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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.

MSC:

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
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