Abstract
Beltrami fields are complex vector fields \({\mathbf {F}}\) which satisfy the equation \({\text {curl}} {\mathbf {F}} + \lambda {\mathbf {F}}=0.\) Such fields appear in astrophysics, electromagnetics and plasma physics. We construct a complete system of solutions to the differential equation \((D+\lambda (x_3)+M^{\gamma (x_3){\mathbf {e}}_3})u=0\) for a complex quaternionic valued function u in a symmetric domain in \({\mathbb {R}}^3\), by means of transmutation operators. We then apply this result to construct Beltrami fields, giving a complete system of fields when \(\lambda \) depends only on \(x_3\).
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The research of Vladislav V. Kravchenko was supported by CONACYT, Mexico via the project 284470 and by the Regional Mathematical Center of Southern Federal University Rostov-on-Don, Russia.
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Kravchenko, V.V., Moreira, P.E. & Porter, R.M. Complete Systems of Beltrami Fields Using Complex Quaternions and Transmutation Theory. Adv. Appl. Clifford Algebras 31, 31 (2021). https://doi.org/10.1007/s00006-021-01131-w
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DOI: https://doi.org/10.1007/s00006-021-01131-w
Keywords
- Complete system of solutions
- Beltrami field
- Force-free field
- Runge type theorem
- Monogenic function
- Transmutation operator
- Complex Quaternion