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

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.

[1]  M. Pavlov,et al.  Integrability of the Manakov--Santini hierarchy , 2009, 0910.2400.

[2]  Yu-Tung Chen,et al.  Hodograph solutions for the generalized dKP equation , 2009, 0904.4595.

[3]  L. V. Bogdanov,et al.  Generalized dKP: Manakov-Santini hierarchy and its waterbag reduction , 2008, 0810.0556.

[4]  L. V. Bogdanov.  A class of multidimensional integrable hierarchies and their reductions , 2008, 0810.2397.

[5]  M. Dunajski.  An interpolating dispersionless integrable system , 2008, 0804.1234.

[6]  S. Manakov,et al.  A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 - dimensional vector fields , 2006, nlin/0611047.

[7]  S. Manakov,et al.  On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking , 2007, 0707.1802.

[8]  L. V. Bogdanov,et al.  Dunajski generalization of the second heavenly equation: dressing method and the hierarchy , 2007, 0707.1675.

[9]  S. Manakov,et al.  A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields , 2007 .

[10]  L. V. Bogdanov,et al.  On the dressing method for Dunajski anti-self-duality equation , 2006, nlin/0612046.

[11]  S. Manakov,et al.  Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation , 2006, nlin/0604023.

[12]  A. Shabat,et al.  Hydrodynamic Reductions and Solutions of a Universal Hierarchy , 2003, nlin/0312043.

[13]  L. V. Bogdanov,et al.  Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type , 2003, nlin/0312013.

[14]  Maciej Dunajski,et al.  A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type , 2003, nlin/0311024.

[15]  Maxim V. Pavlov,et al.  Integrable hydrodynamic chains , 2003, nlin/0301010.

[16]  L. Alonso,et al.  Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type , 2002, nlin/0202008.