Nonlinear Schr\"odinger equation in terms of elliptic and hyperelliptic σ functions

It is known that the elliptic function solutions of the nonlinear Schr\"odinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, ${ [-√-1∂/∂ t -α ∂/∂ u]Ψ +1/2 ∂^2/∂ u^2Ψ +(Ψ^* Ψ) Ψ = 12 (2β+(v)+λ_2-α^2)Ψ }$, where $Ψ(u;v, t):=eα u+√-1β t+c$ ${{e-ζ(v)uσ(u+v)}{σ(u)σ(v)}}$, its dual $Ψ^*(u; v,t)$, and certain complex numbers $α, β$. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions.