Third kind elliptic integrals and 1-motives

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen – Lee – Liu equation, are studied. More precisely, the complex envelope is separa...

Feynman integrals and motives

This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a “bottom-up” approach based on the algebraic geometry of varieties associated to Feynman graphs, and a “top-down” approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint wor...

Integrals involving complete elliptic integrals

We give a closed-form evaluation of a number of Erd elyi-Kober fractional integrals involving elliptic integrals of the rst and second kind, in terms of the 3F2 generalized hypergeometric function. Reduction formulae for 3F2 enable us to simplify the solutions for a number of particular cases. c © 1999 Elsevier Science B.V. All rights reserved.

Elliptic Crystals and Modular Motives

The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves. Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic sett...

Mixed elliptic motives