One Symmetry Does Not Imply Integrability

Abstract

We show that that Bakirov's counterexample which had been checked by computeralgebra methods to order 53 to the conjecture that one nontrivial symmetry of an evolution equation implies in nitely many is indeed a counterexample. To prove this we use the symbolic method of Gel'fand-Dikii and p-adic analysis. W e also formulate a conjecture to the e ect that almost all equations in the family considered by Bakirov h a ve at most nitely many symmetries. This conjecture depends on the solution of a diophantine problem, which w e explicitly state.

Symbolic representation and classification of integrable systems

“…In [7] it has shown that example given in [6] is indeed a counterexample to the conjecture (see also [9]). Even a rectified conjecture [39] that for N-component equations one needs N symmetries is incorrect either.…”

“…If G is a symmetry, then evolutionary equation u τ = G is compatible with (7). There are many other equivalent definitions of symmetry (see for example [10,12]).…”

“…In literature a formal recursion operator is also called a formal symmetry of equation (7). The central result of the Symmetry Approach can be represented by the following Theorem, which we attribute to Shabat: Theorem 1 If equation (7) has an infinite hierarchy of symmetries of arbitrary high order, then a formal recursion operator exists and its coefficients can be found recursively.…”

“…Existence of a finite number of symmetries of a partial differential equation may not secure its integrability [6,7,8,9].…”

Integrable systems and their recursion operators

“…A good source on this is [Zak91]. More recently we were able to classify a large class of scalar equations [SW98,SW00b] and a not so large class of systems of evolution equations [BSW98,BSW01], using the symbolic method and number theory. We are presently working on the classification of homogeneous polynomial systems.…”