English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group L as positive and negative subgroups $${L_\pm}$$L± and a commuting sequence in the Lie algebra $${\mathcal{L}_+}$$L+ of $${L_+}$$L+. Given $${f\in L_-}$$f∈L-, there is a formal inverse scattering solution uf of the hierarchy. When there is a 2 co-cycle on $${\mathcal{L}}$$L that vanishes on both $${\mathcal{L}_+ {\rm and} \mathcal{L}_-}$$L+andL-, Wilson constructed for each $${f\in L_-}$$f∈L- a tau function $${\tau_f}$$τf for the hierarchy. In this third paper, we prove the following results for the n × n KdV hierarchy: (1) The second partials of $${\ln\tau_f}$$lnτf are differential polynomials of the formal inverse scattering solution uf. Moreover, $${u_f}$$uf can be recovered from the second partials of $${\ln\tau_f}$$lnτf.(2)The natural Virasoro action on $${\ln\tau_f}$$lnτf constructed in the second paper is given by partial differential operators in $${\ln\tau_f}$$lnτf.(3)There is a bijection between phase spaces of n × n KdV hierarchy and Gelfand-Dickey (GDn) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection.(4) Our Virasoro action on the n × n KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GDn hierarchy under the bijection.

Tau Function and Virasoro Action for the n × n KdV Hierarchy