Abstract
We investigate the standard binary Darboux transformation of the supersymmetric Heisenberg model and calculate multisoliton solutions of the quasi determinants of the supersolitons of the Heisenberg magnet model by iterating the Darboux binary transformation.
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References
L. A. Takhtajan, “Integration of the continuous Heisenberg spin chain through the inverse scattering method,” Phys. Lett. A, 64, 235–237 (1977).
L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach to the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (2007).
S. J. Orfanidis, “SU(n) Heisenberg spin chain,” Phys. Lett. A, 75, 304–306 (1980).
Z. Amjad and B. Haider, “Darboux transformations of supersymmetric Heisenberg magnet model,” J. Phys. Commun., 2, 035019 (2018).
B. Haider and M. Hassan, “The U(N) chiral model and exact multi-solitons,” J. Phys. A: Math. Theor., 41, 255202 (2008).
J. J. C. Nimmo, “Darboux transformation from reductions of KP hierarchy,” arXiv:solv-int/9410001vl (1994).
J. J. C. Nimmo, “Darboux transformations for discrete systems,” Chaos Solitons Fractals, 11, 115–120 (2000).
J. C. Nimmo, C. R. Gilson, and Ya. Ohta, “Applications of Darboux transformations to the self-dual Yang-Mills equations,” Theor. Math. Phys., 122, 239–246 (2000).
W. Oevel and W. Schief, “Darboux theorems and the KP hierarchy,” in: Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (NATO ASI Ser. C: Math. Phys. Sci., Vol. 413, P. A. Clarkson, ed.), Kluwer, Dordrecht (1993), pp. 193–206. 796
B. Haider and M. Hassan, “Binary Darboux transformation for the supersymmetric principal chiral field model,” J. Nonlinear Math. Phys., 18, 557–581 (2011).
C. Rogers and W. K. Schief, Backlund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge Univ. Press, Cambridge (2002).
G. Darboux, “Sur une proposition relative aux equations lineaires,” C. R. Acad. Sci. Paris, 94, 1456–1459 (1882).
V. B. Matveev and M. A. Salle, Darboux Transformations and Soliton, Springer, Berlin (1991).
V. E. Zakharov and A. V. Mikhailov, “On the integrability of classical spinor models in two-dimensional space-time,” Commun. Math. Phys., 74, 21–40.
C. H. Gu and Z. X. Zhou, “On the Darboux matrices of Backlund transformations for AKNS systems,” Lett. Math. Phys., 13, 179–187.
C. Gu, H. Hu, and Z. Zhou, Darboux Transformation in Integrable Systems: Theory and Their Applications to Geometry (Math. Phys. Stud., Vol. 26), Springer, Dordrecht (2005).
B. Haider and M. Hassan, “Quasideterminant multisoliton solutions of a supersymmetric chiral field model in two dimensions,” J. Phys. A: Math. Theor., 43, 035204 (2010).
Th. Th. Voronov, S. Hill, and E. S. Shemyakova, “Darboux transformations for differential operators on the superline,” Russian Math. Surveys, 70, 1173–1175 (2015).
S. Li, E. Shemyakova and Th. Voronov, “Differential operators on the superline, Berezinians, and Darboux transformations,” Lett. Math. Phys., 107, 1689–1714 (2017); arXiv:1605.07286v3 [math.DG] (2016).
I. Gelfand, S. Gelfand, V. Retakh, and R. L. Wilson, “Quasideterminants,” Adv. Math., 193, 56–141 (2005).
I. M. Gel'fand and V. S. Retakh, “Determinants of matrices over noncommutative rings,” Fund. Anal. Appl., 25, 91–102 (1991).
I. Gelfand, V. Retakh, and R. L. Wilson, “Quaternionic quasideterminants and determinants,” in: Lie Groups and Symmetric Spaces: In Memory of F. I. Karpelevich (Amer. Math. Soc. Transl. Ser. 2, Vol. 210, S. G. Gindikin, ed.), Amer. Math. Soc, Providence, R. I. (2003), pp. 111–123; arXiv:math.QA/0206211vl (2002).
I. M. Gel'fand and V. S. Retakh, “A theory of noncommutative determinants and characteristic functions of graphs,” Fund. Anal. Appl, 26, 231–246 (1992).
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 199, No. 3, pp. 357–371, June, 2019.
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Amjad, Z., Haider, B. Binary Darboux Transformations of The Supersymmetric Heisenberg Magnet Model. Theor Math Phys 199, 784–797 (2019). https://doi.org/10.1134/S0040577919060023
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DOI: https://doi.org/10.1134/S0040577919060023