Lamé function and elliptic deformation of KdV vertex operators

Abstract

In this talk, we will show a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors.τfunctions in Hirota’s form are derived and vertex operators that generate suchτfunctions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational functions. Reductions by dispersion relation are considered by employing the so-called elliptic N-th roots of the unity. τfunctions, vertex operators and bilinear equations of the KdV hierarchy and Boussinesq equation are obtained from those of the KP.