Cnoidal Waves as Exact Sums of Repeated Solitary Waves: New Series for Elliptic Functions

Via Poisson summation of the corresponding Fourier series, it is shown that the cnoidal wave of the Korteweg–de Vries equation can be written as an infinite sum of periodically repeated solitary waves, a result first proved by Toda through a different method. Similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn, sn, and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves. As a by-product, new series are also obtained for the complete elliptic integral K and modulus k.