Camassa–Holm equation

Remarks

Example: for $\kappa = 0$ and $c > 0$, the peakon $u(x, t) = c\,e^{-\abs{x - ct}}$ is a weak solution. It is not differentiable at the crest, so it solves the equation in the weak sense, written as $u_t + uu_x + \partial_x G * \left(u^2 + \tfrac{1}{2}u_x^2\right) = 0$, with $G(x) = \tfrac{1}{2}e^{-\abs{x}}$ the kernel of $(1 - \partial_x^2)^{-1}$. For the peakon, $m = 2c\,\delta(x - ct)$ is a point mass, since $(1 - \partial_x^2)e^{-\abs{x}} = 2\delta$. Non-example: the equation $u_t + 2\kappa u_x - u_{xxt} + 3uu_x = 0$, without the right-hand side, is an equation of Benjamin–Bona–Mahony type; it misses the terms $2u_xu_{xx} + uu_{xxx}$, so it is not the Camassa–Holm equation. The two forms agree because $m_t = u_t - u_{xxt}$ and $um_x + 2mu_x = 3uu_x - uu_{xxx} - 2u_xu_{xx}$.