On the Integrability of Scalar Evolution Equations

This paper was motivated by the observation that after quickly finding a number of hierarchies (mKdV, Sawada-Kotera, Kaup-Kuperschmidt) soon after finding that KdV was integrable, nothing more was found for polynomial scalar evolution equations linear in the highest order derivative. In this paper we prove that under some mild conditions on the equations one can put a uniform bound on the order of the recursion operator of any such hierarchy. We do this using the symbolic method, introduced by Gel’fand-Dikii [GD75]. This method was used in [TQ81] and [QT82] to produce the following results. give results The basic idea is very old, probably dating from the time when the position of index and power were not as fixed as they are today. In fact, the symbolic calculus of classical invariant theory relies on it. The idea is simply to replace ui, where i is an index, in our case counting the number of derivatives, by ξ, where ξ is now a symbol. We see that the basic operation of differentation, i.e. replacing ui by ui+1, is now replaced by multiplation with ξ, as is the case in Fourier transformation theory. If one has multiple u’s, as in uiuj , one replaces this by 1 2 ( ξ 1ξ j 2 + ξ j 1ξ i 2 ) . We have averaged over the permutation group Σ2 to retain complete equality among the symbols, reflecting the fact that uiuj = ujui. Differentiation now becomes multiplication with ξ1 + ξ2. With this method one can readily translate solvability questions into divisibility questions and we can use generating functions to handle infinitely many orders at once.