Nonlinear differential identities for cnoidal waves

Abstract

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^{(\alpha)}u_s^{(\beta)}\big)(x)=\sum_{n=0}^{2+\alpha+\beta}b_{\alpha,\beta}(n)u_s^{(n)}(x)+c_{\alpha,\beta}, {equation*} for any $s>0$ and for all $\alpha,\beta\in\N_0$. We give explicit expressions for the coefficients $b_{\alpha,\beta}(n)$ and $c_{\alpha,\beta}$ for given $s$. Moreover, we show that for any $s$ satisfying $\sinh(\pi/(2s))\geq 1$ the set of functions $\{1,u^{\vphantom{a}}_s,u'_s,u"_s,...\}$ constitutes a basis for $L^2(0,2\pi)$. 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.