Advertisement

Acta Applicandae Mathematicae

, Volume 160, Issue 1, pp 129–167 | Cite as

On the (Non)Removability of Spectral Parameters in \(\mathbb{Z}_{2}\)-Graded Zero-Curvature Representations and Its Applications

  • Arthemy V. Kiselev
  • Andrey O. KrutovEmail author
Article
  • 34 Downloads

Abstract

We generalise to the \(\mathbb{Z}_{2}\)-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over \(\mathbb{Z}_{2}\)-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Frölicher–Nijenhuis bracket.

Keywords

Zero-curvature representation Spectral parameter Removability Supersymmetry Korteweg–de Vries equation Gardner’s deformation Frölicher–Nijenhuis bracket 

Mathematics Subject Classification (2010)

35Q53 37K25 58J72 58A50 

Notes

Acknowledgements

The authors are grateful to I.S. Krasil’shchik, D.A. Leites, M. Marvan, M.A. Nesterenko, P.J. Olver, W.M. Seiler, and A.M. Verbovetsky for helpful correspondence and constructive criticisms. The authors thank P. Mathieu for his attention to this work; the authors are grateful to the anonymous referees for remarks and advice.

This research was done in part while the first author was visiting at the MPIM (Bonn) and the second author was visiting at Utrecht University and New York University Abu Dhabi; the hospitality and support of these institutions are gratefully acknowledged. The research of the first author was partially supported by JBI RUG project 106552 (Groningen); the second author was supported by ISPU scholarship for young scientists and WCMCS post-doctoral fellowship.

References

  1. 1.
    Labelle, P., Mathieu, P.: A new \(N=2\) supersymmetric Korteweg-de Vries equation. J. Math. Phys. 32(4), 923–927 (1991).  https://doi.org/10.1063/1.529351 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Hussin, V., Kiselev, A.V., Krutov, A.O., Wolf, T.: \(N=2\) supersymmetric \(a=4\)-Korteweg–de Vries hierarchy derived via Gardner’s deformation of Kaup–Boussinesq equation. J. Math. Phys. 51(8), 083507 (2010).  https://doi.org/10.1063/1.3447731. arXiv:0911.2681 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  3. 3.
    Kiselev, A.V., Krutov, A.O.: Gardner’s deformations of the graded Korteweg–de Vries equations revisited. J. Math. Phys. 53(10), 103511 (2012).  https://doi.org/10.1063/1.4754288. arXiv:1108.2211 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  4. 4.
    Miura, R.M.: Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202–1204 (1968).  https://doi.org/10.1063/1.1664700 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Miura, R.M., Gardner, C.S., Kruskal, M.D.: Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys. 9, 1204–1209 (1968).  https://doi.org/10.1063/1.1664701 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Laberge, C.A., Mathieu, P.: \(N=2\) superconformal algebra and integrable \(\mathrm{O}(2)\) fermionic extensions of the Korteweg-de Vries equation. Phys. Lett. B 215(4), 718–722 (1988).  https://doi.org/10.1016/0370-2693(88)90048-2 MathSciNetGoogle Scholar
  7. 7.
    Mathieu, P.: Open problems for the super KdV equations. In: Bäcklund and Darboux Transformations. The Geometry of Solitons, Halifax, NS, 1999. CRM Proc. Lecture Notes, vol. 29, pp. 325–334. Am. Math. Soc., Providence (2001). arXiv:math-ph/0005007 Google Scholar
  8. 8.
    Kiselev, A.V., Krutov, A.O.: Gardner’s deformation of the Krasil’shchik–Kersten system. J. Phys. Conf. Ser. 621(1), 012007 (2015).  https://doi.org/10.1088/1742-6596/621/1/012007. Group Analysis of Differential Equations and Integrable Systems (GADEISVII) (June 15–19, 2014, Larnaca, Cyprus), arXiv:1409.6688 [nlin.SI] Google Scholar
  9. 9.
    Krutov, A.O.: Deformations of equations and structures in nonlinear problems of mathematical physics. Ph.D. Thesis. University of Groningen, JBI, The Netherlands (2014) Google Scholar
  10. 10.
    Wahlquist, H.D., Estabrook, F.B.: Prolongation structures of nonlinear evolution equations. J. Math. Phys. 16, 1–7 (1975).  https://doi.org/10.1063/1.522396 MathSciNetzbMATHGoogle Scholar
  11. 11.
    Estabrook, F.B., Wahlquist, H.D.: Prolongation structures of nonlinear evolution equations. II. J. Math. Phys. 17(7), 1293–1297 (1976).  https://doi.org/10.1063/1.523056 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kupershmidt, B.A.: Deformations of integrable systems. Proc. R. Ir. Acad. Sect. A 83(1), 45–74 (1983) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Fordy, A.P.: Projective representations and deformations of integrable systems. Proc. R. Ir. Acad. Sect. A 83(1), 75–93 (1983) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kiselev, A.V.: Algebraic properties of Gardner’s deformations for integrable systems. Theor. Math. Phys. 152(1), 963–976 (2007).  https://doi.org/10.1007/s11232-007-0081-5. arXiv:nlin/0610072 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  15. 15.
    Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer Series in Soviet Mathematics. Springer, Berlin (1987), x+592 pp. zbMATHGoogle Scholar
  16. 16.
    Zakharov, V., Shabat, A.: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl. 13(3), 166–174 (1979).  https://doi.org/10.1007/BF01077483 MathSciNetzbMATHGoogle Scholar
  17. 17.
    González-López, A., Kamran, N., Olver, P.J.: Lie algebras of vector fields in the real plane. Proc. Lond. Math. Soc. 64(2), 339–368 (1992) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Popovych, R.O., Boyko, V.M., Nesterenko, M.O., Lutfullin, M.W.: Realizations of real low-dimensional Lie algebras. J. Phys. A, Math. Gen. 36(26), 7337–7360 (2003).  https://doi.org/10.1088/0305-4470/36/26/309. arXiv:math-ph/0301029 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Shchepochkina, I.M.: How to realize a Lie algebra by vector fields. Theor. Math. Phys. 147(3), 821–838 (2006).  https://doi.org/10.1007/s11232-006-0078-5. arXiv:math/0509472 [math.RT] MathSciNetzbMATHGoogle Scholar
  20. 20.
    Roelofs, M.: Prolongation structures of supersymmetric systems. Ph.D. Thesis. University of Twente, Enschede, The Netherlands (1993) Google Scholar
  21. 21.
    Nesterenko, M.: Realizations of Lie algebras. Phys. Part. Nucl. Lett. 11(7), 987–989 (2014).  https://doi.org/10.1134/S1547477114070346 Google Scholar
  22. 22.
    Marvan, M.: On the horizontal gauge cohomology and nonremovability of the spectral parameter. Acta Appl. Math. 72(1–2), 51–65 (2002).  https://doi.org/10.1023/A:1015218422059 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Marvan, M.: On the spectral parameter problem. Acta Appl. Math. 109(1), 239–255 (2010).  https://doi.org/10.1007/s10440-009-9450-4. arXiv:0804.2031 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sakovich, S.Y.: On zero-curvature representations of evolution equations. J. Phys. A, Math. Gen. 28(10), 2861–2869 (1995).  https://doi.org/10.1088/0305-4470/28/10/016 MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sakovich, S.Y.: Cyclic bases of zero-curvature representations: five illustrations to one concept. Acta Appl. Math. 83(1–2), 69–83 (2004).  https://doi.org/10.1023/B:ACAP.0000035589.61486.a7. arXiv:nlin/0212019 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  26. 26.
    Das, A., Huang, W.-J., Roy, S.: Zero curvature condition of \(\mathrm{OSp}(2/2)\) and the associated supergravity theory. Int. J. Mod. Phys. A 7(18), 4293–4311 (1992).  https://doi.org/10.1142/S0217751X92001915 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Igonin, S., Kersten, P.H.M., Krasil’shchik, I.: On symmetries and cohomological invariants of equations possessing flat representations. Differ. Geom. Appl. 19(3), 319–342 (2003).  https://doi.org/10.1016/S0926-2245(03)00049-4. arXiv:math/0301344 [math.DG] MathSciNetzbMATHGoogle Scholar
  28. 28.
    Igonin, S., Krasil’shchik, J.: On one-parametric families of Bäcklund transformations. In: Lie Groups, Geometric Structures and Differential Equations—One Hundred Years After Sophus Lie, Kyoto/Nara, 1999. Adv. Stud. Pure Math., vol. 37, pp. 99–114. Math. Soc. Japan, Tokyo (2002). arXiv:nlin/0010040 [nlin.SI] Google Scholar
  29. 29.
    Krasil’shchik, I.S.: Algebras with flat connections and symmetries of differential equations. In: Lie Groups and Lie Algebras. Math. Appl., vol. 433, pp. 407–424. Kluwer Academic, Dordrecht (1998) Google Scholar
  30. 30.
    Kiselev, A.V.: On the Bäcklund autotransformation for the Liouville equation. Vestn. Mosk. Univ., Ser. III Fiz. Astronom. 6, 22–26 (2002) zbMATHGoogle Scholar
  31. 31.
    Baran, K., Marvan, M.: A conjecture on nonlocal terms of recursion operators. Fundam. Prikl. Mat. 12(7), 23–33 (2006).  https://doi.org/10.1007/s10958-008-9030-6 zbMATHGoogle Scholar
  32. 32.
    Tian, K., Liu, Q.P.: Supersymmetric fifth order evolution equations. In: Nonlinear and Modern Mathematical Physics, Beijing, China, July 15–21, 2009. AIP Conf. Proc., vol. 1212, pp. 81–88 (2010).  https://doi.org/10.1063/1.3367084 Google Scholar
  33. 33.
    Tian, K., Wang, J.P.: Symbolic representation and classification of \(N=1\) supersymmetric evolutionary equations. Stud. Appl. Math. 138(4), 467–498 (2017).  https://doi.org/10.1111/sapm.12163. arXiv:1607.03947 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  34. 34.
    Dodd, R., Fordy, A.: The prolongation structures of quasipolynomial flows. Proc. R. Soc. Lond. Ser. A 385(1789), 389–429 (1983) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Dodd, R.K., Fordy, A.P.: Prolongation structures of complex quasipolynomial evolution equations. J. Phys. A 17(16), 3249–3266 (1984) MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kaup, D.J.: The Estabrook–Wahlquist method with examples of application. Physica D 1(4), 391–411 (1980).  https://doi.org/10.1016/0167-2789(80)90020-2 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sasaki, R.: Soliton equations and pseudospherical surfaces. Nucl. Phys. B 154(2), 343–357 (1979).  https://doi.org/10.1016/0550-3213(79)90517-0 MathSciNetGoogle Scholar
  38. 38.
    Berezin, F.A.: Introduction to superanalysis. In: Mathematical Physics and Applied Mathematics. Reidel, Dordrecht (1987) Google Scholar
  39. 39.
    Deligne, P., Etingof, P., Freed, D.S., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E. (eds.): Quantum Fields and Strings: A Course for Mathematicians, vols. 1–2. AMS/Institute for Advanced Study (IAS), Providence/Princeton (1999). Vol. 1: xxii+723 pp.; Vol. 2: pp. i–xxiv and 727–1501 pp. zbMATHGoogle Scholar
  40. 40.
    Kersten, P., Krasil’shchik, I., Verbovetsky, A.: Hamiltonian operators and \(\ell ^{*}\)-coverings. J. Geom. Phys. 50(1–4), 273–302 (2004).  https://doi.org/10.1016/j.geomphys.2003.09.010. arXiv:math/0304245 [math.DG] MathSciNetzbMATHGoogle Scholar
  41. 41.
    Krasil’shchik, J., Verbovetsky, A.: Geometry of jet spaces and integrable systems. J. Geom. Phys. 61(9), 1633–1674 (2011).  https://doi.org/10.1016/j.geomphys.2010.10.012. arXiv:1002.0077 [math.DG] MathSciNetzbMATHGoogle Scholar
  42. 42.
    Vinogradov, A.M.: The \({\mathcal{C}}\)-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. J. Math. Anal. Appl. 100(1), 1–40 (1984).  https://doi.org/10.1016/0022-247X(84)90071-4 MathSciNetzbMATHGoogle Scholar
  43. 43.
    Vinogradov, A.M.: The \({\mathcal{C}}\)-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory. J. Math. Anal. Appl. 100(1), 41–129 (1984).  https://doi.org/10.1016/0022-247X(84)90072-6 MathSciNetzbMATHGoogle Scholar
  44. 44.
    Bocharov, A.V., Chetverikov, V.N., Duzhin, S.V., Khor’kova, N.G., Krasil’shchik, I.S., Samokhin, A.V., Torkhov, Y.N., Verbovetsky, A.M., Vinogradov, A.M.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182. Am. Math. Soc., Providence (1999), xiv+333 pp. zbMATHGoogle Scholar
  45. 45.
    Nestruev, J.: Smooth Manifolds and Observables. Graduate Texts in Mathematics, vol. 220. Springer, New York (2003), xiv+222 pp. zbMATHGoogle Scholar
  46. 46.
    Leites, D.A.: Introduction to the theory of supermanifolds. Russ. Math. Surv. 35(1), 1–64 (1980).  https://doi.org/10.1070/RM1980v035n01ABEH001545 MathSciNetzbMATHGoogle Scholar
  47. 47.
    Krasil’shchik, I.S., Kersten, P.H.M.: Symmetries and recursion operators for classical and supersymmetric differential equations. In: Mathematics and Its Applications, vol. 507. Kluwer Academic, Dordrecht (2000), xvi+384 pp. Google Scholar
  48. 48.
    Kiselev, A.V.: The twelve lectures in the (non)commutative geometry of differential equations. Preprint IHÉS/M/12/13 (Bures-sur-Yvette, France) (2012). http://preprints.ihes.fr/2012/M/M-12-13.pdf
  49. 49.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Graduate Texts in Mathematics, vol. 107. Springer, New York (1993).  https://doi.org/10.1007/978-1-4612-4350-2 zbMATHGoogle Scholar
  50. 50.
    Kiselev, A.V., Krutov, A.O.: Non-Abelian Lie algebroids over jet spaces. J. Nonlinear Math. Phys. 21(2), 188–213 (2014).  https://doi.org/10.1080/14029251.2014.900992. arXiv:1305.4598 [math.DG] MathSciNetGoogle Scholar
  51. 51.
    Goldschmidt, H.: Existence theorems for analytic linear partial differential equations. Ann. Math. (2) 86, 246–270 (1967) MathSciNetzbMATHGoogle Scholar
  52. 52.
    Goldschmidt, H.: Integrability criteria for systems of nonlinear partial differential equations. J. Differ. Geom. 1, 269–307 (1967) MathSciNetzbMATHGoogle Scholar
  53. 53.
    Kiselev, A.V.: The geometry of variations in Batalin–Vilkovisky formalism. J. Phys. Conf. Ser. 474(1), 012024 (2013). arXiv:1312.1262 [math-ph] Google Scholar
  54. 54.
    Kiselev, A.V.: The calculus of multivectors on noncommutative jet spaces. J. Geom. Phys. 130, 130–167 (2018).  https://doi.org/10.1016/j.geomphys.2018.03.022. arXiv:1210.0726 [math.DG] MathSciNetzbMATHGoogle Scholar
  55. 55.
    Krasil’shchik, I.S., Vinogradov, A.M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 15(1–2), 161–209 (1989).  https://doi.org/10.1007/BF00131935 MathSciNetzbMATHGoogle Scholar
  56. 56.
    Manin, Y.: Holomorphic supergeometry and Yang-Mills superfields. J. Sov. Math. 30(2), 1927–1975 (1985).  https://doi.org/10.1007/BF02105859 zbMATHGoogle Scholar
  57. 57.
    Kiselev, A.V., Hussin, V.: Hirota’s virtual multisoliton solutions of \(N=2\) supersymmetric Korteweg-de Vries equations. Theor. Math. Phys. 159(3), 490–501 (2009).  https://doi.org/10.1007/s11232-009-0071-x. arXiv:0810.0930 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  58. 58.
    Kiselev, A.V., Wolf, T.: Classification of integrable super-systems using the SsTools environment. Comput. Phys. Commun. 177(3), 315–328 (2007).  https://doi.org/10.1016/j.cpc.2007.02.113. arXiv:nlin/0609065 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  59. 59.
    Kiselev, A.V., Golovko, V.A.: On non-Abelian coverings over the Liouville equation. Acta Appl. Math. 83(1–2), 25–37 (2004).  https://doi.org/10.1023/B:ACAP.0000035587.01597.e0 MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kaup, D.J.: A higher-order water-wave equation and the method for solving it. Prog. Theor. Phys. 54(2), 396–408 (1975) MathSciNetzbMATHGoogle Scholar
  61. 61.
    Borisov, A.B., Pavlov, M.P., Zykov, S.A.: Proliferation scheme for Kaup–Boussinesq system. Physica D 152/153, 104–109 (2001).  https://doi.org/10.1016/S0167-2789(01)00163-4 MathSciNetzbMATHGoogle Scholar
  62. 62.
    Catalano Ferraioli, D., de Oliveira Silva, L.A.: Nontrivial 1-parameter families of zero-curvature representations obtained via symmetry actions. J. Geom. Phys. 94, 185–198 (2015).  https://doi.org/10.1016/j.geomphys.2015.04.001 MathSciNetzbMATHGoogle Scholar
  63. 63.
    Kiselev, A.V., Krutov, A., Wolf, T.: Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment. In: Euler, N. (ed.) Nonlinear Systems and Their Remarkable Mathematical Structures. CRC Press, Boca Raton (2018). arXiv:1805.12397 [nlin.SI] Google Scholar
  64. 64.
    Marvan, M.: On zero-curvature representations of partial differential equations. In: Differential Geometry and Its Applications, Opava, 1992. Math. Publ., vol. 1, pp. 103–122. Silesian Univ. Opava, Opava (1993) Google Scholar
  65. 65.
    Marvan, M.: A direct procedure to compute zero-curvature representations. The case \(\mathfrak{sl}_{2}\). In: The International Conference on Secondary Calculus and Cohomological Physics, Moscow, 1997. Diffiety Inst. Russ. Acad. Nat. Sci., Pereslavl’ Zalesskiy, p. 9 (1997) Google Scholar
  66. 66.
    Berezin, F.A.: Introduction to Superanalysis. MCCME, Moscow (2013). Revised and edited by D. Leites and with appendix “Seminar on Supersymmetry. Vol. 1 1/2” Google Scholar
  67. 67.
    Shander, V.: Invariant functions on supermatrices (1998). Preprint. arXiv:math/9810112 [math.RT]
  68. 68.
    Chern, S.S., Tenenblat, K.: Pseudo-spherical surfaces and evolution equations. Stud. Appl. Math. 74(1), 55–83 (1986) MathSciNetzbMATHGoogle Scholar
  69. 69.
    Marvan, M.: Sufficient set of integrability conditions of an orthonomic system. Found. Comput. Math. 9(6), 651–674 (2009).  https://doi.org/10.1007/s10208-008-9039-8. arXiv:nlin/0605009 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  70. 70.
    Marvan, M.: Reducibility of zero curvature representations with application to recursion operators. Acta Appl. Math. 83(1–2), 39–68 (2004).  https://doi.org/10.1023/B:ACAP.0000035588.67805.0b. arXiv:nlin/0306006 [nlin.SI] MathSciNetzbMATHGoogle Scholar
  71. 71.
    Levi, D., Sym, A., Tu, G.Z.: A working algorithm to isolate integrable surfaces in \(\mathbb{E}^{3}\). Preprint DF INFN 761, Roma, Oct. 10 (1990) Google Scholar
  72. 72.
    Cieśliński, J.: Nonlocal symmetries and a working algorithm to isolate integrable geometries. J. Phys. A, Math. Gen. 26(5), L267–L271 (1993).  https://doi.org/10.1088/0305-4470/26/5/017 MathSciNetzbMATHGoogle Scholar
  73. 73.
    Cieśliński, J.: Group interpretation of the spectral parameter in the case of nonhomogeneous, nonlinear Schrödinger system. J. Math. Phys. 34(6), 2372–2384 (1993).  https://doi.org/10.1063/1.530122 MathSciNetzbMATHGoogle Scholar
  74. 74.
    Cieśliński, J., Goldstein, P., Sym, A.: On integrability of the inhomogeneous Heisenberg ferromagnet model: examination of a new test. J. Phys. A, Math. Gen. 27(5), 1645 (1994).  https://doi.org/10.1088/0305-4470/27/5/028 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Bernoulli Institute for Mathematics, Computer Science and Artificial IntelligenceUniversity of GroningenGroningenThe Netherlands
  2. 2.Institut des Hautes Études Scientifiques (IHÉS)Bures-sur-YvetteFrance
  3. 3.Institute of MathematicsPolish Academy of ScienceWarsawPoland
  4. 4.Independent University of MoscowMoscowRussia

Personalised recommendations