Tau functions and Virasoro actions for soliton hierarchies

2014/05/16 by Terng, Chuu-Lian, Uhlenbeck, Karen · 1 citation
#Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences

paper · doi:10.48550/arxiv.1405.4043

Abstract

There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup 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 the negative subgroup. 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 GL(n,\C)-hierarchy.

Cited by

• On a matrix constrained CKP hierarchy

Related

• Tau function and Virasoro action for the nxn KdV hierarchy
• Flag Spaces in KP Theory and Virasoro Action on det D_j and Segal-Wilson τ-Function
• Vector NLS hierarchy solitons revisited: dressing transformation and tau function approach
• Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
• Aspects of Solitons in Affine Integrable Hierarchies