Formal recursion operators of integrable PDEs of the form Utt = F(U,Ux,Ut,...)

Abstract

We will explain how the symmetry approach to integrability applies to partial differential equations of the form qtt=F(q,qx,…,qn,qt,qtx,…,qtm). Any such equation is integrable if it admits a formal recursion operator, i.e. a pseudodifferential operator R of the form R:=L+MDt where L and M are pseudodifferential operators in the derivation Dx satisfying the symmetry condition F(L+MDt)=(L+MDt)F... We are confronted with solving an equation over pseudodifferential operators in two derivations, a rather nontrivial problem. The equation happens to have a somewhat triangular structure, making its resolution possible. But in the solving process there appear obstructions, written as conditions over the rhs F of the PDE, that are interpreted as integrability conditions. The algebra of formal recursion operators has an interesting structure, and it has important relationships to algebras of commuting (pseudo)-differential operators in two derivations.