Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic σ functions

Abstract

It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the WeierWeierstrass sigma function, (-i∂/∂t‒ α ∂/∂u-(1/2) ∂2/ ∂u2)/Ψ + (Ψ∗Ψ)Ψ =(1/2)(2β + α2 − 3℘(v))Ψ, where Ψ(u;v, t) := exp(αu+iβt+c−ζ(v)u)σ(u + v)/σ(u)σ(v), its dual Ψ∗(u; v, t), and certain complexand certain complex numbers α, β and c. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions.