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.

使用行列式生成非交换KdV方程的四元数解。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。出人意料地，表明了当解决方案基本上是非可交换的时，这种解决方案永远不会是奇异的。当一个1孤子解决方案通过迭代的Darboux变换与另一个解决方案组合时，结果表现为渐近式，就像不同解决方案的组合一样。这种“非线性叠加原理”用于找到一般的2-孤子相互作用中的相移公式。