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

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.


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
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[1] Aratyn K., van de Ler J.: An integrable structure based on the WDVV equations. Theor. Math. Phys. 134, 14-26 (2003) · Zbl 1068.37047
[2] Dickey, L.A.: Soliton Equations and Hamiltonian Systems, 2nd edn. Advanced Series in Mathematical Physics, vol. 26. World Scientific Publishing Co. Inc., River Edge (1986) · Zbl 0753.35075
[3] Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. (Russian) Current problems in mathematics, vol. 24, pp. 81-180 (1984). Itogi Naukii Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow · Zbl 0935.35163
[4] Pressley A., Segal G.B.: Loop Groups. Oxford Science Publ., Clarendon Press, Oxford (1986) · Zbl 0618.22011
[5] Terng C.L., Uhlenbeck K.: Poisson actions and scattering theory for integrable systems. Surv. Differ. Geom. 4, 315-402 (1999) · Zbl 0935.35163
[6] Terng C.L., Uhlenbeck K.: The n × n KdV flows. The Richard S. Palais Festschrift. J. Fixed Point Theory Appl. 10, 37-61 (2011) · Zbl 1251.37070
[7] Terng, C.L., Uhlenbeck, K.: A Tau functions and Virasoro actions for soliton hierarchies, to appear in Commun. Math. Phys · Zbl 1346.37058
[8] van Moerbeke, P.: Integrable foundations of string theory. Lectures on Integrable Systems (Sophia-Antipolis, 1991), pp. 163-267. World Sci. Publ., River Edge, NJ (1994)
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