Tau functions and Virasoro actions for soliton hierarchies. (English) Zbl 1346.37058

Authors’ abstract: There is a general method for constructing a soliton hierarchy from a splitting \(L_\pm\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \(\mathcal{L}_+\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each \(f\) in the negative subgroup \(L_-\) a solution \(u_f\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \(\tau_f\) for each element \(f\in L_-\). In this paper, we give integral formulas for variations of \(\ln\tau_f\) and second partials of \(\ln \tau_f\), discuss whether we can recover solutions \(u_f\) from \(\tau_f\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \(\mathrm{GL}(n,\mathbb{C})\)-hierarchy.


37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
Full Text: DOI arXiv


[1] Ablowitz M.J., Kaup D.J., Newell A.C., Segur H.: The inverse scattering transform—Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249-315 (1974) · Zbl 0408.35068
[2] Adler M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries type equations. Invent. Math. 50, 219-248 (1979) · Zbl 0393.35058
[3] Aratyn K., van de Ler J.: An integrable structure based on the WDVV equations. Theor. Math. Phys. 134, 1426 (2003) · Zbl 1068.37047
[4] Drinfel’d, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type (Russian). Curr. Probl. Math. 24, 81-180 (1984) (Itogi Naukii Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow) · Zbl 1179.37096
[5] Dubrovin B.A.: Geometry of 2D topological field theories, Lecture Notes in Mathematics, vol. 1620. Springer, Berlin (1996) · Zbl 0841.58065
[6] Fordy A.P., Kulish P.P.: Nonlinear Schrödinger equations and simple Lie algebra. Commun. Math. Phys. 89, 427-443 (1983) · Zbl 0563.35062
[7] Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix airy function. Commun. Math. Phys. 147 (1992) · Zbl 0756.35081
[8] Pressley A., Segal G.B.: Loop Groups. Oxford Science Publ., Clarendon Press, Oxford (1986) · Zbl 0618.22011
[9] Reyman A.G., Semenov-Tian-Shansky M.A.: Current algebras and non-linear partial differential equations. Sov. Math. Dokl. 21, 630-634 (1980) · Zbl 0501.58018
[10] Sattinger D.H.: Hamiltonian hierarchies on semi-simple Lie algebras. Stud. Appl. Math. 72, 65-86 (1984) · Zbl 0584.58022
[11] Terng C.L.: Geometries and symmetries of soliton equations and integrable elliptic systems. Adv. Stud. Pure Math. 51, 188-401 (2008) · Zbl 1165.37031
[12] Terng C.L.: Dispersive geometric curve flows. Survey Differ. Geom. 19, 179-230 (2015) · Zbl 1325.37050
[13] Terng C.L., Uhlenbeck K.: Poisson actions and scattering theory for integrable systems. Surveys Differ. Geom. 4, 315-402 (1998) · Zbl 0935.35163
[14] Terng C.L., Uhlenbeck K.: Bäcklund transformations and loop group actions. Commun. Pure Appl. Math. 53, 1-75 (2000) · Zbl 1031.37064
[15] Terng C.L., Uhlenbeck K.: Schrödinger flows on Grassmannians, integrable systems. Geom. Topol. AMS/IP Stud. Adv. Math. 36, 235-256 (2006) · Zbl 1110.37056
[16] 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
[17] Terng, C.L., Uhlenbeck, K.: Tau functions and Virasoro actions for the n × n KdV hierarchy. Commun. Math. Phys. (to appear) · Zbl 1354.37068
[18] Terng C.L., Wang E.: Transformations of flat Lagrangian immersions and Egoroff nets. Asian J. Math. 12, 99-119 (2008) · Zbl 1179.37096
[19] Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surveys Differ. Geom. 1, 243-310 (1990) · Zbl 0757.53049
[20] Wilson G.: The τ-functions of the \[{\mathcal{G}}\] GAKNS equations. Integr. Syst. Verdier Meml. Progress Math. 115, 147-162 (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.