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

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.