Infinite-Dimensional Estabrook-Wahlquist Prolongations for the sine-Gordon Equation

Abstract

We are looking for the universal covering algebra for all symmetries of a given pde, using the sine-Gordon equation as a typical example for a non-evolution equation. For non-evolution equations, Estabrook-Wahlquist prolongation structures for non-local symmetries depend on the choice of a specific sub-ideal, of the contact module, to define the pde. For each inequivalent such choice we determine the most general solution of the prolongation equations, as sub-algebras of the (infinite-dimensional) algebra of all vector fields over the space of non-local variables associated with the pde, in the style of Vinogradov covering spaces. We show explicitly how previously-known prolongation structures, known to lie within the Kac-Moody algebra, $A_1^{(1)}$, are special cases of these general solutions, although we are unable to identify the most general solutions with previously-studied algebras. We show the existence of gauge transformations between prolongation structures, viewed as determining connections over the solution space, and use these to relate (otherwise) distinct algebras. Faithful realizations of the universal algebra allow integral representations of the prolongation structure, opening up interesting connections with algebras of Toeplitz operators over Banach spaces, an area that has only begun to be explored.