On the Jacobi Elliptic functions and Applications

TL;DR

The paper develops Jacobi elliptic function theory via a trigonometric expansion of Jacobi theta functions, deriving a differential system from the heat equation that governs theta-coefficients, and recalling identities such as $sn^2 u + cn^2 u = 1$ and $dn^2 u + k^2 sn^2 u = 1$, alongside the Jacobi Zeta function $Z(u)$. It presents theta-function-based exponential-series expansions for $sn(u)$, $cn(u)$, and $dn(u)$, expressed in terms of theta-function coefficients and related to ratios of theta functions with derivative relations. Under the substitution $u = heta_3^2(0)\,v$, Theorems 4 and 5 give explicit exponential-series representations for $sn\,u$, $cn\,u$, and $dn\,u$ as double sums over $p$ and $k$, including expressions for $\frac{sn\,u}{cn\,u}$ and derivatives like $\frac{\partial sn\,u}{\partial u}$. In closing, the work highlights the continuing relevance of $dn(u,k)$, proposes potential new theta-function characterizations, and notes applications including quantum-mechanical zero modes of periodic SUSY partner potentials $V_+(u)=\frac{2-k+2(k-1)}{dn^2(u,k)}$ and $V_-(u)=2-k+2\,dn^2(u,k)$, as well as a periodic solution to the nonlinear Schrödinger equation $\psi(x,t)=r\,\exp[i(px-p^2-(2-k^2)r^2)t]\, dn(rx-2prt,k^2)$, with cyclic identities and generalized Landen formulas supporting a form of linear superposition in nonlinear contexts.

Abstract

In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57) permits one establish a differential system. This system is derived from the heat equation and is satisfied by their coefficients. Several applications may be deduced. Other types of expansions for the Jacobi elliptic functions as well as for the Zeta function are examined.