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arXiv · 1911.04861

Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions

John Cobb·Alex Kasman·Albert Serna·Monique Sparkman
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Abstract

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 1-soliton 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.

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John Cobb, Alex Kasman, Albert Serna, Monique Sparkman. 2019-11-12. Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions. https://doi.org/10.1080/14029251.2020.1757234

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