Tau function and Virasoro action for the \(n\times n\) KdV hierarchy. (English) Zbl 1354.37068

Terng, Chuu-Lian; Uhlenbeck, Karen

Commun. Math. Phys. 342, No. 1, 81-116 (2016).

The authors published a series of papers to describe a uniform geometric framework in which many integrable systems are included using a splitting of an infinite dimensional group as positive and negative Volterra subgroups. Given a negative Voterra group element, there is a formal inverse scattering solution of the integrable hierarchy. In this paper, they construct a natural Virasoro action on tau functions of \(n\times n\) KdV hierarchy on the space of order \(n\) linear differential operators on the line from a simple Virasoro action on the negative Voterra group. This corresponds to the known Virasoro action on the Gelfand-Dickey hierarchy in terms of pseudo-differential operators under the bijection.

Reviewer: Chuanzhong Li (Ningbo)

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
17B80	Applications of Lie algebras and superalgebras to integrable systems
17B68	Virasoro and related algebras

Keywords:

integrable system; soliton hierarchy; tau function; Virasoro action

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