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Articles

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

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Pages 429-452 | Received 18 Jun 2019, Accepted 03 Nov 2019, Published online: 04 May 2020

References

  • S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, (Oxford University Press, New York, 1995).
  • N. Benes, A. Kasman, and K. Young, On Decompositions of the KdV 2-Soliton, Journal of Nonlinear Science, 16 (2) (2006), 179–200. doi: 10.1007/s00332-005-0709-2
  • S. Carillo, M. Lo Schiavo, E. Porten and C. Schiebold, A novel noncommutative KdV-type equation, its recursion operator, and solitons, J. Math. Phys., 59 (2018), 043501. doi: 10.1063/1.5027481
  • L. Chen, Definition of Determinant and Cramer Solutions over the Quaternion Field, Acta Mathematica Sinica, New Series, 7 (2) (1991), 171–180.
  • J.H. Conway and D.A. Smith, On Quaternions and Octonions, AK Peters/ CRC Press (2003). doi: 10.1201/9781439864180
  • P. Etingof, I. Gelfand, and V. Retakh, Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Mathematical Research Letters, 4 (3) (1997) 413–425. doi: 10.4310/MRL.1997.v4.n3.a10
  • J.D. Gibbon, A quaternionic structure in the three-dimensional Euler and ideal magnetohydrodynamics equation, Physica D, 166 (2002), 17–28. doi: 10.1016/S0167-2789(02)00434-7
  • P.R. Girard, Quaternions, Clifford algebras and relativistic physics, Translated from the 2004 French original. Birkhäuser Verlag, Basel, 2007.
  • M. Hamanaka and H. Okabe, Soliton Scattering in Noncommutative Spaces, Theor. Math. Phys., (2018) 197: 1451. doi: 10.1134/S0040577918100045
  • W.R. Hamilton, On Quaternions; or on a new System of Imaginaries in Algebra, Philosophical Magazine, vol. xxv (1844), pp. 10–13.
  • D.D. Holm, Geometric Mechanics: Part 2, Rotating, Translating and Rolling (Pt. II), Imperial College Press (2008).
  • S. Huang, An operator method for finding exact solutions to vector Korteweg-de Vries equations, Journal of Mathematical Physics, 44 (3) (2003) 1357–1388. doi: 10.1063/1.1544414
  • A. Kasman, Glimpses of Soliton Theory, Vol. 54, American Mathematical Society, 2010.
  • A. Kasman, On factoring an operator using elements of its kernel, Communications in Algebra, 45 (4) (2017), 1443–1451. doi: 10.1080/00927872.2016.1175612
  • K. Kenatani, Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, AK Peters (2015).
  • D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39 (1895), no. 240, 422–443. doi: 10.1080/14786449508620739
  • K.I. Kou, W.-K. Liu, and Y.-H. Xia, Solve the linear quaternion-valued differential equations having multiple eigenvalues, Journal of Mathematical Physics, 60 (2019), 023510.
  • I.I. Kyrchei, Cramer’s rule for quaternionic systems of linear equations, Journal of Mathematical Sciences, 155 (6) (2008), 839–858. doi: 10.1007/s10958-008-9245-6
  • S. Leo, G. Ducati, and C. Nishi, Quaternionic potentials in non-relativistic quantum mechanics, J. Phys. A: Math. Gen., 35 (26) (2001), 5411. doi: 10.1088/0305-4470/35/26/305
  • S. Leo and G. Ducati, Solving simple quaternionic differential equations, J. Math. Phys., 44 (5) (2003), 2224. doi: 10.1063/1.1563735
  • Y.C. Li, Simple explicit formulae for finite time blow up solutions to the complex KdV equation, Chaos Solitons and Fractals, 39 (2009), 369–372. doi: 10.1016/j.chaos.2007.04.015
  • R. Palais, Symmetries of Solitons, Bulletin (New Series) of the American Mathematical Society, Volume 34, Number 4, October 1997, Pages 339–403.
  • L.D. Paniak, Exact Noncommutative KP and KdV Multi-solitons, https://arxiv.org/abs/hep-th/0105185
  • A. Serna, Quaternion-valued Rational Solutions to the KdV Equation, Master’s Thesis, College of Charleston, 2019.
  • P. Wilczynski, Quaternionic-valued ordinary differential equations. II. Coinciding sectors, J. Differ. Equations, 252 (2012), 4503–4528. doi: 10.1016/j.jde.2012.01.005
  • N. Zabusky and M. Kruskal, Interaction of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett., 15 (1965), 240. doi: 10.1103/PhysRevLett.15.240
  • F. Zhang, Quaternions and matrices of quaternions, Linear Algebra and Its Applications, 251 (1997), 21–57.

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