Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
T. Ikeda, K. Takasaki
Abstract
We introduce an extension of the -reduced KP hierarchy, which we call the -Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of =2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra sltor. 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.
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