Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic σ functions

It is known that the elliptic function solutions of the nonlinear Schrödinger 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 = 1/2 (2β+℘(v)+λ_2-α^2)Ψ, where Ψ(u;v, t):=e^α u+√(-1)β 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.