On a class of reductions of the Manakov–Santini hierarchy connected with the interpolating system

Abstract

Using the Lax–Sato formulation of the Manakov–Santini hierarchy, we introduce a class of reductions such that the zero-order reduction of this class corresponds to the dKP hierarchy, and the first-order reduction gives the hierarchy associated with the interpolating system introduced by Dunajski. We present the Lax–Sato form of a reduced hierarchy for the interpolating system and also for the reduction of arbitrary order. Similar to the dKP hierarchy, the Lax–Sato equations for L (the Lax function) split from the Lax–Sato equations for M (the Orlov function) due to the reduction, and the reduced hierarchy for an arbitrary order of reduction is defined by Lax–Sato equations for L only. A characterization of the class of reductions in terms of the dressing data is given. We also consider a waterbag reduction of the interpolating system hierarchy, which defines (1+1)-dimensional systems of hydrodynamic type.