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.

翻译：旋转双分量Camassa-Holm系统由于包含强非线性耦合项和高阶微分项，往往会产生连续的非光滑孤立波解，如尖峰孤子、截断孤子、复合波甚至混沌波。本文利用该系统的哈密顿不变量，推导出一种精确的半离散守恒差分格式。我们证明了该半离散数值格式至少保持质量、动量和能量三个离散守恒律。进一步地，在不破坏任何守恒律的前提下，提出了一种全离散有限差分格式。结合非线性迭代过程和高效阈值策略，数值格式的精度得以保证。同时，该差分格式能够捕捉光滑/非光滑初值下孤立波解的形成与传播过程，并具有良好的长期行为。数值结果揭示了在非零旋转参数下一种新型非对称波浪破碎现象。