Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras. (English) Zbl 1293.35297
Summary: Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
References:
| [1] | Adler, V. E., Nonlinear chains and Painlevé equations, Physica D, 73, 335-351 (1994) · Zbl 0812.34030 |
| [2] | Adler, V. E., Nonlinear superposition formula for Jordan NLS equations, Phys. Lett. A, 190, 53-58 (1994) · Zbl 0959.35509 |
| [3] | Adler, V. E., Classification of discrete integrable equations (2010), Landau Institute: Landau Institute Chernogolovka, DSc thesis |
| [4] | Antonowicz, M.; Fordy, A. P., Super-extensions of energy-dependent Schrödinger operators, Commun. Math. Phys., 124, 487-500 (1989) · Zbl 0702.35217 |
| [5] | Berezin, F. A., Introduction to Superanalysis (1987), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht/Boston/Lancaster/Tokyo · Zbl 0659.58001 |
| [6] | Bonora, L.; Krivonos, S.; Sorin, A., Towards the construction of \(N = 2\) supersymmetric integrable hierarchies, Nucl. Phys. B, 477, 835-854 (1996) · Zbl 0925.81302 |
| [7] | Chaichian, M.; Kulish, P. P., On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations, Phys. Lett. B, 78, 413-416 (1978) |
| [8] | Chen, H. H.; Liu, C. S., Bäcklund transformation solutions of the Toda lattice equation, J. Math. Phys., 16, 1428-1430 (1975) · Zbl 0311.39003 |
| [9] | Cieśliński, J. L., Algebraic construction of the Darboux matrix revisited, J. Phys. A, 42, 40, 404003 (2009) · Zbl 1193.37096 |
| [10] | Dimakis, A.; Müller-Hoissen, F., An algebraic scheme associated with the non-commutative KP hierarchy and some of its extensions, J. Phys. A, 38, 5453-5505 (2005) · Zbl 1075.81035 |
| [11] | Dimakis, A.; Müller-Hoissen, F., Solutions of matrix NLS systems and their discretizations: A unified treatment, Inverse Probl., 26, 095007 (2010) · Zbl 1205.35291 |
| [12] | Dimakis, A.; Müller-Hoissen, F., Binary Darboux transformations in bidifferential calculus and integrable reductions of vacuum Einstein equations, SIGMA, 9, 009 (2013), 31 pages · Zbl 1384.37082 |
| [13] | Drinfelʼd, V.; Sokolov, V. V., Lie algebras and equations of Korteweg-de Vries type, Sov. J. Math., 30, 1975-2036 (1985) · Zbl 0578.58040 |
| [14] | Evans, J.; Hollowood, T., Supersymmetric Toda field theories, Nucl. Phys. B, 352, 723-768 (1991) |
| [15] | Gelfand, I. M.; Gelfand, S. I.; Retakh, V. S.; Wilson, R. L., Quasideterminants, Adv. Math., 25, 56-141 (2005) · Zbl 1079.15007 |
| [16] | Grisaru, M. T.; Penati, S., An integrable noncommutative version of the sine-Gordon system, Nucl. Phys. B, 655, 250-276 (2003) · Zbl 1009.81074 |
| [17] | Gürses, M.; Oğuz, Ö., A super soliton connection, Lett. Math. Phys., 11, 235-246 (1986) · Zbl 0608.35072 |
| [18] | Hamanaka, M.; Toda, K., Towards noncommutative integrable systems, Phys. Lett. A, 316, 77-83 (2003) · Zbl 1031.37046 |
| [19] | Ikeda, K., A supersymmetric extension of the Toda lattice hierarchy, Lett. Math. Phys., 14, 321-328 (1987) · Zbl 0729.35131 |
| [20] | Inami, T.; Kanno, H., Lie superalgebraic approach to super Toda lattice and generalized super KdV equations, Commun. Math. Phys., 136, 519-542 (1991) · Zbl 0738.35087 |
| [22] | Konstantinou-Rizos, S.; Mikhailov, A. V., Yang-Baxter maps and finite reduction groups with degenerated orbits · Zbl 1276.81069 |
| [25] | Kupershmidt, B. A., A super Korteweg-de Vries equation, Phys. Lett. A, 102, 213-215 (1984) |
| [26] | Lechtenfeld, O.; Mazzanti, L.; Penati, S.; Popov, A. D.; Tamassia, L., Integrable noncommutative sine-Gordon model, Nucl. Phys. B, 705, 477-503 (2005) · Zbl 1119.81331 |
| [27] | (Leites, D. A., Seminar on Supersymmetry (2011), Independent University Press: Independent University Press Moscow), (in Russian) |
| [28] | Levi, D., Nonlinear differential difference equations as Bäcklund transformations, J. Phys. A, 14, 1083-1098 (1981) · Zbl 0465.35081 |
| [29] | Li, C. X.; Nimmo, J. J.C., Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. R. Soc. A, Math. Phys. Eng. Sci., 466, 2471-2493 (2010) · Zbl 1194.35374 |
| [30] | Liu, Q. P.; Mañas, M., Darboux transformations for the Manin-Radul supersymmetric KdV equation, Phys. Lett. B, 394, 337-342 (1997) |
| [31] | Liu, Q. P.; Mañas, M., Crum transformation and Wronskian type solutions for supersymmetric KdV equation, Phys. Lett. B, 396, 133-140 (1997) |
| [32] | Manin, Yu. I.; Radul, A. O., A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Commun. Math. Phys., 98, 65-77 (1985) · Zbl 0607.35075 |
| [33] | Mathieu, P., Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys., 29, 2499 (1988) · Zbl 0665.35076 |
| [34] | Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons, Springer Ser. Nonlin. Dyn., vol. 4 (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0744.35045 |
| [35] | Mikhailov, A. V., Integrability of supersymmetric generalization of the classical Chiral model in two-dimensional space-time, JETP Lett., 28, 554-558 (1978) |
| [36] | Mikhailov, A. V.; Wang, J. P.; Xenitidis, P., Cosymmetries and Nijenhuis recursion operators for difference equations, Nonlinearity, 24, 2079-2097 (2011) · Zbl 1416.65521 |
| [37] | Morosi, C.; Pizzocchero, L., A fully supersymmetric AKNS hierarchy, Commun. Math. Phys., 176, 353-381 (1996) · Zbl 0852.35127 |
| [38] | Oevel, W.; Popowicz, Z., The bi-Hamiltonain structure of fully supersymmetric Korteweg-de Vries systems, Commun. Math. Phys., 139, 441-460 (1991) · Zbl 0742.35063 |
| [39] | Olver, P. J.; Sokolov, V. V., Non-abelian integrable systems of the derivative nonlinear Schrödinger type, Inverse Probl., 14, 6, L5-L8 (1998) · Zbl 0915.35097 |
| [40] | Olver, P. J.; Sokolov, V. V., Integrable evolution equations on associative algebras, Commun. Math. Phys., 193, 2, 245-268 (1998) · Zbl 0908.35124 |
| [41] | Olshanetsky, M. A., Supersymmetric two-dimensional Toda lattice, Commun. Math. Phys., 88, 63-76 (1983) · Zbl 0562.35080 |
| [42] | Papageorgiou, V. G.; Tongas, A. G., Yang-Baxter maps and multi-field integrable lattice equations, J. Phys. A, 40, 12677-12690 (2007) · Zbl 1155.35466 |
| [43] | Pickering, A.; Zhu, Z. N., New integrable lattice hierarchies, Phys. Lett. A, 349, 439-445 (2006) · Zbl 1195.37040 |
| [44] | Popowicz, Z., The extended supersymmetrization of the nonlinear Schrödinger equation, Phys. Lett. A, 194, 375-379 (1994) · Zbl 0961.37515 |
| [45] | Shabat, A. B., Inverse scattering problem for a system of differential equations, Funct. Anal. Appl., 9, 244-247 (1975) · Zbl 0352.34020 |
| [46] | Toda, M., Theory of Nonlinear Lattices, Springer Ser. Solid-State Sci., vol. 20 (1981), Springer: Springer Berlin, Heidelberg · Zbl 0465.70014 |
| [47] | Zakharov, V. E.; Manakov, S.; Novikov, S.; Pitaevskii, L., Theory of Solitons: The Inverse Scattering Method (1984), Plenum: Plenum New York · Zbl 0598.35002 |
| [48] | Zakharov, V. E.; Mikhailov, A. V., On the integrability of classical spinor models in two-dimensional space-time, Commun. Math. Phys., 74, 21-40 (1980) |
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