Integrability of the Manakov--Santini hierarchy

Abstract

The first example of the so-called "coupled" integrable hydrodynamic chain is presented. Infinitely many commuting flows are derived. Compatibility conditions of the first two of them lead to the remarkable Manakov--Santini system. Integrability of this four component three dimensional quasilinear system of the first order as well as the coupled hydrodynamic chain is proved by the method of hydrodynamic reductions. In comparision with a general case considered by E.V. Ferapontov and K.R. Khusnutdinova, in this degenerate case N component hydrodynamic reductions are parameterized by N+M arbitrary functions of a single variable, where M is a number of branch points of corresponding Riemann surface. These hydrodynamic reductions also are written as symmetric hydrodynamic type systems. New classes of particular solutions are found.

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Cited by in corpus (8)

• On the Einstein-Weyl and conformal self-duality equations
• A Two-Component Generalization of the Integrable rdDym Equation
• Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
• Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations
• 2D reductions of the equation $u_{yy}=u_{tx}+u_y{u}_{xx}-u_x{u}_{xy}$ and their nonlocal symmetries
• Integrability properties of some symmetry reductions
• 1+1 spectral problems arising from the Manakov-Santini system
• On symmetries of the Gibbons-Tsarev equation