also a solution. We regard {a as defining a one parameter group. There are many more symmetries of the heat equation which can be obtained by Lie's prolonga- tion algorithm. 1 The author thanks Professor A. H. Dooley and the Australian Research Council. Some of this work first appeared in the author's Ph.D. thesis. E-mail: Mark.CraddockÄuts.edu. 107 0022-0396Â00 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 108 MARK CRADDOCK Briefly, the prolongation algorithm works as follows. Consider a PDE in n variables L(x, D:) u(x)= f(x), (1.1) where |:| D:= , :1 :n x1 }}}xn : a multindex, and a vector field n v= : !k (x, u) +,(x, u) . (1.2) k=1 xk u The transformations generated by (1.2) are Lie point symmetries of (1.1) if and only if the following condition holds. prnv[L(x, D:) u(x)&f ]=0 (1.3) whenever u is a solution of (1.1). prnv is called the nth prolongation of v (hence prolongation algorithm). It is defined in such a way as to ensure that the group of transformations it generates take u and its nth derivatives to the nth derivatives of u~, where u~is the function u transformed under the group generated by v. Essentially we just insist that the chain rule hold. The rigorous justification of this method and an explicit formula for prnv is contained in Olver's book [9]. Olver [9] provides probably the best modern account of Lie's theory and its many applications. It also has numerous examples. The important point to keep in mind is that computation of Lie point symmetries is in practice, an entirely straightforward procedure. Indeed many computer packages exist for determining point symmetries. The symmetries that Lie's prolongation algorithm produces are frequently quite complex and unexpected. As such they yield valuable infor- mation about the PDE in question. However, it has usually been thought that they can only be defined as local groups of symmetries and do not lie within the realm of representation theory. The purpose of this paper, and its predecessor is to demonstrate that there is a strong connection between the group of point symmetries that Lie's method produces and the representation theory of the group in which the symmetries are contained. In fact Lie's prolongation algorithm very often yields representations of the underlying groups. In a previous paper [4] we studied the symmetry groups of three impor- tant linear partial differential equations of mathematical physics. These were the Schro dinger, heat and a particular FokkerPlanck equation. SYMMETRY GROUPS 109 In the case of the Schrodinger equation we showed that the group of point symmetries is in fact a semi direct product of the double cover of SL(2, R) and the three dimensional Heisenberg group. Moreover G=H3 < SL(2, R)isaglobal group of symmetries, and if we denote the symmetry operator that Lie's prolongation algorithm gives by _, then _ is a genuine irreducible representation of G. (We refer to _ as the Lie representation of G). Further there exists a unitary representation 2 [T1 , L (R)] of G, and a unitary operator A (built essentially from the fundamental solution of the Schrodinger equation) such that (_(g) Af )(x, t)=(AT1(g) f )(x, t) (1.4) for all g # G and f # L2(R). For the FokkerPlanck and heat equations analagous results hold, although _ is no longer unitary and the representation space is now a space of distributions rather than L2(R). Analogous results hold for the n-dimen- sional versions of the heat and Schro dinger equations, and Schro dinger's equation for several important nonzero potentials. In this paper we present similar results for two different equations. Namely the axially symmetric wave equation and the Laplace equation on R2. These equations are respectively hyperbolic and elliptic. For the axially symmetric wave equation we have complete results. For Laplace's equation we have only partial results. The essential reason for the incompleteness of the results for the Laplace equation on R2 is that the group of point sym- metries is infinite dimensional, whereas the groups for the wave equation, heat equation, etc. are finite dimensional. Nevertheless the results we have suggest strongly that the phenomenon we are describing is in fact of a general nature. Namely that for linear PDEs the group of point symmetries that Lie's algorithm produces is equivalent to the action of a representation of the underlying group on some locally convex topological vector space. This suggests that one may exploit the machinery of representation theory to study solutions of PDEs in a very direct manner. We will present some results which utilise the connection between the Lie symmetries and a representation of the underlying group, to illustrate the basic ideas. 2. THE AXIALLY SYMMETRIC WAVE EQUATION As our first, and most complete case, we consider the axially symmetric wave equation 2u 1 u 2u + = . (2.1) x2 x x t2 110 MARK CRADDOCK Equation (2.1) is a hyperbolic PDE which arises in a number of areas of mathematical physics. It has a four dimensional Lie algebra of symmetries which may readily be computed. We state the result and refer the reader to Bluman and Kumei [1] for the details. Theorem 2.1. The symmetry algebra of (2.1) is spanned by the vector fields v = (2.2) 1 t 1 v =x +t & u (2.3) 2 x t 2 u v =2xt +(x2+t2) &tu (2.4) 3 x t u v =u (2.5) 4 u v =:(x, t) , (2.6) : u where : is any solution of (2.1). We will be concerned with the vector fields [v1 , v2 , v3 , v4]. The infinite dimensional Lie algebra generated by the v: reflects the fact that the equation is linear and one can add solutions together to form new solu- tions. The next result is an elementary calculation with commutator tables. Lemma 2.2. [v1 , v2 , v3 , v4] generate a copy of the Lie algebra of sl2(R)ÄR. Proof. Fix the following basis for sl2 , 1 0 0 2 0 0 &1 X1= , X2= 1 , X3= (2.7) \1 0+ \0 &2+ \0 0+ and let X4 be any basis for the one dimensional Lie algebra of R. If one constructs the commutator tables for [X1 , X2 , X3 , X4] and [v1 , v2 , v3 , v4], we see that there is a Lie algebra isomorphism which send vk to Xk , k=1, ..., 4. K SYMMETRY GROUPS 111 Corollary 2.3. The Lie group of symmetries spanned by v1 , ..., v4 is locally isomorphic to SL(2, R)_R. Our aim is to show that the symmetries generated by the above vector fields is in fact equivalent to an action of a representation of the principal series of SL(2, R). In order to show this we need to know what the Lie point symmetries do to solutions of (2.1). This is the content of the next result. The action of the symmetries on solutions we refer to as the Lie representation. Proposition 2.4. The Lie representation {, obtained from exponentiating v1 , ..., v4 is given by {(exp =v1) u(x, t)=u(x, t&=) &(1Â2) = &= &= {(exp =v2) u(x, t)=e u(e x, e t) 1 {(exp =v3) u(x, t)= - 1+2=t&=2(x2&t2) x t&=(x2&t2) _u , \1+2=t&=2(x2&t2) 1+2=t&=2(x2&t2)+ = {(exp =v4) u(x, t)=e u(x, t), where u is any solution of (2.1). The proof is once more straightforward. The content of this result is that if we have a solution of the axially symmetric wave equation, and we trans- form it according to Propostion 2.4, we will still have a solution, at least if = is not too large.
Load more

Recommended publications
  • On Certain Types of Point Symmetries of Systems of Second-Order Ordinary Differential Equations
    On certain types of point symmetries of systems of second-order ordinary differential equations R. Campoamor-Stursberg Instituto de Matem´atica Interdisciplinar (IMI) and Dpto. de Geometr´ıay Topolog´ıa,Fac. CC. Matem´aticas Universidad Complutense de Madrid Plaza de Ciencias 3, E-28040 Madrid, Spain Abstract Existence criteria for some generic types of point symmetries of systems of n-second order ordinary differential equations are studied, specially in connection with the generation of semisimple subalgebras of symmetries belonging to the simple linear and orthogonal types, as well as their maximal dimension and rank. The structure of certain time-dependent symmetries, in particular scaling symmetries, are also stud- ied, and the structure of the subalgebras they span determined. Generic examples illustrating the procedure are given. Key words: Lie symmetry method, second order systems of ODEs, Lie algebra, PACS: 02.20Qs, 02.30Hq 1 Introduction Far beyond its original motivation and conception to reduce and classify ordinary differen- tial equations (ODEs), as well as to analyze certain properties of their solutions, the Lie symmetry method has been generalized and enlarged to become a powerful and sometimes indispensable tool in Geometry and Mechanics [1,2]. Among the various questions analyzed, the linearization of ordinary differential equations is relevant problem, as it reduces heav- ily the computational complexity. While the linearization problem for scalar second order ODEs has been solved satisfactorily [3{5], for the case of systems there still remains some work to do. In this context, it is known that, under certain symmetry conditions, systems Email address: [email protected] (R.
    [Show full text]
  • A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems
    A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems H. Kötz Institut für Mathematische Physik, Technische Universität Braunschweig, Braunschweig, FRG Z. Naturforsch. 48a, 535-550 (1993); received December 24, 1992 "Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differen- tial equations which admits a finite-dimensional Lie point symmetry group G are an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra 'S of the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of 'S vary in case of 3 < s < r, depending on the solvability of 'S. If the r-dimensional Lie algebra 'S of the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of 'S one has to determine the optimal subsystems of semisimple subalgebras of 'S in order to construct the full optimal systems of s-dimensional subal- gebras of 'S with 3 < s <r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra 'S of the eight-dimensional Lie point symmetry group G admitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case. Key words: Group theory; Statistical plasma physics. Introduction connected subset M of the Euclidean space JR" + m which leave the system & invariant and map (graphs Similarity analysis is a powerful tool for obtaining of) solutions to (graphs of)solutions.
    [Show full text]
  • Lie Symmetries and Singularity Analysis for Generalized Shallow
    Lie symmetries and singularity analysis for generalized shallow-water equations Andronikos Paliathanasis∗ Institute of Systems Science, Durban University of Technology PO Box 1334, Durban 4000, Republic of South Africa June 3, 2020 Abstract We perform a complete study by using the theory of invariant point transformations and the singularity analysis for the generalized Camassa-Holm equation and the generalized Benjamin-Bono-Mahoney equation. From the Lie theory we find that the two equations are invariant under the same three-dimensional Lie algebra which is the same Lie algebra admitted by the Camassa-Holm equation. We determine the one- dimensional optimal system for the admitted Lie symmetries and we perform a complete classification of the similarity solutions for the two equations of our study. The reduced equations are studied by using the point symmetries or the singularity analysis. Finally, the singularity analysis is directly applied on the partial differential equations from where we infer that the generalized equations of our study pass the singularity test and are integrable. Keywords: Lie symmetries; invariants; shallow water; Camassa-Holm; Benjamin-Bono-Mahoney 1 Introduction The Lie symmetry analysis plays a significant role in the study of nonlinear differential equations. The existence arXiv:2006.01133v1 [nlin.SI] 1 Jun 2020 of a Lie symmetry for a given differential equation is equivalent with the existence of one-parameter point transformation which leaves the differential equation invariant. The later property can be used to reduce the number of independent variables on the case of partial differential equations (PDE), or reduce the order of an ordinary differential equation (ODE) [1], that is achieved thought the Lie invariants.
    [Show full text]
  • Symmetry in Nonlinear Mathematical Physics (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 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.
    [Show full text]
  • Lie Symmetries of Differential Equations: Classical Results and Recent Contributions
    Symmetry 2010, 2, 658-706; doi:10.3390/sym2020658 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Review Lie Symmetries of Differential Equations: Classical Results and Recent Contributions Francesco Oliveri Department of Mathematics, University of Messina, Contrada Papardo, Viale Ferdinando Stagno d’Alcontres 31, I–98166 Messina, Italy; Tel.: +39-090-6765065; Fax: +39-090-393502; E-Mail: [email protected] Received: 2 January 2010 / Accepted: 30 March 2010 / Published: 8 April 2010 Abstract: Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems. Keywords: lie point symmetries; invariance of differential equations; invertible mappings between differential equations Classification: MSC 22E70, 34A05, 35A30, 58J70, 58J72. 1. Introduction Symmetry (joined to simplicity) has been, is, and probably will continue to be, an elegant and useful tool in the formulation and exploitation of the laws of nature. The request of symmetry accounts for the regularities of the laws that are independent of some inessential circumstances. For instance, the reproducibility of experiments in different places at different times relies on the invariance of the laws Symmetry 2010, 2 659 of nature under space translation and rotation (homogeneity and isotropy of space), and time translation (homogeneity of time).
    [Show full text]
  • Lie Symmetry Group Methods for Differential Equations
    Lie symmetry group methods for differential equations F. G¨ung¨or∗1 1Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 Istanbul, Turkey Abstract Fundamentals on Lie group methods and applications to differ- ential equations are surveyed. Numerous examples are included to elucidate their extensive applicability for analytically solving both or- dinary and partial differential equations. arXiv:1901.01543v8 [math.CA] 1 Jun 2021 ∗[email protected] 1 Contents 1 Vector fields and integral curves 4 2 Differential equations and their symmetry group 7 3 Differential invariants 18 3.1 Invariant differentiation . 19 4 Reduction of order for ordinary differential equations 24 4.1 Group classification problem . 50 5 Group-invariant solutions 57 5.1 Linearization by symmetry structure . 81 5.2 Lie’s Linearization Theorem . 88 5.3 Concludingremarks........................ 90 2 Introduction This paper reviews a treatment of differential equations using methods from Lie group theory. Symmetry group methods are amongst the most pow- erful universal tools for the study of differential equations. There has been rapid progress on these methods over the last few decades. Methods and algo- rithms for classifying subalgebras of Lie algebras, new results on the structure and classification of abstract finite and infinite dimensional Lie algebras [48] and methods for solving group classification problems for differential equa- tions greatly facilitated to systematically obtain exact analytic solutions by quadratures to ordinary differential
    [Show full text]
  • Lie Symmetry Analysis, Lie-Bäcklund Symmetries, Explicit Solutions, and Conservation Laws of Drinfeld-Sokolov-Wilson System
    Zhang and Zhao Boundary Value Problems (2017)2017:154 DOI 10.1186/s13661-017-0885-7 R E S E A R C H Open Access Lie symmetry analysis, Lie-Bäcklund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system Yufeng Zhang1* and Zhonglong Zhao2 *Correspondence: [email protected] Abstract 1College of Mathematics, China University of Mining and The symmetry analysis method is used to study the Drinfeld-Sokolov-Wilson system. Technology, Xuzhou, 221116, The Lie point symmetries of this system are obtained. An optimal system of P.R. China one-dimensional subalgebras is derived by using Ibragimov’s method. Based on the Full list of author information is available at the end of the article optimal system, similarity reductions and explicit solutions of the system are presented. The Lie-Bäcklund symmetry generators are also investigated. Furthermore, the method of constructing conservation laws of nonlinear partial differential equations with the aid of a new conservation theorem associated with Lie-Bäcklund symmetries is presented. Conservation laws of the Drinfeld-Sokolov-Wilson system are constructed by using this method. MSC: 76M60; 34A05; 70S10 Keywords: Drinfeld-Sokolov-Wilson system; optimal system; Lie-Bäcklund symmetries; explicit solutions; conservation laws 1 Introduction The Lie symmetry method was initiated by Lie [] in the second half of the nineteenth cen- tury. It has become one of the most powerful methods to study nonlinear partial differen- tial equations (NLPDEs). The core idea of the Lie symmetry method is that one-parameter groups of transformations acting on the space of the independent and dependent variables leave the NLPDEs unchanged [–].
    [Show full text]
  • Lie Group Classification of Second-Order Ordinary Difference
    Lie group classification of second-order ordinary difference equations Vladimir Dorodnitsyn∗ Roman Kozlov† Pavel Winternitz‡ CRM-2634 October 1999 ∗Keldysh Institute of Applied Mathematics of Russian Academy of Science, Miusskaya Pl. 4, Moscow, 125047, Russia; [email protected] †Department of Mathematical Sciences, NTNU, N-7491, Trondheim, Norway; [email protected] ‡Centre de recherches math´ematiques, Universit´ede Montr´eal,Montr´eal,QC, H3C 3J7, Canada; [email protected] Abstract A group classification of invariant difference models, i.e. difference equations and meshes, is presented. In the continuous limit the results go over into Lie’s classification of second order ordinary differential equations. The discrete model is a three point one and we show that it can be invariant under Lie groups of dimension 0 ≤ n ≤ 6. R´esum´e Une classification de sch´emasaux diff´erences finies est pr´esent´ees,c.-`a-d. une classification des ´equations aux diff´erences finies et des r´eseaux correspondants. Dans la limite continue les r´esultats co¨ıncident avec la classification des ´equations diff´erentielles ordinaires de second ordre due `aS. Lie. Les mod`eles discrets consid´er´essont des mod`eles `a3 points sur le r´eseauet nous montrons que leur groupe de sym´etrie est de dimension n avec 0 ≤ n ≤ 6. 1 Introduction Lie group theory started out as a theory of transformations of solutions of sets of differential equations [1, 2, 3, 4]. Over the years it has developed into a powerful tool for classifying differential equations and for solving them. These aspects of Lie group theory have been described in many books and lecture notes [5, 6, 7, 8, 9, 10].
    [Show full text]
  • Lie Symmetry, Full Symmetry Group, and Exact Solutions to the (2+1)-Dimensional Dissipative Akns Equation
    LIE SYMMETRY, FULL SYMMETRY GROUP, AND EXACT SOLUTIONS TO THE (2+1)-DIMENSIONAL DISSIPATIVE AKNS EQUATION ZHENG-YI MA1,2,∗, HUI-LIN WU1, QUAN-YONG ZHU1 1Department of Mathematics, Lishui University, Lishui 323000, P.R. China 2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, P.R. China E-mail∗: [email protected] (corresponding author) Received January 13, 2017 Abstract. In this paper, a detailed Lie symmetry analysis of the (2+1)-dimensional dissipative AKNS equation is performed. The general finite transformation group is given via a simple direct method, which is in fact equivalent to Lie point symme- try group. The similarity reductions are considered from the general Lie symmetry and some exact solutions of the (2+1)-dimensional dissipative AKNS equation are ob- tained. Key words: (2+1)-dimensional dissipative AKNS equation, Lie symmetry, full symmetry group, similarity reduction, invariant solutions. PACS: 02.30Jr, 02.20.Sv, 05.45.Pq, 02.20.Tw. 1. INTRODUCTION The study of the symmetries of partial differential equations (PDEs) is a central issue in nonlinear mathematical physics. During the nineteenth century, Sophus Lie [1, 2] proposed a standard technique, the Lie symmetry method, to find the classical Lie point symmetry groups and algebras for differential systems. A lot of universal applications of Lie symmetry groups in differential equation were discussed in the literature [1–19], such as the reduction of order of ordinary differential equations and dimension of PDEs, the construction of invariant solutions, the mapping solutions to other solutions and the detection of linearizing transformations, the derivation of conservation laws and so on.
    [Show full text]
  • Lie Symmetry Analysis, Exact Solutions, Conservation Laws and Bäcklund Transformations of the Gibbons-Tsarev Equation
    S S symmetry Article Lie Symmetry Analysis, Exact Solutions, Conservation Laws and Bäcklund Transformations of the Gibbons-Tsarev Equation Huanhuan Lu and Yufeng Zhang * School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China; [email protected] * Correspondence: [email protected]; Tel.: +1-362-616-6486 Received: 3 August 2020; Accepted: 17 August 2020; Published: 18 August 2020 Abstract: In this paper, we mainly put the Lie symmetry analysis method on the Gibbons-Tsarev equation (GTe) to obtain some new results, including some Lie symmetries, one-parameter transformation groups, explicit invariant solutions in the form of power series. Subsequently, the self-adjointness of the GTe is singled out. It follows that the conservation laws associated with symmetries of GTe are constructed with the aid of Ibragimov’ method. Finally, we present the Bäcklund transformations so that more abundant solutions can be worked out. Keywords: symmetry; exact solutions; conservation laws; Bäcklund transformation 1. Introduction During the last decades, searching for the symmetries of the nonlinear differential equations has attracted increasing attention from researchers for their wide applications in investigating the integrability of the original equation. In early reference documentations, various powerful approaches were put forward, one after another, to obtain the symmetries of equations [1–5]. Among them, the Lie symmetry analysis method proposed by Olver is regarded as the most classic and verified to be extremely effective for studying Lie symmetry and related group transformations, conservation laws and invariant solutions [1]. With the aid of the classical Lie symmetry analysis method, a large number of differential equations have been discussed, such as the new fifth-order nonlinear integrable equation [6], the Heisenberg equation [7] and some others [8–11].
    [Show full text]
  • Abdus Salam United Nations Educational, Scientific and Cultural International XA0102712 Organization Centre
    the 1C72001/55 HOT abdus salam united nations educational, scientific and cultural international XA0102712 organization centre international atomic energy agency for theoretical physics ON LIE POINT SYMMETRY OF CLASSICAL WESS-ZUMINO-WITTEN MODEL •*; >*. •« > Karmadeva Maharana 32/37 ^.?^ !"* e'i Available at: http://www.ictp.trieste.it/~pub-off IC/2001/55 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON LIE POINT SYMMETRY OF CLASSICAL WESS-ZUMINO-WITTEN MODEL Karmadeva Maharana1 Physics Department, Utkal University, Bhubaneswar 751 004, India and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract We perform the group analysis of Witten's equations of motion for a particle moving in the presence of a magnetic monopole, and also when constrained to move on the surface of a sphere, which is the classical example of Wess-Zumino-Witten model. We also consider variations of this model. Our analysis gives the generators of the corresponding Lie point symmetries. The Lie symmetry corresponding to Kepler's third law is obtained in two related examples. MIRAMARE - TRIESTE June 2001 'E-mail: [email protected] 1 Introduction The Wess-Zumino model[l] has the distinction of being applicable via the current algebra con- ditions to explain the process K+K~—>7r+7r°7r~ as well as in setting up the string theory actions through the generalizations made by Witten[2]. To motivate before arriving at the generalized form, Witten had considered an example of the equation of motion of a particle in three dimen- sions constrained to move on the surface of a sphere in the presence of a magnetic monopole.
    [Show full text]
  • Lie Symmetry Method for Partial Differential Equations with Application in finance
    Università degli Studi di Trento Dipartimento di Matematica Corso di Laurea magistrale in Matematica Lie symmetry method for partial differential equations with application in finance FINAL THESIS Supervisor: Graduant: Prof. Luciano Tubaro Francesco Giuseppe Cordoni Co-supervisor: Prof. Luca Di Persio Co-supervisor: Prof. Andrea Pascucci Accademic Year 2012-2013 i "Per cui, quella sera, nell’ansa che il fiume descrive costeggiando la zona degli orti per anziani, alcuni pensionati avevano fiduciosamente messo a mollo le lenze." Preface Preface Partial differential equations (PDE) arise in many areas of scientific research and they have been classically used in describing models spanning from quantum mechanics to biological systems. More recently the theory of PDE have attracted an increasing interest by mathematicians and practitioners working in those areas related to banks and insurances. When we study a particular differential equation, we have to face with two fundamental questions: " Is there (at least) a solution ?" and (just in case) "Is it possible to write it (them) explicitly ?" Unfortunately it does not exist a standard procedure to follow to reach these goals. Even if we are able to show that a given differential problem has solutions, we often cannot show them analytically. This makes the derivation of exact solutions for PDE a corner stone in studying differential equations theory. In fact, although numerical analysis has made impressive strides in recent years, still the quest of find analytic solution is the main goal in studying PDE. It follows that a method able to recursively produce solutions of a given PDE may be very useful. In this thesis we deeply analyse one of the most fruitful technique for finding exact solutions of a given differential equations.
    [Show full text]