Cnoidal wave

Example

A cnoidal wave is a periodic travelling wave solution of the Korteweg–de Vries equation. This equation models long water waves of small amplitude moving in one direction in shallow water. Take the KdV equation in the normalised form $u_t + 6uu_x + u_{xxx} = 0$. Let $e_1 > e_2 > e_3$ be real numbers, let $m = \frac{e_1 - e_2}{e_1 - e_3} \in (0, 1)$ and $\gamma = \sqrt{(e_1 - e_3)/2}$. Then $$u(x, t) = e_2 + (e_1 - e_2)\,\operatorname{cn}^2\big(\gamma(x - ct) \,\big|\, m\big), \qquad c = 2(e_1 + e_2 + e_3),$$ is a solution, where $\operatorname{cn}$ is the Jacobi elliptic function with parameter $m$. Check: for $u = f(\xi)$ with $\xi = x - ct$, the equation integrates twice to $\tfrac{1}{2}(f')^2 = -(f - e_1)(f - e_2)(f - e_3)$. The identities $\operatorname{cn}' = -\operatorname{sn}\operatorname{dn}$ and $\operatorname{dn}^2 = 1 - m + m\operatorname{cn}^2$ apply to the stated $f$. They show that it satisfies this equation exactly when $2\gamma^2 m = e_1 - e_2$ and $2\gamma^2(1 - m) = e_2 - e_3$. These two conditions hold by the choice of $m$ and $\gamma$.