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

Abstract

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, $\displaystyle{ \left[-{\frak{i}}\frac{\partial}{\partial t} +\alpha \frac{\partial}{\partial u}\right]\Psi -\frac{1}{2} \frac{\partial^2}{\partial u^2}\Psi +(\Psi^* \Psi) \Psi = \frac12 (2\beta+\alpha^2-3\wp(v))\Psi }$, where $\Psi(u;v, t):=\mathrm{e}^{\alpha u+{\frak{i}}\beta t+c}$ $\displaystyle{\frac{\mathrm{e}^{-\zeta(v)u}\sigma(u+v)}{\sigma(u)\sigma(v)}}$, its dual $\Psi^*(u; v,t)$, and certain complex numbers $\alpha, \beta$ 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.