Hydrodynamic type integrable equations on a segment and a half-line
  The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions of multi-field systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semi-line are presented. 
Metin Gurses, Ismagil Habibullin and Kostyantyn Zheltukhin