  The rotation-two-component Camassa--Holm system, which possesses strongly
nonlinear coupled terms and high-order differential terms, tends to have
continuous nonsmooth solitary wave solutions, such as peakons, stumpons,
composite waves and even chaotic waves. In this paper an accurate semi-discrete
conservative difference scheme for the system is derived by taking advantage of
its Hamiltonian invariants. We show that the semi-discrete numerical scheme
preserves at least three discrete conservative laws: mass, momentum and energy.
Furthermore, a fully discrete finite difference scheme is proposed without
destroying anyone of the conservative laws. Combining a nonlinear iteration
process and an efficient threshold strategy, the accuracy of the numerical
scheme can be guaranteed. Meanwhile, the difference scheme can capture the
formation and propagation of solitary wave solutions with satisfying long time
behavior under the smooth/nonsmooth initial data. The numerical results reveal
a new type of asymmetric wave breaking phenomenon under the nonzero rotational
parameter.
