2000/04/30 by T. Ikeda, K. Takasaki
Physics and Astronomy · Mathematics · #nlin.SI #hep-th #math.QA
published as Internat. Math. Res. Notices 2001, No. 7, 329-369 · Latex2e with amsmath,amssymb, 35 pages, no figures, (v2) typos corrected, added info on ref.[30] (v3) added a reference and some examples (v4) added some comments in subsection 4.4
arxiv created 2000/09/05 · arxiv updated 2009/11/30
We introduce an extension of the ℓ-reduced KP hierarchy, which we call the ℓ-Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of ℓ=2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra sl_ℓtor. This reproduces essentially the same Hirota bilinear equations as those recently introduced by Billig and Iohara et al. We can further derive these Hirota bilinear equation from a Lax formalism of the hierarchy.This Lax formalism also enables us to construct a family of special solutions that generalize the breaking soliton solutions of Bogoyavlensky. These solutions contain the N-soliton solutions, which are usually constructed by use of vertex operators.