We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities of arbitrary rank. Second, we obtain a generalization to cyclic identities in which successive ter...

In this paper we make a full analysis of the symmetry reductions beam equation by using classical Lie method infinitesimals and nonclassical method. We consider travelling wave depending on form an arbitrary function. have found several new classes solutions that not been considered before: expressed in terms Jacobi elliptic functions, Wadati solitons compactons. Several coherent structures are...

Landen transformation formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by changing integration variables in elliptic integrals. We rediscover known results as well as obtain more generalized Landen formulas from a very different perspective, by making use of the recently obtained periodic solutions of physically ...

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and Edwards curves.

We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hy-pergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.