Bäcklund Transformations: Some Old and New Perspectives

Transcription

1 Bäklund Transformations: Some Old and New Perspetives C. J. Papahristou *, A. N. Magoulas ** * Department of Physial Sienes, Helleni Naval Aademy, Piraeus 18539, Greee papahristou@snd.edu.gr ** Department of Eletrial Engineering, Helleni Naval Aademy, Piraeus 18539, Greee aris@snd.edu.gr Abstrat Bäklund transformations (BTs) are traditionally regarded as a tool for integrating nonlinear partial differential equations (PDEs). Their use has been reently extended, however, to problems suh as the onstrution of reursion operators for symmetries of PDEs, as well as the solution of linear systems of PDEs. In this artile, the onept and some appliations of BTs are reviewed. As an example of an integrable linear system of PDEs, the Maxwell equations of eletromagnetism are shown to onstitute a BT onneting the wave equations for the eletri and the magneti field; plane-wave solutions of the Maxwell system are onstruted in detail. The onnetion between BTs and reursion operators is also disussed. 1. Introdution Bäklund transformations (BTs) were originally devised as a tool for obtaining solutions of nonlinear partial differential equations (PDEs) (see, e.g., [1] and the referenes therein). They were later also proven useful as reursion operators for onstruting infinite sequenes of nonloal symmetries and onservation laws of ertain PDEs [ 6]. In simple terms, a BT is a system of PDEs onneting two fields that are required to independently satisfy two respetive PDEs [say, (a) and (b)] in order for the system to be integrable for either field. If a solution of PDE (a) is known, then a solution of PDE (b) is obtained simply by integrating the BT, without having to atually solve the latter PDE (whih, presumably, would be a muh harder task). In the ase where the PDEs (a) and (b) are idential, the auto-bt produes new solutions of PDE (a) from old ones. As desribed above, a BT is an auxiliary tool for finding solutions of a given (usually nonlinear) PDE, using known solutions of the same or another PDE. But, what if the BT itself is the differential system whose solutions we are looking for? As it turns out, to solve the problem we need to have parameter-dependent solutions of both PDEs (a) and (b) at hand. By properly mathing the parameters (provided this is possible) a solution of the given system is obtained. The above method is partiularly effetive in linear problems, given that parametri solutions of linear PDEs are generally not hard to find. An important paradigm of a BT assoiated with a linear problem is offered by the Maxwell system of equations of eletromagnetism [7,8]. As is well known, the onsisteny of this system demands that both the eletri and the magneti field independently satisfy a respetive wave equation. These equations have known, parameter-dependent solutions; namely, monohromati plane waves with arbitrary amplitudes, frequenies and wave vetors 1

2 C. J. PAPACHRISTOU & A. N. MAGOULAS (the parameters of the problem). By inserting these solutions into the Maxwell system, one may find the appropriate expressions for the parameters in order for the plane waves to also be solutions of Maxwell s equations; that is, in order to represent an atual eletromagneti field. This artile, written for eduational purposes, is an introdution to the onept of a BT and its appliation to the solution of PDEs or systems of PDEs. Both lassial and novel views of a BT are disussed, the former view predominantly onerning integration of nonlinear PDEs while the latter one being appliable mostly to linear systems of PDEs. The artile is organized as follows: In Setion we review the lassial onept of a BT. The solution-generating proess by using a BT is demonstrated in a number of examples. In Se. 3 a different pereption of a BT is presented, aording to whih it is the BT itself whose solutions are sought. The onept of onjugate solutions is introdued. As an example, in Ses. 4 and 5 the Maxwell equations in empty spae and in a linear onduting medium, respetively, are shown to onstitute a BT onneting the wave equations for the eletri and the magneti field. Following [7], the proess of onstruting plane-wave solutions of this BT is presented in detail. This proess is, of ourse, a familiar problem of eletrodynamis but is seen here under a new perspetive by employing the onept of a BT. Finally, in Se. 6 we briefly review the onnetion between BTs and reursion operators for generating infinite sequenes of nonloal symmetries of PDEs.. Bäklund Transformations: Classial Viewpoint Consider two PDEs P[u]= and Q[v]= for the unknown funtions u and v, respetively. The expressions P[u] and Q[v] may ontain the orresponding variables u and v, as well as partial derivatives of u and v with respet to the independent variables. For simpliity, we assume that u and v are funtions of only two variables x, t. Partial derivatives with respet to these variables will be denoted by using subsripts: u x, u t, u xx, u tt, u xt, et. Independently, for the moment, also onsider a pair of oupled PDEs for u and v: B [ u, v] = ( a) B [ u, v] = ( b) (1) 1 where the expressions B i [u,v] (i=1,) may ontain u, v as well as partial derivatives of u and v with respet to x and t. We note that u appears in both equations (a) and (b). The question then is: if we find an expression for u by integrating (a) for a given v, will it math the orresponding expression for u found by integrating (b) for the same v? The answer is that, in order that (a) and (b) be onsistent with eah other for solution for u, the funtion v must be properly hosen so as to satisfy a ertain onsisteny ondition (or integrability ondition or ompatibility ondition). By a similar reasoning, in order that (a) and (b) in (1) be mutually onsistent for solution for v, for some given u, the funtion u must now itself satisfy a orresponding integrability ondition. If it happens that the two onsisteny onditions for integrability of the system (1) are preisely the PDEs P[u]= and Q[v]=, we say that the above system onstitutes a Bäklund transformation (BT) onneting solutions of P[u]= with solutions of

3 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES Q[v]=. In the speial ase where P Q, i.e., when u and v satisfy the same PDE, the system (1) is alled an auto-bäklund transformation (auto-bt) for this PDE. Suppose now that we seek solutions of the PDE P[u]=. Assume that we are able to find a BT onneting solutions u of this equation with solutions v of the PDE Q[v]= (if P Q, the auto-bt onnets solutions u and v of the same PDE) and let v=v (x,t) be some known solution of Q[v]=. The BT is then a system of PDEs for the unknown u, B [ u, v ] =, i= 1, () i The system () is integrable for u, given that the funtion v satisfies a priori the required integrability ondition Q[v]=. The solution u then of the system satisfies the PDE P[u]=. Thus a solution u(x,t) of the latter PDE is found without atually solving the equation itself, simply by integrating the BT () with respet to u. Of ourse, this method will be useful provided that integrating the system () for u is simpler than integrating the PDE P[u]= itself. If the transformation () is an auto-bt for the PDE P[u]=, then, starting with a known solution v (x,t) of this equation and integrating the system (), we find another solution u(x,t) of the same equation. Let us see some examples of the use of a BT to generate solutions of a PDE: 1. The Cauhy-Riemann relations of Complex Analysis, u = v ( a) u = v ( b) (3) x y y x (here, the variable t has been renamed y) onstitute an auto-bt for the Laplae equation, P [ w] w + w = (4) xx Let us explain this: Suppose we want to solve the system (3) for u, for a given hoie of the funtion v(x,y). To see if the PDEs (a) and (b) math for solution for u, we must ompare them in some way. We thus differentiate (a) with respet to y and (b) with respet to x, and equate the mixed derivatives of u. That is, we apply the integrability ondition (u x ) y = (u y ) x. In this way we eliminate the variable u and find the ondition that must be obeyed by v(x,y): yy P [ v] v + v =. xx Similarly, by using the integrability ondition (v x ) y = (v y ) x to eliminate v from the system (3), we find the neessary ondition in order that this system be integrable for v, for a given funtion u(x,y): yy P [ u] u + u =. xx In onlusion, the integrability of system (3) with respet to either variable requires that the other variable must satisfy the Laplae equation (4). Let now v (x,y) be a known solution of the Laplae equation (4). Substituting v=v in the system (3), we an integrate this system with respet to u. It is not hard to yy 3

4 C. J. PAPACHRISTOU & A. N. MAGOULAS show (by eliminating v from the system) that the solution u will also satisfy the Laplae equation (4). As an example, by hoosing the solution v (x,y)=xy, we find a new solution u(x,y)= (x y )/ +C.. The Liouville equation is written u P[ u] u e = u = e (5) xt Due to its nonlinearity, this PDE is hard to integrate diretly. A solution is thus sought by means of a BT. We onsider an auxiliary funtion v(x,t) and an assoiated PDE, We also onsider the system of first-order PDEs, xt u Q[ v] v xt = (6) ( u v)/ ( u+ v)/ x x t t u + v = e ( a) u v = e ( b) (7) Differentiating the PDE (a) with respet to t and the PDE (b) with respet to x, and eliminating (u t v t ) and (u x +v x ) in the ensuing equations with the aid of (a) and (b), we find that u and v satisfy the PDEs (5) and (6), respetively. Thus, the system (7) is a BT onneting solutions of (5) and (6). Starting with the trivial solution v= of (6), and integrating the system u / u / t u = e, u = e, x we find a nontrivial solution of (5): x+ t u ( x, t) = ln C. 3. The sine-gordon equation has appliations in various areas of Physis, e.g., in the study of rystalline solids, in the transmission of elasti waves, in magnetism, in elementary-partile models, et. The equation (whose name is a pun on the related linear Klein-Gordon equation) is written P[ u] u sin u= u = sin u (8) xt The following system of equations is an auto-bt for the nonlinear PDE (8): xt 1 u v 1 1 u+ v ( u+ v) x = a sin, ( u v) t = sin a (9) where a ( ) is an arbitrary real onstant. [Beause of the presene of a, the system (9) is alled a parametri BT.] When u is a solution of (8) the BT (9) is integrable for v, whih, in turn, also is a solution of (8): P[v]=; and vie versa. Starting with the trivial solution v= of v xt = sin v, and integrating the system 4

5 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES u u ux = a sin, u t = sin, a we obtain a new solution of (8): t u ( x, t) = 4artan C exp ax+ a. 3. Conjugate Solutions and Another View of a BT As presented in the previous setion, a BT is an auxiliary devie for onstruting solutions of a (usually nonlinear) PDE from known solutions of the same or another PDE. The onverse problem, where solutions of the differential system representing the BT itself are sought, is also of interest, however, and has been reently suggested [7,8] in onnetion with the Maxwell equations (see subsequent setions). To be speifi, assume that we need to integrate a given system of PDEs onneting two funtions u and v: B [ u, v] =, i= 1, (1) i Suppose that the integrability of the system for both funtions requires that u and v separately satisfy the respetive PDEs P[ u] = ( a) Q[ v] = ( b) (11) That is, the system (1) is a BT onneting solutions of the PDEs (11). Assume, now, that these PDEs possess known (or, in any ase, easy to find) parameter-dependent solutions of the form u= f ( x, y; α, β, ), v= g( x, y ; κ, λ, ) (1) where α, β, κ, λ, et., are (real or omplex) parameters. If values of these parameters an be determined for whih u and v jointly satisfy the system (1), we say that the solutions u and v of the PDEs (11a) and (11b), respetively, are onjugate through the BT (1) (or BT-onjugate, for short). By finding a pair of BT-onjugate solutions one thus automatially obtains a solution of the system (1). Note that solutions of both integrability onditions P[u]= and Q[v]= must now be known in advane! From the pratial point of view the method is thus most appliable in linear problems, sine it is muh easier to find parameter-dependent solutions of the PDEs (11) in this ase. Let us see an example: Going bak to the Cauhy-Riemann relations (3), we try the following parametri solutions of the Laplae equation (4): u ( x, y) = α ( x y ) + β x+ γ y, v ( x, y) = κ xy+ λ x+ µ y. 5

6 C. J. PAPACHRISTOU & A. N. MAGOULAS Substituting these into the BT (3), we find that κ=α, µ=β and λ= γ. Therefore, the solutions u ( x, y) = α ( x y ) + β x+ γ y, v ( x, y) = α xy γ x+ β y of the Laplae equation are BT-onjugate through the Cauhy-Riemann relations. As a ounter-example, let us try a different ombination: u ( x, y) = α xy, v ( x, y) = β xy. Inserting these into the system (3) and taking into aount the independene of x and y, we find that the only possible values of the parameters α and β are α=β=, so that u(x,y)= v(x,y)=. Thus, no non-trivial BT-onjugate solutions exist in this ase. 4. Example: The Maxwell Equations in Empty Spae An example of an integrable linear system whose solutions are of physial interest is furnished by the Maxwell equations of eletrodynamis. Interestingly, as noted reently [7], the Maxwell system has the property of a BT whose integrability onditions are the eletromagneti (e/m) wave equations that are separately valid for the eletri and the magneti field. These equations possess parameter-dependent solutions that, by a proper hoie of the parameters, an be made BT-onjugate through the Maxwell system. In this and the following setion we disuss the BT property of the Maxwell equations in vauum and in a onduting medium, respetively. In empty spae, where no harges or urrents (whether free or bound) exist, the Maxwell equations are written (in S.I. units) [9] B ( a) E= ( ) E= t E ( b) B= ( d) B= ε µ t where E and B are the eletri and the magneti field, respetively. Here we have a system of four PDEs for two fields. The question is: what are the neessary onditions that eah of these fields must satisfy in order for the system (13) to be selfonsistent? In other words, what are the onsisteny onditions (or integrability onditions) for this system? Guided by our experiene from Se., to find these onditions we perform various differentiations of the equations of system (13) and require that ertain differential identities be satisfied. Our aim is, of ourse, to eliminate one field (eletri or magneti) in favor of the other and find some higher-order PDE that the latter field must obey. As an be heked, two differential identities are satisfied automatially in the system (13): (13) 6

7 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES ( E) =, ( B) =, ( E) = E, ( B) = B. t t t t Two others read ( ) = ( ) E E E ( ) = ( ) B B B (14) (15) Taking the rot of (13) and using (14), (13a) and (13d), we find E ε µ t E = Similarly, taking the rot of (13d) and using (15), (13b) and (13), we get ε µ B t B = (16) (17) No new information is furnished by the remaining two integrability onditions, ( E) = E, ( B) = B. t t t t Note that we have unoupled the equations for the two fields in the system (13), deriving separate seond-order PDEs for eah field. Putting 1 1 ε µ = ε µ (18) (where is the speed of light in vauum) we rewrite (16) and (17) in wave-equation form: 1 E t E = (19) 1 B t B = () We onlude that the Maxwell system (13) is a BT relating solutions of the e/m wave equations (19) and (), these equations representing the integrability onditions of the BT. It should be noted that this BT is not an auto-bt! Indeed, although the PDEs (19) and () are of similar form, they onern different fields with different physial dimensions and physial properties. 7

8 C. J. PAPACHRISTOU & A. N. MAGOULAS The e/m wave equations admit plane-wave solutions of the form F ( k r ω t), with ω = where k = k k (1) The simplest suh solutions are monohromati plane waves of angular frequeny ω, propagating in the diretion of the wave vetor k : E ( r, t) = E exp{ i ( k r ω t)} ( a) () B ( r, t) = B exp{ i ( k r ω t)} ( b) where E and B are onstant omplex amplitudes. The onstants appearing in the above equations (amplitudes, frequeny and wave vetor) an be hosen arbitrarily; thus they an be regarded as parameters on whih the plane waves () depend. We must note arefully that, although every pair of fields ( E, B) satisfying the Maxwell equations (13) also satisfies the wave equations (19) and (), the onverse is not true. Thus, the plane-wave solutions () are not a priori solutions of the Maxwell system (i.e., do not represent atual e/m fields). This problem an be taken are of, however, by a proper hoie of the parameters in (). To this end, we substitute the general solutions () into the BT (13) to find the extra onditions the latter system demands. By fixing the wave parameters, the two wave solutions in () will beome BT-onjugate through the Maxwell system (13). Substituting (a) and (b) into (13a) and (13b), respetively, and taking into i k r i k r aount that e = i k e, we have iω t i k r i ( k r ω t) ( E e ) e = ( k E ) e =, iω t i k r i ( k r ω t) ( B e ) e = ( k B ) e =, so that k E =, k B =. (3) Relations (3) reflet the fat that that the monohromati plane e/m wave is a transverse wave. Next, substituting (a) and (b) into (13) and (13d), we find e e E = i B e iω t i k r i ( k r ω t) ( ) ω i ( k r ω t) i ( k r ω t) k E e = ω B e ( ), iω t i k r i ( k r ω t) e ( e ) B = iωε µ E e i ( k r ω t) ω i ( k r ω t ) ( k B ) e = E e, 8

9 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES so that ω k E = ω B, k B = E (4) We note that the fields E and B are normal to eah other, as well as normal to the diretion of propagation of the wave. We also remark that the two vetor equations in (4) are not independent of eah other, sine, by ross-multiplying the first relation by k, we get the seond relation. Introduing a unit vetor ˆ τ in the diretion of the wave vetor k, we rewrite the first of equations (4) as ˆ τ = k / k ( k = k = ω / ), k 1 B = ( ˆ τ E ) = ( ˆ τ E ) ω. The BT-onjugate solutions in () are now written E ( r, t) = E exp{ i( k r ω t)}, 1 1 (5) B ( r, t) = ( ˆ τ E ˆ )exp{ i ( k r ω t)} = τ E As onstruted, the omplex vetor fields in (5) satisfy the Maxwell system (13). Sine this system is homogeneous linear with real oeffiients, the real parts of the fields (5) also satisfy it. To find the expressions for the real solutions (whih, after all, arry the physis of the situation) we take the simplest ase of linear polarization and write = i E E, R e α where the vetor E,R as well as the number α are real. The real versions of the fields (5), then, read (6) E= E, R os ( k r ω t+ α), 1 1 (7) B= ( ˆ τ E ˆ, R )os ( k r ω t+ α ) = τ E We note, in partiular, that the fields E and B osillate in phase. Our results for the Maxwell equations in vauum an be extended to the ase of a linear non-onduting medium upon replaement of ε and µ with ε and µ, respetively. The speed of propagation of the e/m wave is, in this ase, ω 1 υ = =. k εµ 9

10 C. J. PAPACHRISTOU & A. N. MAGOULAS In the next setion we study the more omplex ase of a linear medium having a finite ondutivity. 5. Example: The Maxwell System for a Linear Conduting Medium Consider a linear onduting medium of ondutivity σ. In suh a medium, Ohm s law is satisfied: J f =σ E, where J f is the free urrent density. The Maxwell equations take on the form [9] B ( a) E= ( ) E= t E ( b) B= ( d) B= µσ E+ εµ t By requiring satisfation of the integrability onditions = = ( E) ( E) E, ( B) ( B) B, (8) we obtain the modified wave equations E t B t E t B t E εµ µσ = B εµ µσ = (9) whih must be separately satisfied by eah field. As in Se. 4, no further information is furnished by the remaining integrability onditions. The linear differential system (8) is a BT relating solutions of the wave equations (9). As in the vauum ase, this BT is not an auto-bt. We now seek BTonjugate solutions. As an be verified by diret substitution into equations (9), these PDEs admit parameter-dependent solutions of the form E ( r, t) = E exp{ s ˆ τ r+ i ( k r ωt)} s = E exp i k r exp( iω t), k B ( r, t) = B exp{ s ˆ τ r+ i ( k r ω t)} s = B exp i k r exp( iω t) k where ˆ τ is the unit vetor in the diretion of the wave vetor k : (3) 1

11 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES ˆ τ = k / k ( k = k = ω / υ ) (υ is the speed of propagation of the wave inside the onduting medium) and where, for given physial harateristis ε, µ, σ of the medium, the parameters s, k and ω satisfy the algebrai system s k εµω µσω sk + =, = (31) We note that, for arbitrary hoies of the amplitudes E and B, the vetor fields (3) are not a priori solutions of the Maxwell system (8), thus are not BT-onjugate solutions. To obtain suh solutions we substitute expressions (3) into the system (8). With the aid of the relation s s i k r i k r k s k e = i k e k one an show that (8a) and (8b) impose the onditions k E =, k B = (3) As in the vauum ase, the e/m wave in a onduting medium is a transverse wave. By substituting (3) into (8) and (8d), two more onditions are found: ( k+ is) ˆ τ E = ωb ( k+ is) ˆ τ B = ( εµω + iµσ ) E Note, however, that (34) is not an independent equation sine it an be reprodued by ross-multiplying (33) by ˆ τ, taking into aount the algebrai relations (31). The BT-onjugate solutions of the wave equations (9) are now written (33) (34) s ˆ τ r i ( k r ωt) E( r, t) = E e e, k+ is s ˆ τ r i ( k r ωt) B( r, t) = ( ˆ τ E) e e (35) ω To find the orresponding real solutions, we assume linear polarization of the wave, as before, and set We also put i E = E, R e α. i ϕ iϕ k i s k i s e k s e ; tan ϕ s / k + = + = + =. 11

12 C. J. PAPACHRISTOU & A. N. MAGOULAS Taking the real parts of equations (35), we finally have: s ˆ τ r E( r, t) = E e os( k r ω t+ α),, R k + s s ˆ τ r B( r, t) = ( ˆ τ E, R ) e os( k r ω t+ α+ ϕ). ω As an exerise, the student may show that these results redue to those for a linear non-onduting medium (f. Se. 4) in the limit σ. 6. BTs as Reursion Operators The onept of symmetries of PDEs was disussed in [1]. Let us review the main fats: Consider a PDE F[u]=, where, for simpliity, u=u(x,t). A transformation u (x,t) u (x,t) from the funtion u to a new funtion u represents a symmetry of the given PDE if the following ondition is satisfied: u (x,t) is a solution of F[u]= if u(x,t) is a solution. That is, F [ u ] = when F [ u] = (36) An infinitesimal symmetry transformation is written u = u+ δu = u+ αq [ u] (37) where α is an infinitesimal parameter. The funtion Q[u] Q(x, t, u, u x, u t,...) is alled the symmetry harateristi of the transformation (37). In order that a funtion Q[u] be a symmetry harateristi for the PDE F[u]=, it must satisfy a ertain PDE that expresses the symmetry ondition for F[u]=. We write, symbolially, S ( Q ; u) = when F [ u] = (38) where the expression S depends linearly on Q and its partial derivatives. Thus, (38) is a linear PDE for Q, in whih equation the variable u enters as a sort of parametri funtion that is required to satisfy the PDE F[u]=. A reursion operator ˆR [1] is a linear operator whih, ating on a symmetry harateristi Q, produes a new symmetry harateristi Q = RQ ˆ. That is, S ( Rˆ Q ; u) = when S ( Q ; u) = (39) It is not too diffiult to show that any power of a reursion operator also is a reursion operator. This means that, starting with any symmetry harateristi Q, one may 1

13 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES in priniple obtain an infinite set of harateristis (thus, an infinite number of symmetries) by repeated appliation of the reursion operator. A new approah to reursion operators was suggested in the early 199s [,3] (see also [4-6]). Aording to this view, a reursion operator is an auto-bt for the linear PDE (38) expressing the symmetry ondition of the problem; that is, a BT produing new solutions Q of (38) from old ones, Q. Typially, this type of BT produes nonloal symmetries, i.e., symmetry harateristis depending on integrals (rather than derivatives) of u. As an example, onsider the hiral field equation 1 1 x x t t F [ g] ( g g ) + ( g g ) = (4) (as usual, subsripts denote partial differentiations) where g is a GL(n,C)-valued funtion of x and t (i.e., an invertible omplex n n matrix, differentiable for all x, t). Let Q[g] be a symmetry harateristi of the PDE (4). It is onvenient to put Q [g] = g Φ[g] and write the orresponding infinitesimal symmetry transformation in the form g = g+ δ g = g+ α gφ [ g] (41) The symmetry ondition that Q must satisfy will be a PDE linear in Q, thus in Φ also. As an be shown [4], this PDE is 1 1 x x tt x x t t S ( Φ; g) Φ + Φ + [ g g, Φ ] + [ g g, Φ ] = (4) whih must be valid when F[g]= (where, in general, [A, B] AB BA denotes the ommutator of two matries A and B). For a given g satisfying F[g]=, onsider now the following system of PDEs for the matrix funtions Φ and Φ : Φ = Φ + [, Φ] 1 x t g g t Φ = Φ + [, Φ] 1 t x g gx (43) The integrability ondition ( Φ x) t = ( Φ t ) x, together with the equation F[g]=, require that Φ be a solution of (4): S (Φ ; g) =. Similarly, by the integrability ondition ( Φ t ) x = ( Φ x ) t one finds, after a lengthy alulation: S (Φ ; g) =. In onlusion, for any g satisfying the PDE (4), the system (43) is a BT relating solutions Φ and Φ of the symmetry ondition (4) of this PDE; that is, relating different symmetries of the hiral field equation (4). Thus, if a symmetry harateristi Q=gΦ of (4) is known, a new harateristi Q =gφ may be found by integrating the BT (43); the onverse is also true. Sine the BT (43) produes new symmetries from old ones, it may be regarded as a reursion operator for the PDE (4). 13

14 C. J. PAPACHRISTOU & A. N. MAGOULAS As an example, for any onstant matrix M the hoie Φ=M learly satisfies the symmetry ondition (4). This orresponds to the symmetry harateristi Q=gM. By integrating the BT (43) for Φ, we get Φ =[X, M] and Q =g[x, M], where X is the potential of the PDE (4), defined by the system of PDEs 1 1 x t, t x X = g g X = g g (44) Note the nonloal harater of the BT-produed symmetry Q, due to the presene of the potential X. Indeed, as seen from (44), in order to find X one has to integrate the hiral field g with respet to the independent variables x and t. The above proess an be ontinued indefinitely by repeated appliation of the reursion operator (43), leading to an infinite sequene of inreasingly nonloal symmetries. 7. Summary Classially, Bäklund transformations (BTs) have been developed as a useful tool for finding solutions of nonlinear PDEs, given that these equations are usually hard to solve by diret methods. By means of examples we saw that, starting with even the most trivial solution of a PDE, one may produe a highly nontrivial solution of this (or another) PDE by integrating the BT, without solving the original, nonlinear PDE diretly (whih, in most ases, is a muh harder task). A different use of BTs, that was reently proposed [7,8], onerns predominantly the solution of linear systems of PDEs. This method relies on the existene of parameter-dependent solutions of the linear PDEs expressing the integrability onditions of the BT. This time it is the BT itself (rather than its assoiated integrability onditions) whose solutions are sought. An appropriate example for demonstrating this approah to the onept of a BT is furnished by the Maxwell equations of eletromagnetism. We showed that this system of PDEs an be treated as a BT whose integrability onditions are the wave equations for the eletri and the magneti field. These wave equations have known, parameterdependent solutions monohromati plane waves with arbitrary amplitudes, frequenies and wave vetors playing the roles of the parameters. By substituting these solutions into the BT, one may determine the required relations among the parameters in order that these plane waves also represent eletromagneti fields (i.e., in order that they be solutions of the Maxwell system). The results arrived at by this method are, of ourse, well known in advaned eletrodynamis. The proess of deriving them, however, is seen here in a new light by employing the onept of a BT. BTs have also proven useful as reursion operators for deriving infinite sets of nonloal symmetries and onservation laws of PDEs [-6] (see also [11] and the referenes therein). Speifially, the BT produes an inreasingly nonloal sequene of symmetry harateristis, i.e., solutions of the linear equation expressing the symmetry ondition (or linearization ) of a given PDE. An interesting onlusion is that the onept of a BT, whih has been proven useful for integrating nonlinear PDEs, may also have important appliations in linear problems. Researh on these matters is in progress. 14

15 BÄCKLUND TRANSFORMATIONS: SOME OLD AND NEW PERSPECTIVES Referenes 1. C. J. Papahristou, Symmetry and integrability of lassial field equations, C. J. Papahristou, Potential symmetries for self-dual gauge fields, Phys. Lett. A 145 (199) C. J. Papahristou, Lax pair, hidden symmetries, and infinite sequenes of onserved urrents for self-dual Yang-Mills fields, J. Phys. A 4 (1991) L C. J. Papahristou, Symmetry, onserved harges, and Lax representations of nonlinear field equations: A unified approah, Eletron. J. Theor. Phys. 7, No. 3 (1) 1 ( 5. C. J. Papahristou, B. K. Harrison, Bäklund-transformation-related reursion operators: Appliation to the self-dual Yang-Mills equation, J. Nonlin. Math. Phys., Vol. 17, No. 1 (1) C. J. Papahristou, Symmetry and integrability of a redued, 3-dimensional self-dual gauge field model, Eletron. J. Theor. Phys. 9, No. 6 (1) 119 ( 7. C. J. Papahristou, The Maxwell equations as a Bäklund transformation, Advaned Eletromagnetis, Vol. 4, No. 1 (15), pages 5-58 ( 8. C. J. Papahristou, Aspets of integrability of differential systems and fields, 9. D. J. Griffiths, Introdution to Eletrodynamis, 3rd Edition (Prentie-Hall, 1999). 1. P. J. Olver, Appliations of Lie Groups to Differential Equations (Springer- Verlag, 1993). 11. M. Marvan, A. Sergyeyev, Reursion operators for dispersionless integrable systems in any dimension, Inverse Problems 8 (1)