Elliptic soliton solutions: $\tau$ functions, vertex operators and bilinear identities

TL;DR

This paper develops a bilinear framework for elliptic soliton solutions using $ au$ functions, vertex operators, and bilinear identities, with applications to the KdV and KP hierarchies and their degenerations.

Contribution

It introduces a novel bilinear framework for elliptic solitons, including $ au$ functions, vertex operators, and identities, extending to degenerations and reductions of integrable hierarchies.

Findings

Derived $ au$ functions in Hirota's form for elliptic solitons

Constructed vertex operators generating $ au$ functions

Analyzed degenerations leading to trigonometric, hyperbolic, and rational cases

Abstract

We establish a bilinear framework for elliptic soliton solutions which are composed by the Lam\'e-type plane wave factors. $\tau$ functions in Hirota's form are derived and vertex operators that generate such $\tau$ functions 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. $\tau$ functions, vertex operators and bilinear equations of the KdV hierarchy and Boussinesq equation are obtained from those of the KP. We also formulate two ways to calculate bilinear derivatives involved with the Lam\'e-type plane wave factors, which shows that such type of plane wave factors result in quasi-gauge property of bilinear equations.