A NON-COMMUTATIVE OPERATOR-HIERARCHY OF BURGERS EQUATIONS AND BACKLUND TRANSFORMATIONS

Abstract

An operator equation on a Banach space, which represents the operator analog of Burgers equation, is here considered. The well known Cole-Hopf transformation, a particular case of the wider class of Backlund transformations, which connects the classical nonlinear Burgers equation to the linear heat equation, is extended to the case of operator valued equations. Then, since the operator Burgers equation admits a recursion operator, a whole hierarchy of Burgers operator equations is generated. Notably, each member of such a Burgers operator hierarchy is related, via Cole-Hopf transformation to the corresponding member of a heat operator hierarchy. Indeed, also the recursion operator admitted by the Burgers operator equation, is related, via Cole-Hopf transformation, to the (trivial) recursion operator admitted by the linear heat operator equation. Furthermore, the Burgers recursion operator is not Abelian, hence, the whole hierarchy does not enjoy commutativity properties

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