On recursion operators
Introduction
If we look at the historical development of the theory of integrable systems (with the KdV equation as the most famous example), we see that equations and symmetries which did not explicitly depend on the time or space variables, were first studied. Based on this assumption a theoretical framework was developed. When later on examples were found for both equations and symmetries with explicit time or space dependency, the theoretical framework was slightly adapted to cover the new situation, but by that time the way things were calculated was already somewhat fixed, and these new features were not analyzed in any fundamental way.
In this paper we first describe some of the difficulties one has when one applies the results in the literature to explicitly time-dependent situations. These difficulties came as quite a surprise to us, since they appear at such an elementary level, but they seem to be common to problems related to nonlocal variables (for details see [9]). We then proceed to lay the foundations of a recursion operator formalism in the t-dependent case.
The goal of this paper is not to compute the symmetries of time-dependent equations; for instance one can derive the symmetries of cKdV from those of KdV using the transformation given in [4]. The point is that we want foolproof algorithms so that we can compute symmetries, given any equation and its recursion operator.
To illustrate the problem we consider the well-known (cf. [8, p. 315]) Burgers equationwhich has a time-dependent recursion operatorApplying this operator repeatedly to a symmetry like 1+tu1 is supposed to produce a hierarchy of symmetries. There is, however, one problem with this elementary fact: while is indeed a symmetry, is not! Let LK denote the Lie derivative. If Q is an evolutionary vectorfield, LKQ can be defined asand we say that Q is a symmetry if LKQ=0. Since by definition, , a sufficient condition for an operator to map a symmetry to a new symmetry is that . Thus one often defines a recursion operator by the property that . It is easily checked that the above operator satisfies this condition for the Burgers equation. This seems a fairly paradoxical situation until one realizes how the proof that works: in the computation one uses the relation Dx−1Dx=1. But this relation is obviously not true,1 since it is wrong on , e.g., Dx−1Dxt=0≠t. Instead, we should read Dx−1Dx=1−Π, where Π is the projection on (for example Π(t+t2u1)=t).
Let us now compute for the Burgers equation. We find thatThe combination ΠDx is only nonzero on symmetries that contain a term of the form g(t)x. If we now compute , we see that it equalsand there is indeed a term of the form g(t)x, namely x. This explains why is not a symmetry.
Having located the source of the problem, the next question is: how to adapt the definition of recursion operator so that it will also handle this case correctly? After giving the necessary definitions in Section 2, we answer this question in Section 3. In Section 4 we explicitly compute the u-independent term of all the symmetries of the cKdV equation generated by the corrected recursion operator.
Section snippets
Recursion operators
In order to understand the behavior of recursion operators, it helps to have a good formal setup so that we know clearly in which spaces the objects we study live. Our approach is based on [3], [5] and is described in detail in [10]. We only use one variable u to keep the notation simple, the generalization to systems being fairly obvious.
Let be the space of C∞-functions of t and the space of local functions, that is, formal power series x,u,u1,…uk, for some k<∞ with coefficients in . We
The corrected recursion operator
We call an operator a weak recursion operator of the equation ut=K0 if it satisfies using the rule Dx−1Dx=1. In this section, we give the corrected recursion operator based on a weak recursion operator . Definition 2 We define a projection as follows. Given any f(t,x,u,…,uk) we put (Πf)(t)=f(t,0,0,…,0). Definition 3 We denote all the terms with the projection operator Π in an operator · by [[·]]. For example, we have [[Dx−1Dx]]=−Π. Lemma 4 For any local function P∈imDx and any
More details on symmetries of cKdV
In this section we want to derive the u-independent part of all symmetries. If by K00 we denote the linear part of the equation ut=K0, and by f the u-independent part of the symmetry, then f obeysSo for cKdV this implies thatWe find that if we try the homogeneous expression , j=0,1,2, this gives us the recursion relationThis results in a divergent sequence
Concluding remarks
We have seen the theory of recursion operators is not quite as well settled as we may have believed, but we have shown how the correction can be obtained by a combination of abstract methods and concrete calculations. Although the time-dependent symmetries and recursion operators are not all that common, this has clarified the way things should be defined.
Acknowledgements
J.P. Wang gratefully acknowledges the support from Netherlands Organization for Scientific Research (NWO) for this research.
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