77 Progress of Theoretical Physics, Vol. 105, No. 1, January 2001
On the Relation between Lie Symmetries and Prolongation Structures of Nonlinear Field Equations Non-Local Symmetries M. Leo, R. A. Leo, G. Soliani∗) and P. Tempesta Dipartimento di Fisica dell’Universit` a di Lecce, 73100 Lecce, Italy (Received May 23, 2000) An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra L associated with the equations under consideration. Our approach, which is applied by way of example to the Dym and the Korteweg-de Vries equation, allows us to obtain a general formula for the infinitesimal operator of non-local symmetries expressed in terms of elements of L. The method could be exploited to investigate the symmetry properties of other nonlinear field equations possessing nontrivial prolongations.
§1.
Introduction
One of the most important tools to find exact solutions of ordinary and partial differential equations is Lie group theory. An extensive body of literature on this fundamental topic has been produced in the last century, starting with the original works by Lie (see Refs. 1) – 4) and references therein). If a system ∆ of nonlinear differential equations is invariant under the action of an infinitesimal group of transformations generated by a differential operator V , we can use its invariance properties to construct B¨acklund transformations relating new solutions to old ones, or to reduce the number of independent variables of ∆. This can give rise to a new system ∆ that is in principle simpler than the original one. The solutions of the new system allow one to generate solutions of ∆. We point out that group theoretical methods are of crucial importance for treating systems of nonlinear field equations and often represent the only possible procedure available to find exact solutions. In the last few decades, great effort has been devoted to generalize and significantly extend the Lie approach. The so-called non-standard symmetries3)-11) permit the enlargement of the class of possible solutions obtainable by using the group analysis in two distinct ways. The first way is to weaken the invariance requirement of ∆, under the action of the symmetry transformations. This leads to the nonclassical symmetries. This approach was pioneered by Bluman and Cole12) and has been widely investigated in recent years. (See Ref. 3), Chapter 11, for a review of this approach.) The second way consists of extending the space of independent variables by adding auxiliary variables and looking for symmetry properties of an extended system which is related to the original one and exists in this new space. The new variables can be potentials or ∗)
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M. Leo, R. A. Leo, G. Soliani and P. Tempesta
pseudopotentials associated with the system. The corresponding symmetries are called non-local.3)-11) The importance of the non-local approach has been increasingly recognized in the last 10 years. Potential symmetries, extensively studied by several authors, can be exploited to perform the symmetry reduction of a given system of differential equations. The case of partial differential equations has been dealt with by Bluman and Reid,3),4) and applied to the study of nonlinear evolution systems (see, e.g., Refs. 14) – 16)). Edelen9) and Krasil’shchik and Vinogradov10) introduced the concept of non-local symmetries in which the non-locality character is carried out by variables of the pseudopotential type. In this context, Galas has rederived the onesoliton solutions of the Korteweg-de Vries (KdV) and the Dym equation, and the AKNS system.11) Other interesting results on the theory of non-local symmetries are given in Refs. 5) – 7), 17) and 18). Recently, Guthrie and Hickman8) derived new algebraic structures for the bi-Hamiltonian version of the KdV equation, obtaining generalized symmetries depending on non-local variables. The aim of this article is to propose a coherent framework for symmetries of the non-local type with pseudopotentials, based on a deeper algebraic approach. Following Krasil’shchik and Vinogradov,10) we establish a connection between the theory of Lie symmetries and the prolongation structures of Estabrook and Wahlquist (EW).13) We show how to produce B¨acklund transformations of the nonlocal type in a natural way by using the generators of the prolongation algebra associated with the equation under analysis. The core of the method consists of expressing the generator V of the Lie symmetries and the corresponding determining equations1) in terms of the generators of the EW algebra L, and in finding a nontrivial representation of this algebra. Once the determining equations are solved and the explicit form of the operator V is calculated, one can immediately write the corresponding B¨acklund transformations and build up new interesting solutions of our equations. Since a finite-dimensional subalgebra Lo of L yields trivial symmetries only, our technique is based on the idea of enlarging Lo by adding new generators which are functions of the pseudopotential variables. This provides a new, interesting infinitedimensional algebraic structure, we call “extended algebra,” in which the structure constants are replaced by structure functions depending on the pseudopotentials. Starting with a simple representation of Lo , by resorting to the extended algebra it is possible to build up an original realization of the EW algebra L associated with the equation under study and to find nontrivial symmetries. This procedure enables us to re-obtain and generalize known soliton-like solutions11) for two classical nonlinear evolution equations, which we treated as case studies: the KdV equation and the Dym equation. The paper is organized as follows. Section 2 is devoted to some preliminaries on potential and pseudopotential symmetries. In §§3 and 4 we consider the Dym equation and the KdV equation, respectively. The prolongation algebras allowed by these equations are incomplete (in the sense that not all of the commutators among the generators are known). We apply the algebraic technique described above to obtain the generators of the non-local symmetries (of the pseudopotential type) together
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with examples of interesting solutions of the equations under investigation. Some of these solutions are well known and deriving them does not, of course, represent the main goal of the paper, which is based conversely on a unifying (algebraic) method to find non-local symmetries. Finally, §5 contains some concluding remarks, while in Appendices A and B, details of the calculations are presented. §2.
Formulation of the problem
2.1. Local and non-local symmetries Local symmetries of differential equations are defined by infinitesimal operators which generally are functions of the independent and dependent variables (fields) involved in the equations under consideration. Let us consider the system of nonlinear field equations ∆(x, u) ≡ ut + K(x, t, u, ux , · · · , ux···x ) = 0,
(2.1)
with (x, t) ∈ R2 , u = (u1 , ..., un ) ∈ Rn and ut = Dt (u), ux = D(u), and so on, where Dt and Dx represent the total derivatives with respect to t and x. To find symmetries we use the formalism of evolutionary vector fields,1) which enables us to treat Lie point symmetries, generalized symmetries and non-local symmetries, on the same footing. A generalized (local) symmetry is expressed by a vector field as V =
q
Qα (x, u, ux , uxx , · · ·)∂uα ,
(2.2)
α=1
Qα
of the symmetry depend on derivatives up to order where the characteristics n. The condition for V to generate a symmetry of the equation, i.e. to generate transformations taking solutions into solutions, is pr(n) V [∆] |∆=0 = 0, where
pr(n) V =
DJ Qα ∂uα,J
(2.3) (2.4)
α,J
is an nth order operator (the “nth prolongation”) acting on the nth order jet space that is associated with the manifold M of the dependent and independent variables.1) Here, ∂uα , uα,i = ∂xi and J is a multi-index given by J ≡ (j1 , · · · , jk ),
k ≥ 0,
1 ≤ jk ≤ p.
Equation (2·3) is equivalent to a set of linear equations (the determining equations) for the functions Qα .
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Non-local symmetries are characterized by infinitesimal operators depending also on the global behavior of the fields, expressed for instance by their integrals,1),2) or by variables implicitly defined. In the first case, we have the so-called potential symmetries, while in the second case, non-local symmetries with pseudopotentials. Now let us suppose that the system (2·1) admits conservation laws of the type ∂ i ∂ i F (x, t, u, ux , · · · , ux···x ) = G (x, t, u, ux , · · · , ux···x ). ∂t ∂x
(2.5)
With this in mind, we can introduce a set of potentials y = {y i } such that yxi = F i ,
yti = Gi .
(2.6)
Then, we can consider non-local operators for Eq. (2·1) of the form V = Qα (x, t, u, ux , · · · , ux···x , y i )
∂ . ∂uα
(2.7)
The non-local character of V is due to the fact that it depends on the variables y i , which are defined by quadratures. In contrast to the case of the potential variables, pseudopotential variables 11),13) can be defined by the set of implicit equations yxi = F i (x, t, u, ux , · · · , ux···x , y k ),
yti = Gi (x, t, u, ux , · · · , ux···x , y k ),
(2.8)
where the functional dependences of F i and Gi include the pseudopotential variables also. F i and Gi are explicitly determined by requiring that the compatibility condition i i yxt = ytx (2.9) reproduces the original equation (2·1). Equation (2·9) furnishes the expression for the pseudopotentials in terms of some operators, which satisfy an infinite-dimensional Lie algebra. This is precisely the EW prolongation algebra L associated with Eq. (2·1). The method is outlined below. 2.2. Outline of the procedure—the method of the “extended” algebra The EW technique allows us to associate an integrable nonlinear field equation with an infinite-dimensional Lie algebra which is incomplete. This procedure, which makes use of the theory of exterior differential calculus,19) has been applied to a large variety of equations in order to study their integrability properties and to determine the corresponding spectral problem. For later convenience, we briefly discuss the geometrical construction underlying this approach. Given a system of evolution equations as (2·1), we can introduce in the space of dependent and independent variables a basis of 1-forms, by means of which it is possible to write (2·1) as a set of 2-forms αi . To ensure the equivalence between these forms and (2·1) we must impose the closure condition αi ∧ ηi , (2.10) dαi = i
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where d is the exterior derivative, ∧ denotes the wedge product, and the ηi are elements of the Grassmann algebra of the exterior differential forms αi . Equation (2·10) implies that the exterior differential of each element lies in the ideal itself. Now, let us extend the ideal generated by αi , adding the set of 1-forms ω k = dy k − F k (u, ux, uxx , · · · ; y i )dx − Gk (u, ux, uxx , · · · ; y i )dt
(2.11)
depending on the old variables and on some new variables y k (pseudopotential variables). If we require also that this extended ideal is closed and, therefore, that the compatibility condition (2·9) holds, we get a system of equations involving F k and Gk . In such a manner we obtain yxk = F k =
N
fj (u, ux, · · ·)Akj +
j=1
ytk = Gk =
N
fl,m (u, ux, · · ·)[Akl , Akm ],
(2.12a)
fq,r (u, ux, · · ·)[Akq , Akr ],
(2.12b)
l,m
gj (u, ux, · · ·)Akj +
j=1
q,r
where the Aj are functions depending on the pseudopotential variables only and generating the (incomplete) infinite-dimensional EW prolongation algebra L . The expression [Aki , Akj ] represents the Lie bracket: [Aki , Akj ] = Ali
∂ k ∂ Aj − Alj l Aki . l ∂y ∂y
Hereafter, for simplicity we omit the index k and adopt an operator formalism, defining the operators ∧
∂ F = F l l , ∂y
G = Gl
∂ , ∂y l
∧
Aj = Alj
∂ , ∂y l
and so on. Then, the Lie brackets are transformed into commutators as ∧
∧
[Aj , Ak ] ≡ [Aj , Ak ]l
∂ ,···. ∂y l ∧
Below, to avoid cumbersome expressions, we use F in place of F , etc. Once a representation of L is found, Eqs. (2·12a) and (2·12b) yield the spectral problem associated with the system (2·1) and, in principle, B¨ acklund transformations mapping solutions of (2·1) into new solutions. From the previous considerations, in order to find non-local symmetries with pseudopotentials, it is necessary to unify the Lie-symmetry analysis and the prolongation method in a coherent language. For this purpose, we start from infinitesimal symmetries generated by an infinitesimal operator of the form V = ϕ(u, ux , · · · ; y j )∂u + Λk (u, ux , · · · ; y j )∂yk
(2.13)
for the extended system given by (2·1) and Eqs. (2·12), where ϕ and Λk are assumed to be smooth functions. In the following, we often use Λ in place of Λl ∂yl . For the
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sake of clarity, we limit ourselves to the case of the independent variables u = u (x, t) and y ≡ {y i (x, t)}. The transformations we consider are of the type u = u + ε ϕ(u, ux , · · · ; y), y = y + εΛ(u, ux , · · · ; y),
(2.14a) (2.14b)
where ε is an infinitesimal parameter. If we impose the invariance condition on Eqs. (2·1), (2·12a) and (2·12b), we obtain, formally, relations of the form ϕt = Φ(u, ux , · · · ϕx , ϕxx , · · ·), ux Λu + uxx Λux + · · · + [F, Λ] − (ϕFu + ϕx Fux + · · ·) = 0, ut Λu + uxt Λux + · · · + [G, Λ] − (ϕGu + ϕx Gux + · · ·) = 0,
(2.15a) (2.15b) (2.15c)
ϕt = Dt ϕ = (ut ∂u + uxt ∂ux + · · · + G)ϕ, ϕx = Dx ϕ = (ux ∂u + uxx ∂ux + · · · + F )ϕ,
(2.16a) (2.16b)
where
and so on. The derivatives ut, uxt , uxxt , · · · are not independent and must be calculated with the help of Eq. (2·1). We remark that F ϕ represents the function F k ∂y∂ k ϕ
(analogously for Gϕ), while ϕFu represents the vector field ϕFuk ∂y∂ k . In principle, equating to zero the coefficients of the various powers of u and its derivatives with respect to x, regarded as independent quantities in the nth order jet space, from Eqs. (2·15) we obtain the linear system of determining equations associated with Eq. (2·1). Now, if we substitute into Eqs. (2·15) the expressions for the derivatives of ϕ in terms of the vector fields Aj , taking into account formulas (2·12a) and (2·12b) and the prolongation algebra L, we get an algebraic characterization of the determining equations in terms of the generators of L. We have obtained the following general result: All the possible non-local symmetries of the pseudopotential type for Eq. (2·1) can in principle be calculated as solutions of the overdetermined system obtained from Eqs. (2·15) by equating to zero the coefficients of the various powers of u and its derivatives with respect to x. The crucial problem is now to look for a convenient realization (if such exists) of the prolongation algebra L in such a way that the determining equations cannot be trivially solved. This task can be obviously in different ways tackled. Our approach has the merit of being entirely algebraic. The starting point is to consider a finite-dimensional Lie algebra L0 which “closes” the incomplete EW algebra L, namely a subalgebra L0 ⊂ L in terms of which we can express all the unknown commutation relations among the vector fields Aj generating L as linear combinations of the generators of L0 . Often it turns out that an sl(2, R) algebra is a suitable subalgebra for this purpose. Usually, this procedure is employed to write down the spectral problem (2·8) associated with the equation investigated, once a representation of L0 is known.
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In all the cases we have studied to this time, a finite-dimensional closure L0 furnishes only trivial non-local symmetries. For this reason, we “extend” L0 by adding new vector fields depending on the pseudopotential only. More precisely, if we denote by A0,j the generators of L0 , we formally introduce new vector fields Aj of the form fj,k (y i )A1,k , (2.17) Aj = A0,j + k
where the fj,k (y) are functions only of the pseudopotential y, and the A1,k are new vector fields. We impose the condition that the Aj be the generators of an infinitedimensional Lie subalgebra L ⊂ L (in which the structure constants are replaced by functions of the pseudopotential) in such a way that the relations (2·15) are not trivially satisfied. Surprisingly enough, in the cases presented in this article, the summation contained in (2·17) reduces to at most one term. First of all, we have to build up a representation of L0 . Then, we substitute formulas (2·17) into the algebra L and into the determining equations (2·15). In so doing, we of course obtain an overdetermined set Θ of linear equations involving the functions fj,k (y i ). If we are able to find particular nontrivial solutions of this system, we obtain at the same time a representation of the EW prolongation algebra L and the explicit form of the generator (2·13) of the non-local symmetries associated with Eq. (2·1). By integrating this generator, in principle we can obtain B¨ acklund transformations relating solutions of Eq. (2·1) to other solutions. In the following we consider two case studies concerning the determination of non-local symmetries based on this technique. In order to look for special solutions of the system Θ, we impose some restrictions on the commutation relations of the deformed vector fields Aj . §3.
Non-local symmetries of the Dym equation
The EW prolongation technique applied to the Dym equation ut = u3 uxxx
(3.1)
gives yx = F (u; y) =
A + B, u2
(3.2a)
yt = G(u, ux , uxx ; y) = −2 A uxx − 2 C ux −
2 [A, C] + 2 u [B, C], u
(3.2b)
where A, B and C are vector fields depending on the pseudopotential y = {y i } only, which satisfy the following commutation rules: [A, B] = C, [A, [A, C]] = 0, [B, [B, C]] = 0.
(3.3a) (3.3b) (3.3c)
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Equations (3·3a) – (3·3c) define an incomplete Lie algebra L. In order to investigate the existence of non-local symmetries (of the pseudopotential type) of Eq. (3·1), let us deal with the infinitesimal transformations ∼
u = u + εϕ(u, ux , y),
(3.4a)
y = y + εΛ(u, y),
(3.4b)
∼
where ϕ is a function and Λ is a vector field to be determined and ε is a real parameter. By imposing Eq. (3·4a), Eq. (3·1) yields ϕt = 3u2 ϕuxxx + u3 ϕxxx .
(3.5)
By making Eq. (3·5) explicit, taking into account the prolongation algebra and equating to zero the coefficients of the derivatives of u regarded as independent functions, we obtain (3.6) ϕ = (ux − uB)ϕo (y), where ϕo is a function of y, and the determining equations Aϕo = 0, A2 Bϕo = 0, B 3 ϕo = 0, BCBϕo = 0, 2[A, C]Bϕo + A2 B 2 ϕo + ABABϕo = 0.
(3.7a) (3.7b) (3.7c) (3.7d) (3.7e)
Equations (3·7a) and (3·7b) imply [A, C]ϕo = 0.
(3.8)
We observe that both Λ and ϕ could have functional dependences more complicated than those given here; indeed, the choice expressed by Eqs. (3·4a) and (3·4b) corresponds to a minimal assumption. We note also that if we take ϕ to be independent of ux , only a trivial symmetry arises. Combining (3·4b) and (3·2a) we obtain (3.9) ux Λu + [F, Λ] − ϕFu = 0, that is, 1 2 2 [A, Λ] + [B, Λ] + 3 ux ϕo A − 2 (Bϕo )A = 0. (3.10) 2 u u u Setting the coefficients of the independent functions of u and their partial derivatives appearing in (3·10) to zero, we find ux Λu +
1 ϕo A + Λo , u2 where Λo = Λo (y) is a vector field of integration, and Λ=
[A, ϕo A] = 0, [A, Λo ] + [B, ϕo A] − 2(Bϕo )A = 0, [B, Λ0 ] = 0.
(3.11)
(3.12a) (3.12b) (3.12c)
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85
In a similar manner, starting from (3·2b) and with the help of (3·4b), we obtain u3 uxxx Λu + [G, Λ] −
2 ϕ[A, C] − 2ϕ[B, C] + 2ϕx C + 2ϕxx A = 0. u2
(3.13)
Equation (3·13) can be rendered more explicit to give (ABϕo )A − [C, ϕo A] − ϕo [A, C] = 0, [C, Λo ] + ϕo [B, C] + (B 2 ϕo )A = 0, [[A, C], ϕo A] = 0, [[B, C], Λo ] + (Bϕo )[B, C] − (B 2 ϕo )C = 0, [[B, C], ϕo A] − [[A, C], Λo ] + (Bϕo )[A, C] − A(Bϕo )C −(AB 2 ϕo )A − (BABϕo )A = 0.
(3.14a) (3.14b) (3.14c) (3.14d) (3.14e)
Finally, the infinitesimal generator of the non-local symmetries of the pseudopotential type for the Dym equation is (see (3·6) and (3·11)) VN L = ϕ∂u + Λ = (ux ϕo − uBϕo )∂u + Since
1 A u2
1 ϕo A + Λo . u2
= yx − B from (3·2a), we can write also VN L = −ϕo ∂x − u(Bϕo )∂u − ϕo B + Λo .
(3.15)
The problem of the determination of the non-local symmetries of the Dym equation becomes that of finding suitable realizations of the incomplete prolongation algebra (3·3a) – (3·3c) and the constraints associated with this equation. In this context, we show below that by choosing a finite-dimensional realization of the prolongation algebra (3·3a) – (3·3c) of the sl(2, R) type, we arrive at trivial transformations only. In fact, in our case a possible finite-dimensional subalgebra of the sl(2, R) type of the prolongation algebra L is given by [A, B] = C. [A, C] = −2λA, [B, C] = 2λB,
(3.16a) (3.16b) (3.16c)
where λ is an arbitrary parameter. On the other hand, using (3·12c) and (3·16b), Eq. (3·14d) gives (3.17) 2λ(Bϕo )B = (B 2 ϕo )C, while from (3·7) and (3·14e), we have (2λ(Bϕo ) − (AB 2 ϕo ) − (BA(Bϕo ))A = (ABϕo )C.
(3.18)
Applying (3·19) to ϕo yields (ABϕo )(Cφo ) = 0,
(3.19)
Cϕo = 0.
(3.20)
from which
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As a consequence, the constraint (3·17) provides Bϕo = 0. To conclude, the quotient algebra (3·16a) – (3·16c) leads to the relations Aϕo = 0,
Bϕo = 0,
Cϕo = 0,
(3.21)
which imply that ϕo = const. 3.1. The method of the “extended” algebra We have seen that within a finite-dimensional subalgebra [such as sl(2, R)] of the prolongation algebra L defined by (3·3a) – (3·3c), we have not been able to find nontrivial non-local symmetries. From this reason, we have introduced an infinitedimensional subalgebra of L. As we show below, this approach succeeds. For this purpose, let us introduce the vector fields A = A0 + ϕ0 A1 , B = Bo , C = C0 − (B0 ϕ0 )A1
(3.22) (3.23) (3.24)
into the commutation relations (3·3a) – (3·3c), where [A0 , B0 ] = C0 ,
[A0, C0 ] = −2λA0 ,
[B0 , C0 ] = 2λB0 ,
(3.25)
and A1 denotes a vector field obeying the commutation rules [A1 , A0 ] = 0,
[A1 , B0 ] = 0,
[A1 , C0 ] = 0.
(3.26)
Furthermore, we assume that and
A1 ϕ0 = 0,
(3.27)
A0 ϕ0 = A2o Bo ϕo = Bo3 ϕo = 0.
(3.28)
[We observe that the constraints (3·28) are really special cases of (3·7a), (3·7b) and (3·7c), respectively]. By virtue of (3·22) – (3·28), we can prove directly that the commutators [A, B0 ] = C0 − (B0 ϕ0 )A1 , [A, C] = −2λA0 − {(A0 B0 ϕ0 ) + (C0 ϕ0 )}A1 , [B0 , C] = 2λB0 − (B02 ϕ0 )A1 ,
(3.29a) (3.29b) (3.29c)
realize the prolongation algebra L expressed by (3·3a) – (3·3c). When A1 = 0, we obtain A = A0 , C = C0 , and the commutation rules (3·29a) – (3·29c) reproduce just the sl(2, R) algebra (3·25). In some sense, the algebra defined by Eqs. (3·29a) – (3·29c) plays the role of an “extended” algebra, LE , relative to the sl(2, R) algebra (3·25). Since A1 is multiplied by an arbitrary function, it turns out that LE is an infinite-dimensional subalgebra of L (satisfying all the constraints involved in the theory of non-local symmetries). In the following, we see that the use of LE instead of sl(2, R) enables us to obtain nontrivial non-local symmetries.
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Now let us demand that [Bo , Λo ] = 0,
[Ao ,
Λo ] = 0,
[Co , Λo ] = 0.
(3.30)
A possible bidimensional representation of the algebra (3·25) is Ao = λ∂y1 , Bo = −y12 ∂y1 + y1 ∂y2 , Co = −2λy1 ∂y1 + λ∂y2 .
(3.31a) (3.31b) (3.31c)
From (3·12b) we find Λo ϕo Bo ϕo A1 + Ao + Co , ϕo ϕo
(3.32)
(Λo Bo ϕo ) 2λϕo Bo (Bo2 ϕo )Ao A1 + + . (Bo ϕo ) (Bo ϕo ) (Bo ϕo )
(3.33)
[A1 , Λo ] = while (3·14b) gives [A1 , Λo ] =
Furthermore, from (3·14d) we deduce [A1 , Λo ] =
(Λo Bo2 ϕo ) 2λϕo Bo A1 + − Co . 2 (Bo ϕo ) (Bo2 ϕo )
(3.34)
By comparing (3·32) – (3·34) we have (Λo Bo ϕo ) (Λo ϕo ) (Λo Bo2 ϕo ) = = , ϕo (Bo ϕo ) (Bo2 ϕo ) 2λ(Bo ϕo ) 2λϕo Bo (Bo2 ϕo )Ao (Bo ϕo ) + = Bo − Co . Ao + Co = ϕo (Bo ϕo ) (Bo ϕo ) (Bo2 ϕo )
(3.35) (3.36)
Now we are ready to derive ϕo , A1 and Λo . In so doing, let us suppose that these quantities can be expressed as ϕo = ϕo (y) = ϕo (y1 , y2 , y3 ), A1 = a1 (y)∂y1 + a2 (y)∂y2 + a3 (y)∂y3 , Λo = f1 (y)∂y1 + f2 (y)∂y2 + f3 (y)∂y3 ,
(3.37) (3.38) (3.39)
where aj and fj are functions of y ≡ (y1 , y2 , y3 ) to be determined. From (3·36) we deduce
which yields
Bo ϕo = 2ϕo y1 ,
(3.40)
ϕo = φ(y3 )e2y2 ,
(3.41)
φ(y3 ) being a function of integration depending on y3 only. Then, taking into account (3·40), it turns out that (3·35) and (3·36) are identically satisfied. We are able to demonstrate that
88
M. Leo, R. A. Leo, G. Soliani and P. Tempesta i) a1 = f1 = 0. ii) a2 , a3 , f2 and f3 depend on y3 only. iii) The following relations hold: (3.42)
2a2 φ + a3 (y)φy3 = 0, φy3 + λ = 0, φ φy + a3y3 f3 + 2f2 a3 + f3 a3 3 = 0. φ
f3 a2y3 − a3 f2y3 + 2f2 a2 + f3 a2
(3.43)
−a3 f3y3
(3.44)
Equations (3·42) – (3·44) represent an overdetermined system with unknowns a2 , a3 , f2 , f3 and φ. With these quantities we can determine the function ϕo and the operators A1 and Λo . Then, the generator (3·15) takes the form VN L = −φ(y3 )e2y2 ∂x − 2uφ(y3 )y1 e2y2 ∂u + φ(y3 )e2y2 y12 ∂y1 +[f2 (y3 ) − y1 φ(y3 )e2y2 ]∂y2 + f3 (y3 )∂y3 .
(3.45)
Thus, the corresponding group transformations arise from the differential equations ∼
dx dε ∼ du dε ∼ d y1 dε ∼ d y2 dε ∼ d y3 dε
∼
∼
y
= −φ( y 3 )e2 2 , ∼
∼
(3.46a) ∼
∼
y
= −2 u φ( y 3 ) y1 e2 2 , ∼2
∼
(3.46b)
∼
y = φ( y 3 ) y 1 e2 2 , ∼
∼
∼
(3.46c) ∼
y = −φ( y 3 ) y 1 e2 2 + f2 ( y 3 ), ∼
(3.46d) (3.46e)
= f3 ( y 3 ),
∼
where t = t, ε is the group parameter, and the boundary conditions ∼
∼
∼
∼
∼
x|ε=0 = x, u|ε=0 = u, y1 |ε=0 = y1 , y2 |ε=0 = y2 , y3 |ε=0 = y3 ,
(3.47)
are considered. Now, in order to illustrate how our method works, let us consider the trivial solution u = −1 to Eq. (3·1). Consequently, the prolongation equations (3·2a) and (3·2b) provide y1x y1t y2x y2t y3x y3t
= λ − y12 , = −4λ2 + 4λy12 , = y1 + a2 φ(y3 )e2y2 , = −8λa2 φ(y3 )e2y2 − 4λy1 + 4a2 y12 φ(y3 )e2y2 , = a3 φ(y3 )e2y2 , = −8λa3 φ(y3 )e2y2 + 4a3 y12 φ(y3 )e2y2 ,
(3.48a) (3.48b) (3.48c) (3.48d) (3.48e) (3.48f)
On the Relation between Lie Symmetries and Prolongation Structures
89
where y1 , y2 and y3 have not been regarded as vector fields, but functions of (x, t). After some manipulations, from Eqs. (3·46a) – (3·46e) and (3·48a) – (3·48f) we obtain y2 u = u ∼12 , y1
∼
(3.49)
where y1 can be derived by solving the pair of Riccati equations (3·48a) and (3·48b). By choosing, for example λ > 0, we obtain √
√
√ e λξ − ae− λξ √ , y1 = λ √ e λξ + ae− λξ
(3.50)
where a is a constant of integration, and ξ = x − 4λt. By scrutinizing the remaining equations involving the prolongation variables, (3·48c) – (3·48f), we have ∼ y1
= y1
√ √ εχ λ 2 λξ 2√ (e √ εχ λ 2 λξ 2 (e
1 − 2εχλa(x − 12λt) + 1 − 2εχλa(x − 12λt) −
− a2 e−2 − a2 e−2
√ λξ )
(3.51)
. √ λξ )
In order to find interesting solutions to Eq. (3·1) (starting from the trivial solution u = −1), we have to use Eq. (3·49) and write the variables x and t in terms of ∼ ∼ x and t. In so doing, since t = t , it is sufficient to consider Eqs. (3·46a) and (3·46c). These yield ∼
x =x+
1 ∼ y1
−
1 . y1
(3.52)
∼
Then, by using (3·50) and (3·51), we have (t = t ) εχ(e
∼
x =x−
√ λξ
+ ae−
1 − 2εχλa(x − 12λt) +
√ λξ )2
√ √ εχ λ 2 λξ 2 (e
− a2 e−2 ∼
(3.53)
. √ λξ ) ∼
∼ ∼
From (3·53), we can formally derive x as a function of x and t : x = τ (x, t ). Thus, from Eq. (3·49) (with u = −1) we obtain ∼
u=−
∼ ∼ 1 − 2εχλa[τ (∼ x, t ) −12λ t ] −
∼ ∼
∼
1 − 2εχλa[τ (x, t ) −12λ t ] +
∼ ∼
√ √ ∼ εχ λ 2 λ ξ (e 2 √ √ ∼ εχ λ 2 λ ξ (e 2
− a2 e−2 −
√ ∼ 2 λξ )
√ ∼ a2 e−2 λ ξ )
,
(3.54)
∼
where ξ = τ (x, t ) − 4λ t . We remark that for a = 0, χ = −1 and ε < 0, Eq. (3·54) produces the (formal) solitary wave solution √ ∼ u = sec h2 λ( ξ +δ) − 1,
∼
√
√
(3.55)
where δ is a constant defined by |ε|2 λ = e2 λδ . This solution corresponds to that found in Ref. 11). By choosing λ < 0 in the Riccati equations (3·48a) and (3·48b),
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M. Leo, R. A. Leo, G. Soliani and P. Tempesta
a procedure similar to that employed to derive formula (3·54) leads to the solution (A·3) given in Appendix A. To conclude this section, we observe that the solutions (3·54) and (A·3) are a consequence of the choice u = −1 (i.e. the trivial solution of the Dym equation) in the B¨acklund transformation (3·49). Of course, in principle, other choices should be possible and, correspondingly, other solutions should be derivable. §4.
Non-local symmetries of the Korteweg-de Vries equation
Another interesting case which constitutes a good context for checking the validity of our approach to non-local symmetries with pseudopotentials is given by the KdV equation (4.1) ut + 6uux + uxxx = 0. The pseudopotential for Eq. (4·1) is defined by 1 (4.2a) yx = F = u2 A + uB + C,
2 1 1 yt = G = −uuxx + u2x − 2u3 A − (uxx + 3u2 )B − u2 [B, D] 2 2 (4.2b) −u[C, D] + ux D + E, where A, B, C, D and E are vector fields (depending on y ≡ {y i } only) satisfying the commutation relations [A, B] = [A, C] = [A, D] = 0, [A, [B, D]] = 0, [B, [B, D]] = [A, [D, C]] = 0, [A, E] = 3[C, [B, D]] + 6D, [B, E] = [C, [C, D]], [C, E] = 0, [C, [B, D]] = [B, [C, D]], D = [C, B].
(4.3a) (4.3b) (4.3c) (4.3d) (4.3e) (4.3f) (4.3g) (4.3h)
The prolongation algebra (4·3a) – (4·3h) is incomplete. Now let us carry out the infinitesimal transformations ∼
u = u + εϕ(u, y), ∼
(4.4a)
y = y + εΛ(u, y).
(4.4b)
ϕt + 6uϕx + 6ϕux + ϕxxx = 0.
(4.5)
Then, Eq. (4·1) provides
Equation (4·5) can be used to obtain the following set of constraints: Aϕ = 0,
(4.6a)
On the Relation between Lie Symmetries and Prolongation Structures B 2 ϕ = 0, CBϕ = −2ϕ, 1 −3Bϕ − [B, D]ϕ + 6Bϕ + B 2 Cϕ + BCBϕ = 0, 2 −[C, D]ϕ + 6Cϕ + BC 2 ϕ + CBCϕ + C 2 Bϕ = 0, Eϕ + C 3 ϕ = 0,
91
(4.6b) (4.6c) (4.6d) (4.6e) (4.6f)
where ϕ = ϕ(y) (i.e. ϕ depends on the pseudopotential only). By virtue of (4·4b), Eq. (4·2a) yields ux Λu + [F, Λ] − ϕFu = 0.
(4.7)
Equation (4·7) tells us that Λu = 0, i.e. Λ = Λ(y). Moreover, the conditions [B, Λ] = 0, [C, Λ] − ϕB = 0, [D, Λ] + (Bϕ)B = 0, [[B, D], Λ] = 0, −[[C, D], Λ] + 6ϕB + ϕ[B, D] − (Bϕ)D + (BCϕ)B + (CBϕ)B = 0, [E, Λ] + ϕ[C, D] − (Cϕ)D + (C 2 ϕ)B = 0
(4.8a) (4.8b) (4.8c) (4.8d) (4.8e) (4.8f)
hold. Then, our task is to find the infinitesimal generator of the non-local symmetries for Eq. (4·1), which reads (4.9) VN L = ϕ∂u + Λ. For this purpose, first we look for a finite-dimensional realization of the prolongation algebra (4·3a) – (4·3h). Assuming A = 0, a representation of this kind is [Co , Bo ] = Do , [Bo , Do ] = −2Bo , [Co , Do ] = 4λBo + 2Co , Eo = −4λCo ,
(4.10a) (4.10b) (4.10c) (4.10d)
where λ is an arbitrary parameter. 4.1. The method of the “extended algebra” for the Korteweg-deVries equation We have seen that limiting ourselves to starting from the closed algebra (4·10a) – (4·10d), namely taking A = 0, B = Bo , C = Co and E = Eo , only trivial non-local symmetries arise. As a possible way to remedy this problem, we employ a procedure similar to that applied in §3 for the case of the Dym equation. In other words, we search for a realization of the prolongation algebra (4·3a) – (4·3h) such that A = 0, and
B = Bo ,
(4.11)
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M. Leo, R. A. Leo, G. Soliani and P. Tempesta
C = Co + µC1 ,
(4.12)
where µ = µ(y) is a function of the pseudopotential to be determined, Bo , Co and Do satisfy the relations (4·10a) – (4·10c), and the operator C1 = 0 is assumed to obey the commutation rules [C1 , Bo ] = [C1 , Do ] = [C1 , Co ] = 0.
(4.13)
For simplicity, we take E1 = C1 . Now, let us introduce the operator E = Eo + νE1 ,
(4.14)
where Eo is given by (4·10d) and ν is a function depending on the pseudopotential. Let us assume also that the relations [E1 , Bo ] = [E1 , Co ] = [E1 , Do ] = 0,
(4.15)
E1 ϕ = E1 µ = 0,
(4.16)
are valid. When employing the commutation rule (4·8a), it is natural to set Λ = γµBo + Λ1 ,
(4.17)
where γ is a constant, and the operator Λ1 is chosen in such a way that [Λ1 , Bo ] = [Λ1 , Co ] = [Λ1 , Do ] = 0.
(4.18)
In order to obtain the infinitesimal generator of the non-local symmetries, we need to know the field ϕ and the operator Λ (see (4·9)). With this aim, first we exploit a specific realization of the algebra (4·10a) – (4.10c). Precisely, let us consider Bo = −∂y1 , Co = (λ − y12 )∂y1 + y1 ∂y2 , Do = −2y1 ∂y1 + ∂y2 .
(4.19a) (4.19b) (4.19c)
Second, we make the hypothesis that the field ϕ(y) and the functions µ(y) and ν(y) depend on a pseudopotential vector having at least three components, say y ≡ (y1 , y2 , y3 ). Consequently, in this context it is reasonable to suppose that the operators C1 and Λ1 , appearing in (4·12) and (4·17), respectively, take the form C1 = f1 (y)∂y1 + f2 (y)∂y2 + f3 (y)∂y3 ,
(4.20)
Λ1 = X1 (y)∂y1 + X2 (y)∂y2 + X3 (y)∂y3 ,
(4.21)
and where fj (y) and X(y) are functions to be determined. Starting from the prolongation algebra (4·3a) – (4·3h), and keeping in mind the previous assumptions and the algebraic constraints (4·6a) – (4·6f) and (4·8a) –
On the Relation between Lie Symmetries and Prolongation Structures
93
(4·8f), in analogy with the case of the Dym equation, we can deduce another set of constraints which we omit for brevity. By using all these relations, after some manipulations, we obtain the following results: µ = µ1 (y3 )e2y2 , ϕ = −αµ1 (y3 )y1 e2y2 , ν = 4µ1 (y3 )y12 e2y2 + ν1 (y2 , y3 ), C1 = f2 (y3 )∂y2 + f3 (y3 )∂y3 , Λ1 = X2 (y3 )∂y2 + X3 (y3 )∂y3 ,
(4.22) (4.23) (4.24) (4.25) (4.26)
where the functions µ1 = µ1 (y3 ), ν1 = ν1 (y2 , y3 ), f2 = f2 (y3 ), f3 = f3 (y3 ), X2 = X2 (y3 ) and X3 = X3 (y3 ) satisfy the system of linear differential equations 2f2 µ1 + f3 µ1y3 = 0, α 1 −f2y3 X3 + f3 X2y3 = + 2f2 X2 + f2 X3 µ1y3 , 4 µ1 1 f3 X3y3 − f3y3 X3 = f3 X3 µ1y3 + 2f3 X2 , µ1 1 X2 ν1y2 + X3 ν1y3 = 2X2 ν1 + X3 ν1 µ1y3 , µ1 2y2 16λµ1 e + ν1y2 = 0, f2 ν1y2 + f3 ν1y3 = 0.
(4.27a) (4.27b) (4.27c) (4.27d) (4.27e) (4.27f)
This system can be handled by means of the procedure used for the Dym equation. In applying this procedure, let us make the infinitesimal generator (4·9) explicit, employing the form VN L = −αy1 µ1 (y3 )e2y2 ∂u +
α µ1 (y3 )e2y2 ∂y1 + X2 (y3 )∂y2 + X3 (y3 )∂y3 . 4
(4.28)
Then, from the the group transformations corresponding to the operator (4·28), we easily find ∼
∼2
u = −2( y 1 −y12 ) + u.
(4.29)
In our scheme, if we choose, for instance u = 0, the prolongation equations (4·2a) and (4·2b) become y1x y1t y2x y2t y3x y3t
= λ − y12 , = −4λ2 + 4λy12 , = y1 + f2 (y3 )µ1 (y3 )e2y2 , = −4λy1 + (4y12 − 8λ)f2 (y3 )µ1 (y3 )e2y2 , = f3 (y3 )µ1 (y3 )e2y2 , = (4y12 − 8λ)f3 (y3 )µ1 (y3 )e2y2 .
(4.30a) (4.30b) (4.30c) (4.30d) (4.30e) (4.30f)
94
M. Leo, R. A. Leo, G. Soliani and P. Tempesta
Then, for λ > 0, from Eqs. (4·30a) and (4·30b) we obtain √
√
√ e λξ − ae− λξ √ , y1 = λ √ e λξ + ae− λξ
(4.31)
where ξ = x − 4λt and a is a constant. By integrating Eqs. (4·30c) – (4·30f), from Eq. (4·29) we obtain (for u = 0) √ √ √ √ √ (e λξ − ae− λξ ) 2λ(e λξ − ae− λξ )2 ∼ √ √ √ u= −2 λ √ (e λξ + ae− λξ )2 (e λξ + ae− λξ )
2 √ √ (e λξ + ae− λξ )2 √ √ +αε . (4.32) 4 − εα[2a(x − 12λt) + 2√1 λ (e2 λξ − a2 e−2 λξ )] ∼
∼
We note that since x= x, t = t, this solution is explicit and contains the well-known soliton solution11)
√ |ε| 1 ∼ u = 2λ sec h2 √ , (4.33) λξ + ln 2 8 λ which emerges for a = 0 and α = −1. Another solution to the KdV equation can be obtained for the choice λ < 0. This is given in Appendix B. §5.
Conclusions
We have developed a procedure to obtain non-local symmetries of the pseudopotential type of nonlinear field equations whose prolongation algebra L is incomplete. We have applied this procedure to two equations: the Dym equation and the KdV equation. For both equations, first we have found a finite-dimensional subalgebra (quotient algebra) L0 of the related (incomplete) prolongation algebra L. Then, we have attempted to use L0 to look for non-local symmetries. Unfortunately, through L0 only trivial symmetries emerge. For this reason, we have “extended” the subalgebra L0 by introducing new operators to be determined by the requirement that the commutation relations defining L and the constraints involved in the infinitesimal transformations for the non-local symmetries are satisfied. The determination of these new operators is crucial, since they appear in the generator of the non-local symmetries. For the two equations under investigation our approach is successful. From the generator of the non-local symmetries, expressed in terms of pseudopotential variables, one can write the corresponding group transformations which enable us to obtain exact solutions of the Dym and the KdV equation. Some of these solutions are well known. Notwithstanding, they serve as paradigms to probe and to illustrate the potentiality of our algebraic approach. As we can argue on the basis of the results achieved for the above-mentioned applications, our method could be exploited to treat other nonlinear field equations admitting nontrivial prolongations. However, some aspects of the method remain to
On the Relation between Lie Symmetries and Prolongation Structures
95
be elucidated, and only the consideration of a number of additional examples could allow us to see the appropriate method of its implementation. To be precise, for instance we remark that a basic role is played by the realization of the “extended” algebra in terms of vector fields depending on pseudopotential variables. In our calculations, we have chosen simple but nonlinear realizations (of the polynomial type). Different algebraic realizations (say, polynomial realizations in higher dimensions or realizations which are not of the polynomial type) might produce different non-local symmetries and, correspondingly, different solutions to the original equations. In any case, this possibility should be explored. Another interesting approach which deserves to be considered is the use of an infinite-dimensional realization (of the prolongation algebra L) of the Kac-Moody type. This kind of realization may have a significant role in the search for non-local symmetries of the pseudopotential type. Finally, we point out that an important problem is to obtain an extension of our approach to nonlinear field equations in more than 1+1 dimensions. However, due to the fact that at present only a few applications in higher dimensions have been carried out within the prolongation scheme,20),21) this program is still far being completed. Acknowledgements Support from MURST of Italy and INFN-Sezione di Lecce is gratefully acknowledged. Appendix A Here we sketch the calculation used to solve the Dym equation (3·1) corresponding to the choice λ < 0 in the Riccati equations (3·48a) and (3·48b). These give y1 =
|λ|
cos sin
| λ |θ − b sin
| λ |θ + b cos
| λ |θ | λ |θ
,
(A.1)
where θ = x + 4 | λ | t and b is a constant. This formula is the analogous to (3·50). Carrying out the same type of calculations leading to (3·53), we obtain the transformation x = x − [εχ(sin | λ |θ + b cos | λ |θ)2 ] 2 − 1) sin 2 | λ |θ − 2b cos 2 | λ |θ (b εχ | λ | 2(b2 + 1)(x + 12 | λ | t) + × 1+ 4 |λ|
−1 , (A.2) + εχ | λ | (cos | λ |θ − b sin | λ |θ)(sin | λ |θ + b cos | λ |θ)
∼
where χ = ±1. ∼ ∼ With x = γ(x, t ) representing a formal expression of the inverse of the expression
96
M. Leo, R. A. Leo, G. Soliani and P. Tempesta
(A·2), keeping in mind (A·1) from (3·49), we have ∼ εχ | λ | ∼ ∼ ∼ u=− 1+ 2(b2 + 1)(γ(x, t ) + 12 | λ | t ) 4 ∼ ∼ (b2 − 1) sin 2 | λ | θ −2b cos 2 | λ | θ + |λ| ∼ εχ | λ | ∼ ∼ × 1+ 2(b2 + 1)(γ(x, t ) + 12 | λ | t ) 4 ∼ ∼ (b2 − 1) sin 2 | λ | θ −2b cos 2 | λ | θ + |λ| +εχ | λ | (cos
∼
| λ | θ −b sin
∼
| λ | θ )(sin
∼
| λ | θ +b cos
∼
| λ | θ)
−1 , (A.3)
∼
∼ ∼
∼
where θ = γ(x, t ) + 4 | λ | t . Appendix B By choosing λ < 0, Eqs. (4·30a) and (4·30b) give rise to the solution expressed by (A·1). Following the same procedure adopted to find the solution (4·33), we obtain ∼ u = [−4εα | λ |(cos | λ |θ − b sin | λ |θ)(sin | λ |θ + b cos | λ |θ)] (b2 − 1) sin 2 | λ |θ − 2b cos 2 | λ |θ εα 2 2(1 + b )θ + × 4− 4 |λ| −1 − 2ε2 α2 (sin | λ |θ + b cos | λ |θ)4 +16 | λ |(1 + b2 )t (b2 − 1) sin 2 | λ |θ − 2b cos 2 | λ |θ εα 2 2(1 + b )(x + 4 | λ | t) + × 4− 4 |λ| −2 +16 | λ |(1 + b2 )t , (B.1)
with θ = x + 4 | λ | t, α ∈ R. References 1) J. Olver, Applications of Lie groups to differential equations (Springer-Verlag, New York, 1986). 2) G. W. Bluman and S. Kumei, Symmetries and differential equations (Springer-Verlag, New York, 1989). 3) CRC Handbook of Lie group analysis of differential equations, ed. N. H. Ibragimov (CRC press, New York, 1996).
On the Relation between Lie Symmetries and Prolongation Structures 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
97
G. W. Bluman and G. J. Reid, J. Math. Phys. 29 (1988), 806. B. G. Konopeltchenko and V. G. Mokhnachev, J. of Phys. A13 (1980), 3113. O. V. Kapkov, Sov. Math. Dokl. 25 (1982), 173. V. V. Pukhnachev, Sov. Math. Dokl. 35 (1987), 555. G. A. Guthrie and M. Hickman, J. Math. Phys. 34 (1993), 193. D. G. B. Edelen, Isovector methods for equations of balance (Alphen ann den Rijn: Sijthoff and Noordhoff, 1980). I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math. 15 (1989), 161. F. Galas, J. of Phys. A25 (1992), L981. G. N. Bluman and J. D. Cole, J. Math. Mech. 18 (1969), 1025. H. D. Wahlquist and F. B. Estabrook, J. Math. Phys. 16 (1975), 1. F. B. Estabrook and H. D. Wahlquist, J. Math. Phys. 17 (1976), 1293. E. Pucci and G. Saccomandi, J. of Phys. A26 (1993), 681. G. Saccomandi, J. of Phys. A30 (1997), 2211. M. Senthilvelan and M Torrisi, J. of Phys. A33 (2000), 405. J. Schiff, e-Preprint, solv-int/9606004. W. M. Sluis and P. H. M. Kersten, J. of Phys. A23 (1990), 2195. E. Cartan, Les Syst` emes Diff´erentiels Ext´erieurs et leurs Applications G´eometriques (Herman, Paris, 1946). H. C. Morris, J. Math. Phys. 17 (1976), 1870. M. Palese, R. A. Leo and G. Soliani, “The prolongation problem for the 3D Toda field equation,” Quaderni del Dipartimento di Matematica Universit` a di Torino, n. 23 and references therein (1998).
