Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy

Abstract

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-component KP hierarchy is (re)formulated to depend on several discrete variables called ``charges''. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multi-component version of the so called differential Fay identity are derived from the Hirota equations of the $\tau$-function of this ``charged'' multi-component KP hierarchy. These multi-component differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton-Jacobi equations for the $S$-functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multi-component KP hierarchy.

latex2e (a4paper, 12pt) using packages "amssymb,amsmath,amsthm", 44 pages, no figure; (v2) a few typos corrected, bibliographic data updated, final form for publication in special issue of Physica D (2007)

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Cited by in corpus (16)

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• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• The multicomponent 2D Toda hierarchy: dispersionless limit
• The Multicomponent KP Hierarchy: Differential Fay Identities and Lax Equations
• Differential Fay identities and auxiliary linear problem of integrable hiearchies
• An extension of the Kadomtsev-Petviashvili hierarchy and its Hamiltonian structures
• Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries
• Dispersionful analogue of the Whitham hierarchy
• Spin generalization of the Calogero-Moser hierarchy and the matrix KP hierarchy
• Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
• Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy
• Elliptic solutions to matrix KP hierarchy and spin generalization of elliptic Calogero-Moser model
• Frobenius Manifolds, Spectral Curves, and Integrable Hierarchies
• Multi-component Toda lattice hierarchy
• Kernel Formula Approach to the Universal Whitham Hierarchy
• Non-degenerate solutions of universal Whitham hierarchy