Linear reciprocal transformations of bihamiltonian integrable hierarchies

Abstract

For an integrable hierarchy which possesses a bihamiltonian structure with semisimple hydrodynamic limit, we prove that the linear reciprocal transformation with respect to any of its symmetry transforms it to another bihamiltonian integrable hierarchy. Moreover, we show that the central invariants of the bihamiltonian structure are preserved under the reciprocal transformation. The main tools that we use to obtain this result are the bihamiltonian and variational bihamiltonian cohomologies defined for a bihamiltonian structure of hydrodynamic type. We also apply this result to study the problem of classification of bihamiltonian integrable hierarchies.