Product and rational decompositions of theta functions representations for nonlinear periodic waves

A class of periodic solutions of nonlinear evolution equations is expressed as products and rational expressions of theta/elliptic functions. Examples of equations treated include a coupled system of nonlinear Schrodinger (NLS) equations. the (2 + 1) dimensional sine Gordon system and the Sasa-Satsuma equation. Coupled modified Korteweg-de Vries and NLS systems show that these product periodic waves can be expanded as an infinite sum of solitary waves arising from the coupling. Brief consideration of discrete evolution equations show similar trends but some quantitative difference with the continuous counterpart.