Well-posedness and convergence of a variational discretization of the Camassa-Holm equation

Sondre Tesdal Galtung, Xavier Raynaud

Published 2020-03-06Version 1

We show how the two-component Camassa-Holm (2CH) system, for which the Camassa-Holm equation is a special case, can be derived from variational principles by introducing a potential energy. The variation is done in Lagrangian coordinates, but the Euler-Lagrange equations have a pure Eulerian formulation. After discretizing the kinetic and potential energies, we use the same variational principles to derive a semi-discrete system of equations as an approximation of the 2CH system. In the discrete case, the Euler$-$Lagrange equations are only available in Lagrangian variables. In this derivation there naturally appears a discrete Sturm-Liouville operator whose coefficients depend on the initial data. We show the existence of a decaying Green's function for this operator. Then, we prove that the semi-discrete system admits globally defined unique solutions, and preserves discrete versions of the energy and momentum for the 2CH system. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions which converges to the solution of the 2CH system.