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

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ functions. We then demonstrate the differential identities of the $\mathrm{al}_{ab}$ 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 $\mathrm{al}_a$ function. Thus, we also show that by the identities, the $\mathrm{al}_{ab}$ function has the capability to be the hyperelliptic solution to the nonlinear Schr\"odinger and complex modified Korteweg-de Vries equations.