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Energy-dependent potentials revisited: A universal hierarchy of hydrodynamic type. (English) Zbl 0997.37045
Summary: A hierarchy of infinite-dimensional systems of hydrodynamic type is considered and a general scheme for classifying its reductions is provided. Wide families of integrable systems including, in particular, those associated with energy-dependent spectral problems of Schrödinger type, are characterized as reductions of this hierarchy. $$N$$-phase type reductions and their corresponding Dubrovin equations are analyzed. A symmetry transformation connecting different classes of reductions is formulated.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q55 NLS equations (nonlinear Schrödinger equations)
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