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 Weierstrass sigma function, $\displaystyle{ \left[-{\frak{i}}\frac{\partial}{\partial t} +α\frac{\partial}{\partial u}\right]Ψ-\frac{1}{2} \frac{\partial^2}{\partial u^2}Ψ+(Ψ^* Ψ) Ψ= \frac12 (2β+α^2-3\wp(v))Ψ}$, where $Ψ(u;v, t):=\mathrm{e}^{αu+{\frak{i}}βt+c}$ $\displaystyle{\frac{\mathrm{e}^{-ζ(v)u}σ(u+v)}{σ(u)σ(v)}}$, its dual $Ψ^*(u; v,t)$, and 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.

Theorems & Definitions (17)

• Definition 3.1
• Lemma 3.2
• Definition 4.1
• Theorem 4.2
• Corollary 4.3
• Lemma 4.4
• Proposition 4.5
• Lemma 4.6
• Lemma 4.7
• Lemma 4.8