The Lam\'e functions and elliptic soliton solutions: Bilinear approach

TL;DR

This paper reviews recent advances in the Hirota bilinear method for elliptic solitons of the KdV and KP equations, focusing on Lamé functions, tau functions, and vertex operators, including discrete versions.

Contribution

It provides a comprehensive review of bilinear calculations involving Lamé functions and extends the bilinear framework to discrete potential KdV and KP equations.

Findings

Derivation of bilinear forms for discrete potential KdV and KP equations.

Explicit expressions for tau functions of elliptic solitons.

Shared vertex operators between discrete and continuous hierarchies.

Abstract

The Lam\'e function can be used to construct plane wave factors and solutions to the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) hierarchy. The solutions are usually called elliptic solitons. In this chapter, first, we review recent development in the Hirota bilinear method on elliptic solitons of the KdV equation and KP equation, including bilinear calculations involved with the Lam\'e type plane wave factors, expressions of $\tau$ functions and the generating vertex operators. Then, for the discrete potential KdV and KP equations, we give their bilinear forms, derive $\tau$ functions of elliptic solitons, and show that they share the same vertex operators with the KdV hierarchy and the KP hierarchy, respectively.