S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies

Abstract

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.

35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel packages

References in corpus (7)

• Classical R-matrix theory of dispersionless systems: I. (1+1)-dimension theory
• Classical R-matrix theory of dispersionless systems: II. (2+1)-dimension theory
• Boyer-Finley equation and systems of hydrodynamic type
• On the Whitham Hierarchies: Reductions and Hodograph Solutions
• From dispersionless to soliton systems via Weyl-Moyal like deformations
• Geometrical optics in nonlinear media and integrable equations
• On the r-th dispersionless Toda hierarchy I: Factorization problem, symmetries and some solutions

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• Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
• Dispersionless DKP hierarchy and elliptic Lowner equation
• Growth of fat slits and dispersionless KP hierarchy
• Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
• Multi-variable reductions of the dispersionless DKP hierarchy
• Loewner equations and reductions of dispersionless hierarchies
• Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation
• Old and New Reductions of Dispersionless Toda Hierarchy