Prolongation structure of the Kdv equation in the bilinear form of Hirota

The prolongation structure of the KdV equation in the bilinear form of Hirota is determined, the resulting Lie algebra is realised and the Backlund transformation obtained from the prolongation structure is derived. The results are compared with those found by Wahlquist and Estabrook and by Hirota.


Introduction
In their well known article [l] Wahlquist and Estabrook developed a prolongation technique for nonlinear evolution equations and applied it to the Korteweg-de Vries (KdV) equation.This resulted in a Lie algebra structure, which has been obtained independently by Estabrook [2] and van Eck [3], the latter merely using the algebraic foundations of this Lie algebra.
In [4] Hirota shows how nonlinear evolution equations may be transformed into bilinear differential equations.For this purpose it is convenient to write the KdV equation in potential form From equation (1) we obtain the bilinear equation where T ( x + y , t + t')T (xy , tt') l y=O, r , =O ( In this paper we will investigate the prolongation structure of the KdV equation in Hirota's bilinear form (2). Equation (2) is easily seen to admit an additional symmetry with group action ( x , t , T ) H ( x , t, A T ) which has no counterpart for the ordinary KdV.
This investigation is motivated by the conjecture that this additional symmetry might be reflected in the prolongation structure.Moreover, equation ( 2) is directly connected to the T -function approach of nonlinear evolution equations [5].In a certain sense these T functions are more primitive than the corresponding functions U, namely as an element of a group orbit T stands closer to the underlying symmetry group than U itself.This might also be reflected in the prolongation structure.
In section 2 we carry out the actual prolongation, after which in section 3 we obtain the resulting Lie algebra, in the sense that we give a faithful representation of it.Finally, in section 4 we derive the Backlund transformation obtained from the prolongation structure, which appears to be equal to the one found by Hirota.

Writing out equation (2) yields
In order to express equation ( 4) by means of an ideal of differential forms we introduce whereupon equation ( 4) may be written as the first-order equation The set of first-order equations ( 5) and (6) may now be expressed by the following set of differential 2-forms where a solution manifold of equations ( 5) and (6) satisfies the condition and vice versa.In order to ensure integrability of this system Cartan's theory of differential forms [6] imposes the condition d l c I on I = I ( a l , ..., g4), the ideal generated by the ai in the exterior algebra with basis {dx, dt, d r , dp, dq, dr} ; this condition is easily seen to be satisfied.
The prolongation method of Wahlquist and Estabrook may be described as follows: add a Lie algebra valued prolongation form to the original system of differential forms and demand w satisfies the condition do + [G, F ] dx A dt E Z(a,, . . ., u4).

(8)
Condition (8) gives rise to the following system of overdetermined differential equations T ~G , Using the method of characteristics, one derives from equations ( 9), ( 11) and ( 12) that The remaining equations were solved using a computer package for computations in prolongation theory described in [7].If we define we find that where we have introduced In addition we find the commutation relations In the next section we will realise the Lie algebra determined by relations (17) and (18).
Remark.From definition (14) we see that whereupon equation (6) may be written as follows It is easily checked that this set of equations leads to the same prolongation structure as derived for equations ( 5 ) and (6).We see from this that both the KdV in Hirota's bilinear form and the prolongation structure belonging to it, are really determined by T , / t = (logs), = U and not by T itself.In fact, as we will see in section 5, the results found here are the same as for the KdV equation in potential form.Equations ( 19) and (20), however, will be useful in section 4, where we derive the Backlund transformation belonging to the prolongation structure.

Realisation of the prolongation algebra
In this section we will determine the structure of the prolongation algebra of the KdV in Hirota's bilinear form and give a realisation of it, using the theory of Kac-Moody algebras.Henceforth we will denote this prolongation algebra, i.e. the Lie algebra generated by the letters x l r . . .,x, and subjected to relations (17) and (Ill), by HIR.An important tool for our purpose is a gradation.The following proposition is immediate.Using the computer package already mentioned and introducing new letters x8, ..., x,, we have computed (part of) the commutator table of HIR, which can be found in table 1 and in principle may be checked by hand using the Jacobi identity.Using this table and the gradation of proposition 1 we find that HIR contains a subalgebra isomorphic to A , = s1(2), with standard basis { e , f , h } , where and standard relations Moreover we see that HIR contains a two-dimensional Abelian subalgebra commuting with all generators of HIR, hence consisting of central elements.
We recall from van Eck To prove this we will need some results from the theory of Kac-Moody algebras, in particular the defining relations and the realisation of the infinite-dimensional Lie algebra Ai')', i.e. the derived algebra of the infinite-dimensional Lie algebra Ai') is].
We have the following.
is given by (b) Ai"' is isomorphic to the algebra (CC[i.-',A] €3 A , ) @ C c where the isomorphism and the commutator on (CC[,i-I, A] €3 A , ) @ C c by with (.I.) an invariant bilinear symmetric form on A , with Equating to zero fo and h, yields the following.Proposition 3. The Lie algebra with generators eo, e , , f , , h, and defining relations is isomorphic to the algebra C[1.] €3 A , where the isomorphism is given by We are now in a position to give a realisation of HIR.

Backlund transformation
Backlund transformations can be obtained from prolongation structures by realising the prolongation algebra in a appropiate way and putting In this way new solutions for the differential equation for which the prolongation structure was computed may be constructed [l].
Using the nonlinear representation of sl (2) e = -y f = 1 h = 2 y we find for F and G which leads to the equations The compatibility condition y,, = y,, is satisfied if P , Q and R are solutions of the KdV in Hirota's bilinear form (( 19) and (20)).Solving P from equation (27) and substituting the result into equation (28) we find that y has to satisfy On the other hand we can take the realisation of sl (2) yielding In the same way as before we find that v has to satisfy (32) Now a simple observation shows that U = -y transforms equation (32) into equation (29).Differentiating with respect to x gives U , = -y, from which we conclude that where P is another solution of equations ( 19) and (20).From this we find that r' = P or E? = y -P .One easily checks by substitution that r?. = y -P indeed satisfies equations

Conclusions
If we take a closer look at equation (20), which is equivalent to the equation (6) for T , and use equation ( 19) to eliminate Q and R, we find that P has to satisfy 4P, = P,,, + 6P2 which is the KdV equation in potential form (1). Now the potential form of the KdV equation has already been discussed from the prolongation point of view by Kaup [9].In fact, the present equations (15) and ( 16) are the same as equation ( 5 ) in Kaup's paper, but in different notation.Also the realisation that (30) and (31) are the same as Kaup's equation (12).However, in Kaup's paper the explicit form of the prolongation algebra has not been determined.Hence we have found in a natural context that from the prolongation point of view Hirota's formalism is in fact nothing more than a way to obtain results, which could have been obtained without applying Hirota's formalism.As a consequence of this,

[ 3 ]
that the prolongation algebra of the ordinary Kdv is isomorphic to H , , x (cC[i.]&I AI), the direct product of a five-dimensional Heisenberg algebra H,, and the tensor product cC[E.]63 A , .HIR will appear to have a similar structure, namely HHir x (C[E.]&I A I ) , with HHir and A , as defined in (21) and (22).

(
19) and (20) iff P does, hence we have found the auto-Backlund transformation -P = y -P .(33)Differentiatingequation (33) with respect to x and using equation (27) to eliminate y , we obtain the same Backlund transformation in a more familiar form which is the same as the one found by Wahlquist and Estabrook [i] and by Hirota[4].