Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters that lead to non-singular 1soliton and periodic solutions is given. Surprisingly, it is shown that such solutions are never singular when the solution is essentially non-commutative. When a 1-soliton solution is combined with another solution through an iterated Darboux transformation, the result behaves asymptotically like a combination of different solutions. This “non-linear superposition principle” is used to find a formula for the phase shift in the general 2-soliton interaction. A concluding section compares these results with other research on non-commutative soliton equations and lists some open questions.

中文翻译：

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四元数值呼吸孤子、有理和周期 KdV 解决方案

非交换 KdV 方程的四元数值解是使用行列式产生的。以这种方式产生的解是（呼吸）孤子解、有理解、空间周期解和这三种基本类型的混合。给出了导致非奇异 1 孤子和周期解的参数的完整表征。令人惊讶的是，当解决方案本质上是非交换的时，这些解决方案永远不会是奇异的。当 1-孤子解通过迭代 Darboux 变换与另一个解组合时，结果表现得像不同解的组合一样渐近。这种“非线性叠加原理”用于找到一般 2 孤子相互作用中相移的公式。