Difficulties with the SDiff(2) Toda Equation

The SDiff(2) Toda equation appears in several fields of physics. Its algebra of symmetries is the area-preserving diffeomorphisms of a 2-surface, isomorphic to the algebra of Poisson brackets generated by Hamiltonian vector fields in 2 variables. These facts have allowed various researchers to create “zero-curvature representations” and even to create a formal presentation of a general solution; however, none of this seems particularly useful in the search for new, explicit solutions. A new attempt to find a more direct method to “buy” new solutions from old ones begins with the Bäcklund transformation for the (2-dimensional) Toda lattice built over the Cartan matrix for the Lie algebra An or A (1) n , and considers the limit as n →∞. While this limit gives the desired (second-order) pde’s, the associated limit of the (first-order) pde’s that define the Bäcklund transformation gives only trivial results. Reasons for this are analyzed, and deplored. As well the known “Lax pairs” are analyzed in this language, suggesting that they are not useful for this purpose since the representations of SDiff(2) involved are over the defining manifold rather than over (infinite-dimensional) tangent manifolds in additional (pseudopotential) variables. 1. The Equation Itself, and the Goals An equation of interest in several fields of theoretical physics is the SDiff(2) Toda equation, for one unknown function of three variables, with differentiation shown by a subscript: u,zz̃ = e,tt ⇐⇒ v,zz̃ = (e),tt , v ≡ u,tt . (1.1) One derivation was given by myself and Charles Boyer[BF] in 1982. It begins with the Plebański equation for general self-dual vacuum solutions of the Einstein field equations. All such solutions that admit at least one rotational Killing vector may be determined by solutions of this equation. The angular variable associated with that Killing vector combines with the other three above to constitute a local coordinate chart for the 4-dimensional space in question. The name I have chosen has also been used by Mikhail Saveliev[KS] and by Kanehisa Takasaki and T. 1991 Mathematics Subject Classification. Primary 58G37, 35Q75; Secondary 35Q58.