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

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