The Lamé functions and elliptic soliton solutions: Bilinear approach

Source abstract

The Lamé function can be used to construct plane wave factors and solutions to the Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) hierarchies. The solutions are usually called elliptic solitons. In this paper, 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 Lamé type plane wave factors, expressions of τ \tau functions and the generating vertex operators. Then, for the discrete potential KdV and KP equations, we give their bilinear forms, derive τ \tau functions of elliptic solitons, and show that they share the same vertex operators with the KdV hierarchy and the KP hierarchy, respectively.