Skip to main content
SpringerLink
Log in

Recursion operators and hierarchies of \(\text{mKdV}\) equations related to the Kac–Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\)

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We construct three nonequivalent gradings in the algebra \(D_4\simeq so(8)\). The first is the standard grading obtained with the Coxeter automorphism \(C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}\) using its dihedral realization. In the second, we use \(C_2=C_1R\), where \(R\) is the mirror automorphism. The third is \(C_3=S_{\alpha_2}S_{\alpha_1}T\), where \(T\) is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces \( \mathfrak{g} ^{(k)}\), the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg–de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For \(D_4^{(1)}\), these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in \( \partial _x\). For \(D_4^{(2)}\), we have a system of three equations of third order in \( \partial _x\). For \(D_4^{(3)}\), we have a system of two equations of fifth order in \( \partial _x\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1.

Similar content being viewed by others

Notes

  1. Not every Coxeter automorphism can be realized in this way. For example, Coxeter automorphisms of the form \(C(X)=cF(X) c^{-1}\), where \(c\) is a diagonal matrix and \(F\) is some properly chosen outer automorphism, can never be constructed from a Weyl group element.

References

  1. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg–de Vries equation,” Phys. Rev. Lett., 19, 1095–1097 (1967).

    ADS  MATH  Google Scholar 

  2. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Commun. Pure Appl. Math., 21, 467–490 (1968).

    MathSciNet  MATH  Google Scholar 

  3. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP, 34, 62–69 (1972).

    ADS  MathSciNet  Google Scholar 

  4. M. Wadati, “The exact solution of the modified Korteweg–de Vries equation,” J. Phys. Soc. Japan, 32, 1681–1687 (1972).

    ADS  Google Scholar 

  5. V. E. Zakharov and L. D. Faddeev, “Korteweg–de Vries equation: A completely integrable Hamiltonian system,” Funct. Anal. Appl., 5, 280–287 (1971).

    MATH  Google Scholar 

  6. L. A. Takhtadjan, “Hamiltonian systems connected with the Dirac equation,” J. Sov. Math., 8, 219–228 (1973).

    Google Scholar 

  7. R. K. Bullough and P. J. Caudrey, eds., Solitons (Topics Curr. Phys., Vol. 17), Springer, Berlin (1980).

    MATH  Google Scholar 

  8. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform–Fourier analysis for nonlinear problems,” Stud. Appl. Math., 53, 249–315 (1974).

    MathSciNet  MATH  Google Scholar 

  9. D. J. Kaup, “Closure of the squared Zakharov–Shabat eigenstates,” J. Math. Anal. Appl., 54, 849–864 (1976).

    MathSciNet  MATH  Google Scholar 

  10. V. S. Gerdjikov and E. Kh. Khristov, “On the evolution equations, solvable through the inverse scattering method: I. Spectral theory,” Bulgar. J. Phys., 7, 28–41 (1980); “On the evolution equations, solvable through the inverse scattering method: II. Hamiltonian structure and Backlund transformations,” Bulgar. J. Phys., 7, 119–133 (1980).

    MathSciNet  Google Scholar 

  11. D. J. Kaup and A. C. Newell, “An exact solution for a derivative nonlinear Schrödinger equation,” J. Math. Phys., 19, 798–801 (1978).

    ADS  MATH  Google Scholar 

  12. I. D. Iliev, E. Kh. Khristov, and K. P. Kirchev, Spectral Methods in Soliton Equations (Pitman Monogr. Surv. Pure Appl. Math., Vol. 73), Longman Scientific and Technical, Harlow (1994).

    MATH  Google Scholar 

  13. V. E. Zakharov and S. V. Manakov, “The theory of resonant interactions of wave packets in nonlinear media,” Soviet Phys. JETP, 42, 842–850 (1975).

    ADS  Google Scholar 

  14. 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 (1980).

    ADS  MathSciNet  Google Scholar 

  15. E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys., 30, 193–200 (1977).

    MathSciNet  Google Scholar 

  16. C. Athorne and A. Fordy, “Generalised KdV and MKdV equations associated with symmetric spaces,” J. Phys. A: Math. Gen., 20, 1377–1386 (1987).

    ADS  MathSciNet  MATH  Google Scholar 

  17. I. T. Gadzhiev, V. S. Gerdzhikov, and M. I. Ivanov, “Hamiltonian structures of nonlinear evolution equations connected with a polynomial pencil,” J. Math. Sci., 34, 1923–1932 (1986).

    MATH  Google Scholar 

  18. V. S. Gerdjikov and M. I. Ivanov, “A quadratic pencil of general type and nonlinear evolution equations: I. Expansions in ‘squares’ of solutions – generalized Fourier transforms,” Bulgar. J. Phys., 10, 13–26 (1983).

    MathSciNet  Google Scholar 

  19. V. S. Gerdjikov and M. I. Ivanov, “A quadratic pencil of general type and nonlinear evolution equations: II. Hierarchies of Hamiltonian structures,” Bulgar. J. Phys., 10, 130–143 (1983).

    MathSciNet  Google Scholar 

  20. V. S. Gerdjikov, M. I. Ivanov, and P. P. Kulish, “Quadratic bundle and nonlinear equations,” Theor. Math. Phys., 44, 784–795 (1980).

    Google Scholar 

  21. I. M. Gel’fand and B. M. Levitan, “On the determination of a differential equation from its special function,” Izv. Akad. Nauk SSR. Ser. Mat., 15, 309–360 (1951).

    MATH  Google Scholar 

  22. A. B. Shabat, “Inverse-scattering problem for a system of differential equations,” Funct. Anal. Appl., 9, 244–247 (1975).

    MathSciNet  MATH  Google Scholar 

  23. A. B. Shabat, “The inverse scattering problem,” Differ. Equ., 15, 1824 (1979).

    MathSciNet  Google Scholar 

  24. V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem: I,” Funct. Anal. Appl., 8, 226–235 (1974).

    MATH  Google Scholar 

  25. V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering: II,” Funct. Anal. Appl., 13, 166–174 (1979).

    MathSciNet  MATH  Google Scholar 

  26. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York (1984).

    MATH  Google Scholar 

  27. V. S. Gerdjikov and P. P. Kulish, “The generating operator for the \(n{\times}n\) linear system,” Phys. D, 3, 549–564 (1981).

    MathSciNet  MATH  Google Scholar 

  28. V. S. Gerdjikov, “Generalised Fourier transforms for the soliton equations: Gauge-covariant formulation,” Inverse Problems, 2, 51–74 (1986).

    ADS  MathSciNet  Google Scholar 

  29. A. P. Fordy and P. P. Kulish, “Nonlinear Schrödinger equations and simple Lie algebras,” Commun. Math. Phys., 89, 427–443 (1983).

    ADS  MATH  Google Scholar 

  30. V. S. Gerdjikov, “Basic aspects of soliton theory,” in: Geometry, Integrability, and Quantization (Varna, Bulgaria, 3–10 June 2004, I. M. Mladenov and A. C. Hirshfeld, eds.), Softex, Sofia (2005), pp. 78–125; arXiv:nlin.SI/0604004v1 (2006).

    MathSciNet  MATH  Google Scholar 

  31. A. V. Zhiber and A. B. Shabat, “Systems of equations \(u_x=p(u,\,v)\), \(v_y=q(u,\,v)\) that possess symmetries,” Sov. Math. Dokl., 30, 23–26 (1984).

    MATH  Google Scholar 

  32. A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, “The symmetry approach to the classification of non-linear equations: Complete lists of integrable systems,” Russian Math. Surveys, 42, 1–63 (1987).

    ADS  MATH  Google Scholar 

  33. A. V. Mikhailov, V. S. Novikov, and J. P. Wang, “Symbolic representation and classification of integrable systems,” in: Algebraic Theory of Differential Equations (London Math. Soc. Lect. Note Ser., Vol. 357, M. A. H. MacCallum and A. Mikhailov, eds.), Cambridge Univ. Press, Cambridge (2008), pp. 156–216.

    MATH  Google Scholar 

  34. G. Zhao, “Integrability of two-component systems of partial differential equations,” Doctoral dissertation, University of Loughborough, Loughborough, UK (unpublished).

  35. G. Tzitzéica, “Sur une nouvelle classe de surfaces,” C. R. Acad. Sci. Paris, 150, 955–956, 1227–1229 (1910).

    MATH  Google Scholar 

  36. R. K. Dodd and R. K. Bullough, “Polynomial conserved densities for the sine-Gordon equations,” Proc. Roy. Soc. London Ser. A, 352, 481–503 (1977).

    ADS  MathSciNet  Google Scholar 

  37. A. V. Mikhailov, “The reduction problem and the inverse scattering method,” Phys. D, 3, 73–117 (1981).

    MATH  Google Scholar 

  38. A. V. Mikhailov, M. A. Olshanetsky, and A. M. Perelomov, “Two-dimensional generalized Toda lattice,” Commun. Math. Phys., 79, 473–488 (1981).

    ADS  MathSciNet  Google Scholar 

  39. R. Beals and R. Coifman, “Inverse scattering and evolution equations,” Commun. Pure Appl. Math., 38, 29–42 (1985).

    MathSciNet  MATH  Google Scholar 

  40. V. S. Gerdjikov and A. B.Yanovski, “Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system,” J. Math. Phys., 35, 3687–3725 (1994).

    ADS  MathSciNet  MATH  Google Scholar 

  41. V. S. Gerdjikov, G. Vilasi, and A. B. Yanovski, Integrable Hamiltonian Hierarchies: Spectral and Geometric Methods (Lect. Notes Phys., Vol. 748), Springer, Berlin (2008).

    MATH  Google Scholar 

  42. V. S. Gerdjikov and A. B. Yanovski, “On soliton equations with \( \mathbb{Z} _{ {h}}\) and \(\mathbb{D}_{ {h}}\) reductions: Conservation laws and generating operators,” J. Geom. Symmetry Phys., 31, 57–92 (2013).

    MathSciNet  MATH  Google Scholar 

  43. V. S. Gerdjikov and A. B. Yanovski, “CBC systems with Mikhailov reductions by Coxeter automorphism: I. Spectral theory of the recursion operators,” Stud. Appl. Math., 134, 145–180 (2015).

    MathSciNet  MATH  Google Scholar 

  44. V. G. Drinfel’d and V. V. Sokolov, “Equations of Korteweg–de Vries type and simple Lie algebras,” Sov. Math. Dokl., 23, 457–462 (1981).

    MATH  Google Scholar 

  45. V. G. Drinfeld and V. V. Sokolov, “Lie algebras and equations of Korteweg–de Vries type,” J. Soviet Math., 30, 1975–2036 (1985).

    MATH  Google Scholar 

  46. V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, and S. K. Varbev, “Soliton equations related to the Kac–Moody algebra \(D_4^{(1)}\),” Eur. Phys. J. Plus, 130, 106 (2015).

    Google Scholar 

  47. V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, and S. K. Varbev, “MKdV equations related to the \(D_4^{(2)}\) algebra,” Roman. J. Phys., 61, 110–123 (2016).

    Google Scholar 

  48. S. Helgasson, Differential Geometry, Lie Groups, and Symmetric Spaces (Grad. Stud. Math., Vol. 34), Amer. Math. Soc., Providence, R. I. (2012).

    Google Scholar 

  49. R. Carter, Lie Algebras of Finite and Affine Type (Cambridge Stud. Adv. Math., Vol. 96), Cambridge Univ. Press, Cambridge (2005).

    MATH  Google Scholar 

  50. V. G. Kac, Infinite Dimensional Lie Algebras, Cambridge Univ. Press, Cambridge (1995).

    Google Scholar 

  51. A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, “Extension of the module of invertible transformations: Classification of integrable systems,” Commun. Math. Phys., 115, 1–19 (1988).

    ADS  MathSciNet  MATH  Google Scholar 

  52. L. A. Takhtadjan and L. D. Faddeev, Hamiltonian Approach to the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl.: L. D. Faddeev and L. A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (2007).

    Google Scholar 

  53. V. S. Gerdjikov, “Algebraic and analytic aspects of \(N\)-wave type equations,” in: The Legacy of the Inverse Scattering Transform in Applied Mathematics (Contemp. Math., Vol. 301, J. Bona, R. Choudhury, and D. Kaup, eds.), Amer. Math. Soc., Providence, R. I. (2002), pp. 35–68.

    MATH  Google Scholar 

  54. Y. Nutku and M. V. Pavlov, “Multi-Lagrangians for integrable systems,” J. Math. Phys., 43, 1441–1460 (2002); arXiv:hep-th/0108214v2 (2001).

    ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

One of the authors (V. S. G.) is grateful to Professors A. V. Mikhailov and V. S. Novikov for the useful discussions and comments. Two of the authors (V. S. G. and A. A. S.) are grateful to the organizing committee of the IXth International Conference “Solitons, Collapses, and Turbulence: Achievements, Developments, and Prospects,” held in Yaroslavl, Russia, 5–9 August 2019, and dedicated to Vladimir Zakharov’s 80th birthday, for the support and hospitality.

Funding

This research was supported by the Bulgarian Science Foundation (Grant No. NTS-Russia 02/101 from 23 October 2017) and the Russian Foundation for Basic Research (Grant No. 18-51-18007).

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    V. S. Gerdjikov, A. A. Stefanov, I. D. Iliev & G. P. Boyadjiev

  2. National Research Nuclear University MEPHI, Moscow, Russia

    V. S. Gerdjikov

  3. Institute for Advanced Physical Studies, New Bulgarian University, Sofia, Bulgaria

    V. S. Gerdjikov

  4. Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria

    A. A. Stefanov

  5. St. Petersburg State University of Aerospace Instrumentation, St. Petersburg, Russia

    A. O. Smirnov

  6. St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

    V. B. Matveev

  7. Institut de Mathématiques de Bourgogne (IMB), Université de Bourgogne – France Comté, Dijon, France

    V. B. Matveev

  8. Lebedev Physical Institute, RAS, Moscow, Russia

    M. V. Pavlov

Authors
  1. V. S. Gerdjikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Stefanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. D. Iliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. P. Boyadjiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. O. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. V. B. Matveev
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. M. V. Pavlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. S. Gerdjikov.

Ethics declarations

The authors declare no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gerdjikov, V.S., Stefanov, A.A., Iliev, I.D. et al. Recursion operators and hierarchies of \(\text{mKdV}\) equations related to the Kac–Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\). Theor Math Phys 204, 1110–1129 (2020). https://doi.org/10.1134/S0040577920090020

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577920090020

Keywords

Search

Navigation