Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions
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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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Cite this article
TY - JOUR AU - John Cobb AU - Alex Kasman AU - Albert Serna AU - Monique Sparkman PY - 2020 DA - 2020/05 TI - Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions JO - Journal of Nonlinear Mathematical Physics SP - 429 EP - 452 VL - 27 IS - 3 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1757234 DO - https://doi.org/10.1080/14029251.2020.1757234 ID - Cobb2020 ER -