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Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic function $u_s(x)$, which is essentially the Jacobian elliptic function $\cn^2(z;m(s))$ with one non-trivial parameter $s$. More precisely, we show that this function $u_s$ fulfills equations of the form {equation*} \big(u_s^{(α)}u_s^{(β)}\big)(x)=\sum_{n=0}^{2+α+β}b_{α,β}(n)u_s^{(n)}(x)+c_{α,β}, {equation*} for any $s>0$ and for all $α,β\in\N_0$. We give explicit expressions for the coefficients $b_{α,β}(n)$ and $c_{α,β}$ for given $s$. Moreover, we show that for any $s$ satisfying $\sinh(π/(2s))\geq 1$ the set of functions $\{1,u^{\vphantom{a}}_s,u'_s,u"_s,...\}$ constitutes a basis for $L^2(0,2π)$. 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.