Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic $\sigma$ 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, ${\displaystyle [-\mathfrak{i}\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\alpha \frac{\mathrm{\partial }}{\mathrm{\partial }u}]\mathrm{\Psi }-\frac{1}{2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{u}^2}\mathrm{\Psi }+({\mathrm{\Psi }}^{\ast }\mathrm{\Psi })\mathrm{\Psi }=\frac{1}{2}(2\beta +{\alpha }^2-3\mathrm{\wp }(v))\mathrm{\Psi }}$, where $\mathrm{\Psi }(u;v,t):={\mathrm{e}}^{\alpha u+\mathfrak{i}\beta t+c}$ ${\displaystyle \frac{{\mathrm{e}}^{-\zeta (v)u}\sigma (u+v)}{\sigma (u)\sigma (v)}}$, its dual ${\mathrm{\Psi }}^{\ast }(u;v,t)$, 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.