Abstract
We propose an N=1 supersymmetric generalization of the coupled Korteweg-de Vries (KdV) equation and use the Hirota superoperator to obtain a superfield bilinear form of the supersymmetric coupled KdV equation. Using the Hirota method, we obtain explicit expressions for superfield soliton solutions of the supersymmetric coupled KdV equation. We also find superfield one- and two-soliton solutions.
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References
J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, N. J. (1992).
J. Wess and B. Zumino, “Supergauge transformations in four dimensions,” Nucl. Phys. B, 70, 39–50 (1974).
A. Salam and J. Strathdee, “Super-gauge transformations,” Nucl. Phys. B, 76, 477–482 (1974).
P. Di Vecchia and S. Ferrara, “Classical solutions in two-dimensional supersymmetric field theories,” Nucl. Phys. B, 130, 93–104 (1977).
M. Chaichain and P. P. Kulish, “On the method of inverse scattering problem and Bäckhand transformations for supersymmetric equations,” Phys. Lett. B, 78, 413–416 (1978).
A. Mirza and M. ul Hassan, “Bilinearization and soliton solutions of the N=1 supersymmetric coupled dispersionless integrable system,” J. Nonlinear Math. Phys., 24, 107–115 (2017).
A. Mirza and M. Hassan, “Integrability properties of a supersymmetric coupled dispersionless integrable system,” Theor. Math. Phys., 195, 825–833 (2018).
B. A. Kupershmidt, “A super Korteweg-de Vries equation: An integrable system,” Phys. Lett. A, 102, 213–215 (1984).
T. G. Khovanova, “Korteweg-de Vries superequation related to the Lie superalgebra of Neveu-Schwarz-2 string theory,” Theor. Math. Phys., 72, 899–904 (1987).
P. P. Kulish, “An analogue of the Korteweg-de Vries quation for the superconformal algebra,” J. Soviet Math., 41, 970–975 (1988).
P. Mathieu, “Supersymmetric extension of the Korteweg-de Vries equation,” J. Math. Phys., 29, 2499–2506 (1988).
Yu. I. Manin and A. O. Radul, “A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy,” Commun. Math. Phys., 98, 65–77 (1985).
G. H. M. Roelofs and P. H. M. Kersten, “Supersymmetric extensions of the nonlinear Schrodinger equation: Symmetries and coverings,” J. Math. Phys., 33, 2185–2206 (1992).
C. M. Yung, “The N=2 supersymmetric Boussinesq hierarchies,” Phys. Lett. B, 309, 75–84 (1993).
R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Phys. Lett. A, 85, 407–408 (1981).
P. Kersten and J. Krasil’shchik, “Complete integrability of the coupled KdV-mKdV system,” in: Lie Groups, Geometric Structures, and Differential Equations — One Hundred Years after Sophus Lie (Adv. Stud. Pure Math., Vol. 37, T. Morimoto, H. Sato, and K. Yamaguchi, eds.), Math. Soc. Japan, Tokyo (2002), pp. 151–171.
J.-R. Yang and J.-J. Mao, “Soliton solutions of coupled KdV system from Hirota’s bilinear direct method,” Commun. Theor. Phys., 49, 22–26 (2008).
S. Y. Lou, B. Tong, H.-C. Hu, and X.-Y. Tang, “Coupled KdV equations derived from two-layer fluids,” J. Phys. A: Math. Gen., 39, 513–527 (2005); arXiv:nlin/0508029v1 (2005).
A. Sotomayor and A. Restuccia, “Integrability properties of a coupled KdV system and its supersymmetric extension,” J. Phys.: Conf. Ser., 720, 012017 (2016).
A. S. Carstea, “Extension of the bilinear formalism to supersymmetric KdV-type equations,” Nonlinearity, 13, 1645–1656 (2000).
A. S. Carstea, A. Ramani, and B. Grammaticos, “Constructing the soliton solutions for the N=1 supersymmetric KdV hierarchy,” Nonlinearity, 14, 1419–1424 (2001).
B. Grammaticos, A. Ramani, and A. S. Carstea, “Bilinearization and soliton solutions of the N=1 supersymmetric sine-Gordon equation,” J. Physics A: Math. Gen., 34, 4881–4886 (2001).
Q. P. Liu, X.-B. Hu, and M.-X. Zhang, “Supersymmetric modified Korteweg-de Vries equation: Bilinear approach,” Nonlinearity, 18, 1597–1604 (2005).
Q. P. Liu and X.-X. Yang, “Supersymmetric two-boson equation: Bilinearization and solutions,” Phys. Lett. A, 351, 131–135 (2006).
S. Ghosh and D. Sarma, “Bilinearization of supersymmetric KP hierarchies associated with non-trivial flows,” arXiv:nlin/0107019v1 (2001).
S. Ghosh and D. Sarma, “Bilinearization of N=1 supersymmetric modified KdV equations,” Nonlinearity, 16, 411–418 (2002); arXiv:nlin/0202031v1 (2002).
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 1, pp. 14–19, January, 2020.
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Mirza, A., ul Hassan, M. Bilinearization and Soliton Solutions of the Supersymmetric Coupled KdV Equation. Theor Math Phys 202, 11–16 (2020). https://doi.org/10.1134/S004057792001002X
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DOI: https://doi.org/10.1134/S004057792001002X