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

16 pages