The Heisenberg supermagnet model is an important supersymmetric integrable system which is the super extension of the Heisenberg ferromagnet model. By virtue of introducing the general auxiliary matrix variables, we construct a new $(2 + 1)$ -dimensional generalized integrable Heisenberg supermagnet models under two constraints. Meanwhile, we establish their corresponding gauge equivalent counterparts. Moreover, we derive new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

Keywords:

AMSC: 17B65, 17B80, 37K10

References

1. F. H. De Leeuw, R. van den Doel and U. Enz, Dynamic properties of magnetic domain walls and magnetic bubbles, Rep. Prog. Phys. 43 (1980) 689–783. Web of Science, Google Scholar
2. M. Kruczenski, Spin chains and string theory, Phys. Rev. Lett. 93 (2004) 161602. Web of Science, Google Scholar
3. J. S. Hazboun, Constructing an explicit AdS/CFT correspondence with Cartan geometry, Nucl. Phys. B 929 (2018) 254–265. Web of Science, Google Scholar
4. L. Martina, O. K. Pashaev and G. Soliani, Bright solitons as black holes, Phys. Rev. D 58 (1998) 084025. Web of Science, Google Scholar
5. M. Lakshmanan, Continuum spin system as an exactly solvable dynamical system, Phys. Lett. A 61 (1977) 53–54. Web of Science, Google Scholar
6. L. A. Takhtajan, Integration of the continuous Heisenberg spin chain through the inverse scattering method, Phys. Lett. A 64 (1977) 235–237. Web of Science, Google Scholar
7. Y. Ishimori, Article navigation multi-vortex solutions of a two-dimensional nonlinear wave equation, Prog. Theor. Phys. 72 (1984) 33–37. Google Scholar
8. R. Myrzakulov, G. K. Mamyrbekova, G. N. Nugmanova and M. Lakshmanan, Integrable $(2 + 1)$ -dimensional spin models with self-consistent potentials, Symmetry 7 (2015) 1352–1375. Web of Science, Google Scholar
9. L. Wulff, Integrability of the superstring in $A d S_{3} \times S^{2} \times S^{2} \times T^{3}$ , J. Phys. A 50 (2017) 23LT01. Google Scholar
10. D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly $γ$ -deformed $N = 4$ supersymmetric Yang–Mills theory as an integrable conformal field theory, Phys. Rev. Lett. 120 (2018) 111601. Web of Science, Google Scholar
11. P. Padmanabhan, S. Rey, D. Teixeirad and D. Trancanelli, Supersymmetric many-body systems from partial symmetries integrability, localization and scrambling, J. High Energy Phys. 05 (2017) 136. Web of Science, Google Scholar
12. K. Hosomichi, S. Lee and T. Okuda, Supersymmetric vortex defects in two dimensions, J. High Energy Phys. 01 (2018) 033. Web of Science, Google Scholar
13. P. Mathieu, Supersymmetric extension of the Korteweg–de Vries equation, J. Math. Phys. 29 (1988) 2499–2506. Web of Science, Google Scholar
14. S. Bellucci, E. Ivanov, S. Krivonos and A. Pichugin, $N = 2$ super Boussinesq hierarchy: Lax pairs and conservation laws, Phys. Lett. B 312 (1993) 463–470. Web of Science, Google Scholar
15. D. Sarma, Constructing a supersymmetric integrable system from the Hirota method in superspace, Nucl. Phys. B 681 (2004) 351–358. Web of Science, Google Scholar
16. Yu. Manin and A. Radul, A supersymmetric extension of the Kadomtsev–Petviashvili hierarchy, Commun. Math. Phys. 98 (1985) 65–77. Web of Science, Google Scholar
17. V. Kac and E. Medina, On the super-KP hierachy, Lett. Math. Phys. 37 (1996) 435–448. Web of Science, Google Scholar
18. A. LeClair, Supersymmetric KP hierarchy: Free field construction, Nucl. Phys. B 314 (1989) 425–438. Web of Science, Google Scholar
19. L. M. Alonso and E. M. Reus, Tau-function formalism for supersymmetric KP hierarchies, J. Math. Phys. 36 (1995) 4898–4913. Web of Science, Google Scholar
20. Z. Popowicz, The extended supersymmetrization of the nonlinear Schrödinger equation, Phys. Lett. A 194 (1994) 375–379. Web of Science, Google Scholar
21. V. G. Makhankov and O. K. Pashaev, Continual classical Heisenberg models defined on graded $s u (2, 1)$ and $s u (3)$ algebras, J. Math. Phys. 33 (1992) 2923–2936. Web of Science, Google Scholar
22. J. F. Guo, S. K. Wang, K. Wu, Z. W. Yan and W. Z. Zhao, Integrable higher order deformations of Heisenberg supermagnetic model, J. Math. Phys. 50 (2009) 113502. Web of Science, Google Scholar
23. Z. W. Yan, M. N. Zhang and J. F. Cui, Higher-order inhomogeneous generalized Heisenberg supermagnetic model, Chin. Phys. Lett. 35 (2018) 050201. Web of Science, Google Scholar
24. V. G. Makhankov and O. K. Pashaev, Supersymmetric integrable $s u (2 / 1)$ Heisenberg models, Phys. Lett. A 141 (1989) 285–288. Web of Science, Google Scholar
25. M. Saha and A. Roy Chowdhury, Supersymmetric integrable systems in $(2 + 1)$ dimensions and their Bäcklund transformation, Int. J. Theor. Phys. 38 (1999) 2037–2047. Web of Science, Google Scholar
26. Z. W. Yan, M. L. Li, K. Wu and W. Z. Zhao, Fermionic covariant prolongation structure theory for multidimensional super nonlinear evolution equation, J. Math. Phys. 54 (2013) 033506. Web of Science, Google Scholar
27. Z. W. Yan, Tala, F. Chen, T. R. Liu and J. M. Han, $(2 + 1)$ -dimensional supersymmetric integrable equations, Int. J. Geom. Methods Mod. Phys. 15 (2018) 1850114. Link, Web of Science, Google Scholar
28. Z. W. Yan, M. R. Chen, K. Wu and W. Z. Zhao, $(2 + 1)$ -dimensional integrable Heisenberg supermagnet model, J. Phys. Soc. Jpn. 81 (2012) 094006. Google Scholar
29. Z. W. Yan, On the Heisenberg supermagnet model in $(2 + 1)$ -dimensions, Z. Naturforsch A 72 (2017) 331–337. Web of Science, Google Scholar
30. L. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994) 295–315. Web of Science, Google Scholar
31. V. Abramov, Super 3-Lie algebras induced by super Lie algebras, Adv. Appl. Clifford Algebr. 27 (2017) 9–16, arXiv: 1410.5923. Web of Science, Google Scholar
32. V. Abramov, Quantum super Nambu bracket of cubic supermatrices and 3-Lie superalgebra, Adv. Appl. Clifford Algebr., published online, arXiv:1802.05881. Google Scholar
33. M. R. Chen, S. K. Wang, K. Wu and W. Z. Zhao, Infinite-dimensional 3-algebra and integrable system, J. High Energy Phys. 12 (2012) 030. Web of Science, Google Scholar
34. V. Abramov, Matrix 3-Lie superalgebras and BRST supersymmetry, Int. J. Geom. Methods Mod. Phys. 14 (2017) 1750160, arXiv: 1707.01270. Link, Web of Science, Google Scholar