Elliptic Soliton Solutions: $$\tau $$ Functions, Vertex Operators and Bilinear Identities

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 derived and vertex operators that generate such presented. Bilinear identities constructed an algorithm to calculate residues equations is formulated. These investigated detail KdV equation sketched KP hierarchy. Degenerations periods of investigated, giving rise associated with trigonometric/hyperbolic rational functions. Reductions dispersion relation considered employing so-called $N$-th roots unity. functions, hierarchy Boussinesq obtained from those KP. also formulate two ways derivatives involved factors, shows type factors result quasi-gauge property equations.