A generalization of the spatially one-dimensional parallel pipe-line algorithm for solution of the initial-boundary-value problem using explicit difference method to the two-dimensional case is presented. The suggested algorithm has been verified by implementation on a workstation-cluster running under PVM (Parallel Virtual Machine). Theoretical estimates of the speed-up are presented.

We show that the periodic Camassa–Holm equation $u_t-u_{xxt}+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ possesses a global continuous semigroup of weak conservative solutions for initial data ${u|}_{t=0}$ in $H_{\mathrm{per}}^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure $\mu$ with ${\mu }_{\mathrm{ac}}=(u^2+u_x^2)dx$. The total energy is preserved by the solution.