Nonlinear differential identities for cnoidal waves

TL;DR

This paper derives nonlinear differential identities for cnoidal wave functions, providing explicit formulas and basis properties, which facilitate solving nonlinear wave equations like KdV using these identities.

Contribution

It introduces explicit nonlinear differential identities for cnoidal waves and establishes their basis properties in L^2, enabling simplified solutions to nonlinear wave equations.

Findings

Explicit formulas for differential identities of cnoidal waves.

The set {1, u_s, u'_s, ...} forms a basis for L^2(0,2π) under certain conditions.

Finite and infinite ansatz solutions for nonlinear wave equations like KdV.

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.