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Complete Systems of Beltrami Fields Using Complex Quaternions and Transmutation Theory

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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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References

  1. Aleksidze, M.: Fundamental Functions in Approximate Solutions of Boundary Value Problems. Nauka, Moscow (1991). (in Russian)

    MATH  Google Scholar 

  2. Alkauskas, G.: Beltrami vector fields with an icosahedral symmetry. J. Geom. Phys. (2020). https://doi.org/10.1016/j.geomphys.2020.103655

  3. Aschwanden, M.: Physics of the solar corona: an introduction with problems and solutions. Springer, Berlin (2005)

  4. Athanasiadis, C., Costakis, G., Stratis, I.: On some properties of Beltrami fields in chiral media. Rep. Math. Phys. 45, 257–271 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  5. Begehr, H., Gilbert, R.P.: Transformations, Transmutations, and Kernel Functions, vol. 1. Longman Scientific & Technical, Harlow (1992)

    MATH  Google Scholar 

  6. Campos, H.M., Kravchenko, V.V., Mendez, L.M.: Complete families of solutions for the Dirac equation using bicomplex function theory and transmutations. Adv. Appl. Clifford Algebras 22, 577–594 (2012)

    Article  MathSciNet  Google Scholar 

  7. Campos, H.M., Kravchenko, V.V., Torba, S.M.: Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane. J. Math. Anal. Appl. 389(2), 1222–1238 (2012)

    Article  MathSciNet  Google Scholar 

  8. Chandrasekhar, S., Kendall, P.C.: On force-free magnetic fields. Astrophys. J. 126, 457 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  9. Colton, D.L.: Solution of Boundary Value Problem by the Method of Integral Operator. Pitman Publishing, London (1976)

    Google Scholar 

  10. Delgado, B.B., Kravchenko, V.V.: A right inverse operator for \(\operatorname{curl}+\lambda \) and applications. Adv. Appl. Clifford Algebras, 29, 40 (2019). https://doi.org/10.1007/s00006-019-0958-z

  11. Delsarte, J.: Sur certaines transformations fonctionnelles rélatives aux équations linéaires aux dérivées partielles du second ordre. C. R. Acad. Sci. Paris 206, 1780–1782 (1938)

  12. Delsarte, J., Lions, J.L.: Transmutations d’opérateurs différentiels dans le domaine complexe. Comment. Math. Helv. 32, 113–128 (1957)

    Article  MathSciNet  Google Scholar 

  13. Doicu, A., Eremin, Yu., Wriedt, T.: Acoustic and Electromagnetic Scattering Analysis. Acad. Press, London (2000)

    MATH  Google Scholar 

  14. Evans, M., Kielich, S., Prigogine, I., Rice, S.: Modern Nonlinear Optics Part III. Wiley, New York (2001)

    Book  Google Scholar 

  15. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)

    Article  MathSciNet  Google Scholar 

  16. Fisanov, V.: Representations of the Beltrami fields in an isotropic chiral medium with the Drude-Born-Fedorov constitutive relations. Russ. Phys. J. 55, 1022–1027 (2013)

    Article  Google Scholar 

  17. Gal, S.G., Sabadini, I.: Quaternionic approximation. Birkhäuser, Basel (2019)

  18. Gilbert, R.P.: Function Theoretic Methods in Partial Differential Equations. Academic Press, New York (1969)

    MATH  Google Scholar 

  19. Grigor’ev, Y.M.: Regular quaternionic polynomials and their properties. Complex Var. Elliptic Equations 62(9), 1343–1363 (2017)

    Article  MathSciNet  Google Scholar 

  20. Grigor’ev, Y.M.: Quaternionic functions and their applications in a viscous fluid flow. Complex Anal. Oper. Theory 12, 491–508 (2018). https://doi.org/10.1007/s11785-017-0715-z

  21. Grigor’ev, Y.M., Naumov, V.V.: Approximation theorems for the Moisil-Theodorescu system. Sib. Math. J. 25, 693–701 (1984). https://doi.org/10.1007/BF00968681

  22. Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and N-Dimensional Space. Springer, New York (2008)

    MATH  Google Scholar 

  23. Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1998)

    MATH  Google Scholar 

  24. Ilyinsky, A.S., Kravcov, V.V., Sveshnikov, A.G.: Mathematical Models of Electrodynamics. Vysshaya Shkola, Moscow (1991)

    Google Scholar 

  25. Jette, A.D.: Force-free magnetic fields in resistive magnetohydrostatics. J. Math. Anal. Appl. 29, 109–122 (1970)

    Article  Google Scholar 

  26. Khmelnytskaya, K.V., Kravchenko, V.V., Rabinovich, V.S.: Quaternionic fundamental solutions for the numerical analysis of electromagnetic scattering problems. Z. Anal. Anwend. 22, 147–166 (2003). https://doi.org/10.4171/ZAA/1136

  27. Kravchenko, V.G., Kravchenko, V.V.: Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics. J. Phys. A Math. Gen. 36(44), 11285–11297 (2003)

  28. Kravchenko, I.V., Kravchenko, V.V., Torba, S.M.: Solution of parabolic free boundary problems using transmuted heat polynomials. Math. Methods Appl. Sci. 42, 5094–5105 (2019)

  29. Kravchenko, V.V.: On Beltrami fields with nonconstant proportionality factor. J. Phys. A Math. Gen. 36(5), 1515–1522 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  30. Kravchenko, V.V.: Applied pseudoanalytic function theory. Front. Math., Birkhäuser, Basel (2009)

  31. Kravchenko, V.V.: Applied Quaternionic Analysis, vol. 28 of Research and Exposition in Mathematics. Heldermann, Lemgo (2003)

  32. Kravchenko, V.V.: Direct and Inverse Sturm–Liouville Problems. A Method of Solution, Frontiers in Mathematics. Birkhäuser, Basel (2020)

  33. Kravchenko, V.V., Morelos, S., Tremblay, S.: Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations. Math. Methods Appl. Sci. 35(6), 704–715, (2012)

  34. Kravchenko, V.V., Navarro, L.J., Torba, S.M.: Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. Appl. Math. Comput. 314, 173–192 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Kravchenko, V.V., Otero, J.A., Torba, S.M.: Analytic approximation of solutions of parabolic partial differential equations with variable coefficients. Adv. Math. Phys. (2017). https://doi.org/10.1155/2017/2947275

  36. Kravchenko, V.V., Torba, S.M.: Transmutations for Darboux transformed operators with applications. J. Phys. A 45(7), 075201 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  37. Kravchenko, V.V., Torba, S.M.: Transmutations and spectral parameter power series in eigenvalue problems. In: Operator Theory, Pseudo-Differential Equations, and Mathematical Physics, Springer, Basel, 209–238, (2013)

  38. Kravchenko, V.V., Vicente-Benítez, V.A.: Transmutation operators and complete systems of solutions for the radial Schrödinger equation. Math. Methods Appl. Sci. 43, 9455–9486, (2020). https://doi.org/10.1002/mma.6322

  39. Kress, R.: A remark on a boundary value problem for force-free fields. Z. Angew. Math. Phys. 28, 715–722 (1977)

    Article  MathSciNet  Google Scholar 

  40. Kress, R.: The treatment of a Neumann boundary value problem for force-free fields by an integral equation method. Proc. R. Soc. Edinb. 71–86 (1978)

  41. Kress, R.: A boundary integral equation method for a Neumann boundary problem for force-free fields. J. Eng. Math. 15, 29–48 (1981)

    Article  MathSciNet  Google Scholar 

  42. Lanczos, C.: Die tensoranalytischen Beziehungen der Diracschen Gleichung. Z. Phys. 57, 447–473 (1929). https://doi.org/10.1007/BF01340274

  43. Mackay, T.G., Lakhtakia, A.: Simultaneous amplification and attenuation in isotropic chiral materials. J. Opt. 18(5), 055104 (2016)

    Article  ADS  Google Scholar 

  44. Marchenko, V.: A Sturm–Liouville Operators and Applications. Birkhäuser, Basel (1986)

    Book  Google Scholar 

  45. Marsh, G.E.: Force-free Magnetic Fields: Solutions, Topology And Applications. World Scientific, Singapore (1996)

    Book  Google Scholar 

  46. Shafranov, V.D.: On magnetohydrodynamical equilibrium configurations. Zh. Eksp. Teor. Fiz. 33(3), 710–722 (1957)

  47. Vänskä, S.: Solving exterior Neumann boundary value problems for Beltrami fields through the Beltrami system. J. Integr. Equations Appl. 22, 591–629 (2010)

    Article  MathSciNet  Google Scholar 

  48. Wiegelmann, T., Sakurai, T.: Solar Force-free Magnetic Fields, vol. 9. Springer Science and Business Media LLC, New York (2012)

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Correspondence to Pablo E. Moreira.

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Communicated by Roldão da Rocha.

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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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