Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t)+A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on , then p takes a quadratic form of x. 2) If A = f (t)I + D with DT = −D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.

中文翻译：

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多分量 Camassa-Holm 方程的解析笛卡尔解

在这里，我们给出了多分量 Camassa-Holm (MCCH) 方程的解析笛卡尔解的存在性。这样的解决方案可以明确表达，其中速度函数由 u = b(t)+A(t)x 给出，并且不需要对维度 N 的额外约束。我们方法的优点是我们将解析求解 MCCH 方程的过程转化为代数构造合适的矩阵 A(t)。作为应用，我们得到了一些有趣的结果： 1) 如果 u 是 上的线性变换，则 p 采用 x 的二次形式。2) 如果 A = f (t)I + D 且 DT = -D，我们得到螺旋解。当 N = 2 时，该解可用于描述上自由表面的“呼吸式”振荡运动。3) 如果我们得到广义椭圆对称解。当 N = 2 时，