Geometric, algebraic and analytic properties of hyperelliptic ${\mathrm{a}\mathrm{l}}_{ab}$ function of genus $g$

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic ${\mathrm{a}\mathrm{l}}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the ${\mathrm{a}\mathrm{l}}_{ab}$ functions together with the ${\mathrm{a}\mathrm{l}}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{s}\mathrm{n}$, $\mathrm{c}\mathrm{n}$, and $\mathrm{d}\mathrm{n}$ functions. We then demonstrate the differential identities of the ${\mathrm{a}\mathrm{l}}_{ab}$ function. These identities are the novel integrable partial nonlinear differential equations as a natural extension of the hyperelliptic solutions of the modified Korteweg-de Vries equation in terms of the ${\mathrm{a}\mathrm{l}}_a$ function. Thus, we also show that by the identities, the ${\mathrm{a}\mathrm{l}}_{ab}$ function has the capability to be the hyperelliptic solution to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations.