Skip to main content
SpringerLink
Log in

From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

The generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited. Our aim is to adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously. To this end we consider the potential Q of the Lax operator L as local coordinate on some symmetric space depending on an infinite number of variables t, and \(z_k\), \(k=1, 2, \ldots \). The dependence on \(z_k\) is determined by the k-th higher flow (conserved density) of the hierarchy. Thus we have an infinite set of commuting operators with common fundamental analytic solution. We analyze the properties of the resolvent and thus determine the spectral properties of L. Then we derive the generalized Fourier transforms that linearize this hierarchy of NLEE and establish their fundamental properties as well as dynamical compatibility of each two pairs of such flows. Using the classical R-matrix approach we derive the Poisson brackets between all conserved quantities first assuming that Q is a quasi-periodic function of t. Next taking the limit when the period tends to \(\infty \), we derive the Poisson brackets between the scattering data of L. In addition we analyze a possible relation of our approach to the one based on the \(\tau \)-function that could lead to multi-dimensional integrable equations.

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

Fig. 1

Similar content being viewed by others

Notes

  1. In this section, we will often omit the variables \(\mathbf {z}\) when they are insignificant.

References

  1. M. Ablowitz, D. Kaup, A. Newell, H. Segur, The inverse scattering transform—Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974)

    MathSciNet  Google Scholar 

  2. M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, vol. 302, London Mathematical Society, Lecture Notes Series (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  3. M.J. Ablowitz, S. Chakravarty, A.D. Trubatch, J. Villaroel, Novel class of solutions of the non-stationary Schrödinger and the Kadomtcev–Petviaschvily I equations. Phys. Lett. A 267, 132–146 (2000)

    ADS  MathSciNet  Google Scholar 

  4. M. Adler, T. Shiota, P. Van Moerbeke, Random matrices, Virasoro algebras and non-commutative KP. Duke Math. J. 94, 379–431 (1998)

    MathSciNet  Google Scholar 

  5. A. Arnaudon, D.D. Holm, R.I. Ivanov, G-Strands on symmetric spaces. Proc. R. Soc. A 473, 201 (2017). https://doi.org/10.1098/rspa.2016.0795

    Article  MathSciNet  Google Scholar 

  6. B. Bakalov, E. Horozov, M. Yakimov, Tau-functions as highest weight vectors for \(W_{1+\infty }\) algebra. J. Phys. A Math. Gen. 29, 5565–5573 (1996)

    ADS  Google Scholar 

  7. R. Beals, R.R. Coifman, Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math. 37, 39–90 (1984)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its, V.B. Matveev, Algebro-Geometrical Approach to Nonlinear Evolution Equations, Springer Ser. Nonlinear Dynamics (Springer, Berlin, 1994)

    Google Scholar 

  10. M. Boiti, F. Pempinelli, A. Pogrebkov, Properties of solutions of the Kadomtsev–Petviaschvili-I equation. J. Math. Phys. 35, 4683–4718 (1994)

    ADS  MathSciNet  Google Scholar 

  11. M. Boiti, F. Pempinelli, A. Pogrebkov, B. Prinari, The equivalence of different approaches for generating multi-solitons solutions of the KP-II equation. Theor. Math. Phys. 165, 12371255 (2010)

    Google Scholar 

  12. F. Calogero, A. Degasperis, Spectral Transform and Solitons, vol. I (North Holland, Amsterdam, 1982)

    Google Scholar 

  13. S. Chakravarty, B. Prinari, M.J. Ablowitz, Inverse scattering transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening. Physica D 278–279, 58–78 (2014)

    ADS  MathSciNet  Google Scholar 

  14. V. Drinfel’d, V.V. Sokolov, Lie algebras and equations of Korteweg–de Vries type. Sov. J. Math. 30, 1975–2036 (1985)

    Google Scholar 

  15. P. Dubard, V.B. Matveev, Multi-rogue wave solutions to the focusing NLS equation and the KP-I equation. J. Nat. Hazards Earth Syst. Sci. 11, 667672 (2011). https://doi.org/10.5194/nhess-11-667-2011

    Article  Google Scholar 

  16. P. Dubard, V.B. Matveev, Multi-rogue waves solutions: from the NLS to the KP-I equation. Nonlinearity 26, R93–R125 (2013). https://doi.org/10.1088/0951-7715/26/12/R93

    Article  ADS  MathSciNet  Google Scholar 

  17. B.A. Dubrovin, Matrix finite-zone operators. J. Sov. Math. 28(1), 20–50 (1985)

    Google Scholar 

  18. B.A. Dubrovin, Theta functions and nonlinear equations. Usp. Mat. Nauk 36(2), 1180 (1981)

    MathSciNet  Google Scholar 

  19. B.A. Dubrovin, I.M. Krichever, S.P. Novikov, The Schrödinger equation in a periodic field and Riemann surfaces. Dokl. Akad. Nauk SSSR 229(1), 1518 (1976)

    Google Scholar 

  20. B.A. Dubrovin, V.B. Matveev, S.P. Novikov, Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds. Usp. Mat. Nauk 31(1), 55136 (1976)

    Google Scholar 

  21. L.D. Faddeev, L.A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons (Springer, Berlin, 1987)

    Google Scholar 

  22. A.P. Fordy, P.P. Kulish, Nonlinear Schrodinger equations and simple lie algebras. Commun. Math. Phys. 89(4), 427–443 (1983)

    ADS  MathSciNet  Google Scholar 

  23. H. Flaschka, A.C. Newell, T. Ratiu, Kac-Moody Lie algebras and soliton equations II. Lax equations associated with \(A_1^{(1)}\). Physica 9D, 300–323 (1983)

    ADS  Google Scholar 

  24. N.C. Freeman, J.J.C. Nimmo, Soliton-solutions of the Korteweg–deVries and Kadomtsev–Petviashvili equations: the Wronskian technique. Phys. Lett. A 95, 1–3 (1983)

    ADS  MathSciNet  Google Scholar 

  25. S. Gaiarin, A.M. Perego, E.P. da Silva, F. Da Ros, D. Zibar, Experimental demonstration of dual polarization nonlinear frequency division multiplexed optical transmission system. Preprint arXiv:1708.00350 (2017)

  26. S. Gaiarin, A.M. Perego, E.P. da Silva, F. Da Ros, D. Zibar, Dual polarization nonlinear Fourier transform-based optical communication system. Preprint arXiv:1802.10023 (2018)

  27. V.S. Gerdjikov, Generalized Fourier transforms for the soliton equations. Gauge covariant formulation. Inverse Probl. 2(1), 51–74 (1986)

    ADS  Google Scholar 

  28. V.S. Gerdjikov, Algebraic and analytic aspects of soliton type equations. Contemp. Math. 301, 35–68 (2002). (nlin.SI/0206014)

    MathSciNet  Google Scholar 

  29. V.S. Gerdjikov, On the spectral theory of the integro-differential operator \(\Lambda \), generating nonlinear evolution equations. Lett. Math. Phys. 6(6), 315–324 (1982)

    ADS  MathSciNet  Google Scholar 

  30. V.S. Gerdjikov, Complete integrability, gauge equivalence and Lax representations of the inhomogeneous nonlinear evolution equations. Theor. Math. Phys. 92, 374–386 (1992)

    MathSciNet  Google Scholar 

  31. V.S. Gerdjikov, Basic aspects of soliton theory, in Geometry, integrability and quantization, ed. by I.M. Mladenov, A.C. Hirshfeld (Softex, Sofia, 2005), pp. 78–125. (nlin.SI/0604004)

    Google Scholar 

  32. V.S. Gerdzhikov, M.I. Ivanov, P.P. Kulish, Quadratic bundle and nonlinear equations. Theor. Math. Phys. 44(3), 784–795 (1980)

    Google Scholar 

  33. V.S. Gerdjikov, M.I. Ivanov, P.P. Kulish, Expansions over the squared solutions and difference evolution equations. J. Math. Phys. 25(1), 25–34 (1984)

    ADS  MathSciNet  Google Scholar 

  34. V.S. Gerdjikov, E.K. Khristov, On the evolution equations solvable with the inverse scattering problem. I. The spectral theory. Bulgarian J. Phys. 7(1), 28–41 (1980)

    MathSciNet  Google Scholar 

  35. V.S. Gerdjikov, EKh Khristov, On the evolution equations solvable with the inverse scattering problem. II. Hamiltonian structures and bäcklund transformations. Bulgarian J. Phys. 7(2), 119–133 (1980)

  36. V.S. Gerdjikov, P.P. Kulish, The generating operator for the \(n\times n\) linear system. Physica D 3(3), 549–564 (1981)

  37. V.S. Gerdjikov, A.B. Yanovsky, Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys. 35(7), 3687–3725 (1994)

  38. V.S. Gerdjikov, A.B. Yanovsky, CBC systems with Mikhailov reductions by Coxeter automorphism. I. Spectral theory of the recursion operators. Stud. Appl. Math. 134(2), 145–180 (2015)

    MathSciNet  Google Scholar 

  39. B. Guo, L. Tian, Z. Yan, L. Ling, Y.-F. Wang, Rogue Waves, Mathematical Theory and Applications in Physics (Walter de Gruyter Gmbh, Berlin/Boston, 2017)

    Google Scholar 

  40. J.V. Goossens, M. Yousefi, Y. Jaouën, H. Haffermann, Polarization-division multiplexing based on the nonlinear Fourier transform. Preprint arXiv:1707.08589 (2017)

  41. J. Harnad, V.Z. Enol’skii, Schur function expansions of KP \(\tau \)-functions associated to algebraic curves. Usp. Mat. Nauk 66(4(400)), 137–178 (2011)

    Google Scholar 

  42. S. Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, Cambridge, 1978)

    Google Scholar 

  43. R. Hirota, Exact solution of the Kortewegde Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1193 (1971)

    ADS  Google Scholar 

  44. R. Hirota, Y. Ohta, J. Satsuma, Wronskian structures of solutions for soliton equations. Prog. Theor. Phys. Suppl. 94, 59–72 (1988)

    ADS  MathSciNet  Google Scholar 

  45. R. Hirota, The Direct Method in Soliton Theory, vol. 155, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2004)

    Google Scholar 

  46. R. Hirota, Discretization of coupled soliton equationa, in Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete, ed. by L. Faddeev, P. van Moerbecke, F. Lambert (Springer, Berlin, 2006), pp. 113–122

    Google Scholar 

  47. T. Hollowood, J.L. Miramontes, Tau-functions and generalized integrable hierarchies. Commun. Math. Phys. 157, 99–117 (1993)

    ADS  MathSciNet  Google Scholar 

  48. A.R. Its, V.B. Matveev, Schrodinger operators with finite-zone spectrum and \(N\)-soliton solutions of the Korteweg-de Vries equation. Teor. Mat. Fiz. 23(1), 5168 (1975)

    Google Scholar 

  49. M. Jimbo, T. Miwa, Solitons and infinite dimensional Lie algebras. Publ RIMS Kyoto Univ. 19, 943–1001 (1983)

    MathSciNet  Google Scholar 

  50. D.J. Kaup, Closure of the squared Zakharov-Shabat eigenstates. J. Math. Anal. Appl. 54(3), 849–864 (1976)

    MathSciNet  Google Scholar 

  51. D.J. Kaup, A.C. Newell, Soliton equations, singular dispersion relations and moving eigenvalues. Adv. Math. 31, 67–100 (1979)

    Google Scholar 

  52. Y. Kodama, KP Solitons and the Grassmannians. Combinatorics and Geometry of Two-Dimensional Wave Patterns, vol. 22, Springer Briefs in Mathematical Physics (Springer, Berlin, 2017)

    Google Scholar 

  53. I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations. Russian Math. Surveys 32(6), 185–213 (1977), From Uspekhi Mat. Nauk 32(6), 183–208 (1977)

  54. S.T. Le, J.E. Prilepsky, K.S. Turitsyn, Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers. Opt. Express 22, 26720–26741 (2014)

    ADS  Google Scholar 

  55. C. Li, J. He, W. Ke, Y. Cheng, Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy. J. Math. Phys. 51, 043514 (2010)

    ADS  MathSciNet  Google Scholar 

  56. S.V. Manakov, On the theory of two-dimensional stationary self-focusing electromagnetic waves. JETP 65, 505–516 (1973), S.V. Manakov, Sov. Phys. JETP 38, 248–253 (1974)

  57. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its, V.B. Matveev, Two-dimensional solitons of the Kadomtsev–Petviaschvili equation and their interaction. Phys. Lett. A 63(3), 205–206 (1977)

    ADS  Google Scholar 

  58. V.B. Matveev, Abelian functions and Solitons. Preprint No. 373, pp. 1–90. Wroclav University (1976)

  59. V.B. Matveev, Darboux transformation and explicit solutions of the Kadomtsev–Petviashvily equation, depending on functional parameters. Lett. Math. Phys 3, 213–216 (1979). https://doi.org/10.1007/BF00405295

    Article  ADS  MathSciNet  Google Scholar 

  60. V.B. Matveev, Some comments on the rational solutions of the Zakharov–Schabat equations. Lett. Math. Phys. 3(6), 503–512 (1979)

    ADS  MathSciNet  Google Scholar 

  61. V.B. Matveev, Darboux transformation and the explicit solutions of differential–difference and difference–difference evolution equations I. Lett. Math. Phys. 3(3), 217–222 (1979)

    ADS  MathSciNet  Google Scholar 

  62. V.B. Matveev, M.A. Salle, Differential-difference evolution equations. II: Darboux transformation for the Toda lattice. Lett. Math. Phys. 3(5), 425–429 (1979)

    ADS  MathSciNet  Google Scholar 

  63. V.B. Matveev, 30 years of finite-gap integration theory. Phil. Trans. R. Soc. A 366, 837–875 (2008). https://doi.org/10.1098/rsta.2007.2055

    Article  ADS  MathSciNet  Google Scholar 

  64. V.B. Matveev, M. A. Salle, New families of the explicit solutions of the Kadomtsev–Petviashvily equation and their applications to Johnson equation, in Proceedings of the Conference Some Topics on Inverse Problems, ed. by P.C. Sabatier (World Scientific Publishing, 1988), pp 304–315. ISBN 9971-50-647-5

  65. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1991), pp. 1–131

    Google Scholar 

  66. V.B. Matveev, A.O. Smirnov, AKNS hierarchy, MRW solutions, \({P}_{n}\) breathers, and beyond. J. Math. Phys. 59(9), 091419 (2018)

    ADS  MathSciNet  Google Scholar 

  67. A.V. Mikhailov, The reduction problem and the inverse scattering problem. Physica D 3D, 73–117 (1981)

    ADS  Google Scholar 

  68. J. Moser, Integrable Hamiltonian Systems and Spectral Theory Series, (Lezioni Fermiane) (Accademia Nazionale dei Lincei, Pisa, 1981)

    Google Scholar 

  69. M. Sato, T. Miwa, M. Jimbo, Holonomic quantum fields. I. Publ. RIMS Kyoto Univ. 14, 223–267 (1977)

  70. M. Sato, T. Miwa, M. Jimbo, Holonomic quantum fields. II. The Riemann-Hilbert problem. Publ. RIMS Kyoto Univ. 15, 201–278 (1979)

    MathSciNet  Google Scholar 

  71. M. Sato, Soliton equations as dynamical systems on an infinite dimensional Grassmannian manifold. RIMS Kokyuroku (Kyoto University) 439, 30–46 (1981)

    Google Scholar 

  72. A.B. Shabat, The inverse scattering problem for a system of differential equations. Func. Ann. Appl. 9(3), 75–78 (1975)

    MathSciNet  Google Scholar 

  73. A.B. Shabat, The inverse scattering problem. Differ. Equ. 15, 1824–1834 (1979). (In Russian)

    MathSciNet  Google Scholar 

  74. A.O. Smirnov, V.B. Matveev, Y.A. Gusman, N.V. Landa, Spectral curves for the rogue waves. Preprint arXiv:1712.09309 (2017)

  75. M.I. Yousefi, F.R. Kschischang, Information transmission using the nonlinear Fourier transform, part I: mathematical tools. IEEE Trans. Inf. Theory 60, 43124328 (2014)

    MathSciNet  Google Scholar 

  76. M.I. Yousefi, F.R. Kschischang, Information transmission using the nonlinear Fourier transform, part II: numerical methods. IEEE Trans. Inf. Theory 60, 43294345 (2014)

    MathSciNet  Google Scholar 

  77. M.I. Yousefi, F.R. Kschischang, Information transmission using the nonlinear Fourier transform, part III: spectrum modulation. IEEE Trans. Inf. Theory 60, 43464369 (2014)

    MathSciNet  Google Scholar 

  78. G. Wilson, Infinite-dimensional Lie groups and algebraic geometry in soliton theory. Philos. Trans. R. Soc. Lond. A 315, 393–404 (1985)

    ADS  MathSciNet  Google Scholar 

  79. A.V. Zabrodin, Hirota equation and Bethe ansatz. Theor. Math. Phys. 116, 782–819 (1998)

    MathSciNet  Google Scholar 

  80. V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.I. Pitaevskii, Theory of Solitons: The Inverse Scattering Method (Consultants Bureau, Plenum, N.Y., 1984)

    Google Scholar 

  81. V.E. Zakharov, E.A. Kuznetsov, Hamiltonian formalism for nonlinear waves. Phys. Usp. 40(11), 1087–1116 (1997)

    ADS  Google Scholar 

  82. V.E. Zakharov, S.V. Manakov, Resonant interaction of wave packets in nonlinear media. ZheTF Pis. Red. 18, 413–417 (1973)

    ADS  Google Scholar 

  83. V.E. Zakharov, 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 

  84. V.E. Zakharov, A.V. Mikhailov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Zh. Eksp. Teor. Fiz. 74, 1953–1973 (1978)

    MathSciNet  Google Scholar 

  85. V.E. Zakharov, A.B. Shabat, A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. I. Funct. Ann. Appl. 8(3), 43–53 (1974)

    Google Scholar 

  86. V.E. Zakharov, A.B. Shabat, A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. II. Funct. Anal. Appl. 13(3), 13–23 (1979)

    Google Scholar 

  87. V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz. 61, 118–134 (1971)

    Google Scholar 

Download references

Acknowledgements

We are grateful to an anonymous referee for useful suggestions and for careful reading of the manuscript. This work has been supported by the Bulgarian Science Foundation (Grant NTS-Russia 02/101 from 23.10.2017) and by the RFBR (Grant 18-51-18007).

Author information

Authors and Affiliations

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

    V. S. Gerdjikov

  2. Sankt-Petersburg State University of Aerospace Instrumentation, Sankt-Petersburg, Russia

    V. S. Gerdjikov & A. O. Smirnov

  3. Sankt-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Sankt-Petersburg, Russia

    V. B. Matveev

  4. Institut de Mathématiques de Bourgogne - UMR 5584 CNRS, Université de Bourgogne - Franche Comté, Dijon, France

    V. B. Matveev

Authors
  1. V. S. Gerdjikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. O. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. B. Matveev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. S. Gerdjikov.

Additional information

In memory of Boris Dubrovin and Viktor Enol’skii.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gerdjikov, V.S., Smirnov, A.O. & Matveev, V.B. From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms. Eur. Phys. J. Plus 135, 659 (2020). https://doi.org/10.1140/epjp/s13360-020-00668-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00668-2

Search

Navigation