Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions. Issue 3 (2nd July 2020)

• Record Type: Journal Article
• Authors: Cobb, John; Kasman, Alex; Serna, Albert; Sparkman, Monique
• Is Part Of: Journal of nonlinear mathematical physics. Volume 27:Issue 3(2020)
• Page Start: 429
• Page End: 452
• Publication Date: 2020-07-02
• Subjects: Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2020.1757234
• Languages: English
• ISSNs: 1402-9251

Abstract

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 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.