Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic functionu s ( x ) , which is essentially the Jacobian elliptic functioncn 2 ( z ; m ( s ) )with one non‐trivial parameter s > 0 . More precisely, we show that this function u s fulfills equations of the formu s ( α )u s ( β )( x ) = ∑ n = 0 2 + α + βb α , β( n ) u s ( n )( x ) + c α , β , for all α , β ∈ N 0 . We give explicit expressions for the coefficientsb α , β( n )and c α , βfor given s . Moreover, we show that for any s the set of functions { 1 , u s a, u s ′ , u s ′ ′ , ⋯ } constitutes a basis forL 2 ( 0 , 2 π ) . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.