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The rotation-two-component Camassa--Holm system, which possesses stronglynonlinear coupled terms and high-order differential terms, tends to havecontinuous nonsmooth solitary wave solutions, such as peakons, stumpons,composite waves and even chaotic waves. In this paper an accurate semi-discreteconservative difference scheme for the system is derived by taking advantage ofits Hamiltonian invariants. We show that the semi-discrete numerical schemepreserves at least three discrete conservative laws: mass, momentum and energy.Furthermore, a fully discrete finite difference scheme is proposed withoutdestroying anyone of the conservative laws. Combining a nonlinear iterationprocess and an efficient threshold strategy, the accuracy of the numericalscheme can be guaranteed. Meanwhile, the difference scheme can capture theformation and propagation of solitary wave solutions with satisfying long timebehavior under the smooth/nonsmooth initial data. The numerical results reveala new type of asymmetric wave breaking phenomenon under the nonzero rotationalparameter.

Fully Conservative Difference Schemes for the Rotation-Two-Component Camassa-Holm System with Smooth/Nonsmooth Initial Data