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Letters in Mathematical Physics

, Volume 106, Issue 8, pp 1139–1179 | Cite as

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

  • Oleksandr Chvartatskyi
  • Aristophanes Dimakis
  • Folkert Müller-Hoissen
Open Access
Article

Abstract

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima–Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.

Keywords

bidifferential calculus Darboux transformation integrable systems self-consistent sources KP discrete KP nonlinear Schrödinger Davey–Stewartson Toda lattice Yajima–Oikawa 

Mathematics Subject Classification

35C08 37K10 70H06 

References

  1. 1.
    Mel’nikov V.: On equations for wave interactions. Lett. Math. Phys. 7, 129–136 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mel’nikov V.: Some new nonlinear evolution equations integrable by the inverse problem method. Math. USSR Sbornik 49, 461–489 (1984)CrossRefzbMATHGoogle Scholar
  3. 3.
    Mel’nikov V.: Integration method of the Korteweg-de Vries equation with a self-consistent source. Phys. Lett. A 133, 493–496 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mel’nikov V.: Capture and confinement of solitons in nonlinear integrable systems. Commun. Math. Phys. 120, 451–468 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mel’nikov V.: Interaction of solitary waves in the system described by the Kadomtsev-Petviashvili equation with a self-consistent source. Commun. Math. Phys. 126, 201–215 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mel’nikov V.: Integration of the nonlinear Schrödinger equation with a source. Inverse Probl. 8, 133–147 (1992)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Zakharov V., Kuznetsov E.: Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D 18, 455–463 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Konopelchenko B., Sidorenko J., Strampp W.: (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems. Phys. Lett. A 157, 17–21 (1991)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Oevel W.: Darboux theorems and Wronskian formulas for integrable systems I: constrained KP flows. Phys. A 195, 533–576 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krichever I.: Linear operators with self-consistent coefficients and rational reductions of KP hierarchy. Phys. D 87, 14–19 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Aratyn H., Nissimov E., Pacheva S.: Constrained KP hierarchies: additional symmetries, Darboux-Bäcklund solutions and relations to multi-matrix models. Int. J. Mod. Phys. A 12, 1265–1340 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Samoilenko A., Samoilenko V., Sidorenko Y.: Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: many-dimensional generalizations and exact solutions of reduced system. Ukr. Math. J. 51, 86–106 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helminck, G., van der Leur, J.: Constrained and rational reductions of the KP hierarchy. In: Aratyn, H., Imbo, T., Keung, W.-Y., Sukhatme, U. (eds.) Supersymmetry and integrable models, Lecture Notes in Physics, vol. 502, pp. 167–182. Springer, Singapore (2007)Google Scholar
  14. 14.
    Latifi A., Leon J.: On the interaction of Langmuir waves with acoustic waves in plasmas. Phys. Lett. A 152, 171–177 (1991)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Claude C., Latifi A., Leon J.: Nonlinear resonant scattering and plasma instability: an integrable model. J. Math. Phys. 32, 3321–3330 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doktorov E., Vlasov R.: Optical solitons in media with resonant and non-resonant self-focusing nonlinearities. Opt. Acta 30, 223–232 (1983)ADSCrossRefGoogle Scholar
  17. 17.
    Nakazawa M., Yamada E., Kubota H.: Coexistence of self-induced transparency soliton and nonlinear Schrödinger soliton. Phys. Rev. Lett. 66, 2625–2628 (1991)ADSCrossRefGoogle Scholar
  18. 18.
    Matveev V., Salle M.: Darboux transformations and solitons, Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Dimakis A., Müller-Hoissen F.: Bi-differential calculi and integrable models. J. Phys. A Math. Gen. 33, 957–974 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dimakis A., Müller-Hoissen F.: Bidifferential graded algebras and integrable systems. Discrete Continuous Dyn. Syst. Suppl. 2009, 208–219 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dimakis A., Müller-Hoissen F.: Binary Darboux transformations in bidifferential calculus and integrable reductions of vacuum Einstein equations. SIGMA 9, 009 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yajima N., Oikawa M.: Formation and interaction of sonic-Langmuir solitons. Progr. Theor. Phys. 56, 1719–1739 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mel’nikov V.: A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the x, y plane. Commun. Math. Phys. 112, 639–652 (1987)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Deng S.-F., Chen D.-Y., Zhang D.-J.: The multisoliton solutions of the KP equation with self-consistent sources. J. Phys. Soc. Jpn. 72, 2184–2192 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xiao T., Zeng Y.: Generalized Darboux transformations for the KP equation with self-consistent sources. J. Phys. A Math. Gen. 37, 7143–7162 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu X., Zeng Y., Lin R.: A new extended KP hierarchy. Phys. Lett. A 372, 3819–3823 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lin R., Liu X., Zeng Y.: The KP hierarchy with self-consistent sources: construction, Wronskian solutions and bilinear identities. J. Phys. Conf. Ser. 538, 012014 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    Chvartatskyi O., Sydorenko Y.: Darboux transformations for (2+1)-dimensional extensions of the KP hierarchy. SIGMA 11, 028 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sakhnovich A.: Matrix Kadomtsev-Petviashvili equation: matrix identities and explicit non-singular solutions. J. Phys. A Math. Gen. 36, 5023–5033 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hase Y., Hirota R., Ohta Y., Satsuma J.: Soliton solutions of the Mel’nikov equations. J. Phys. Soc. Jpn. 58, 2713–2720 (1989)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Kumar C., Radha R., Lakshmanan M.: Exponentially localized solutions of Mel’nikov equation. Chaos Soliton Fractals 22, 705–712 (2004)ADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Chvartatskyi O., Müller-Hoissen F., Stoilov N.: “Riemann equations” in bidifferential calculus. J. Math. Phys. 56, 103512 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Leon J., Latifi A.: Solution of an initial-boundary problem for coupled nonlinear waves. J. Phys. A Math. Gen. 23, 1385–1403 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lin R., Zeng Y., Ma W.-X.: Solving the KdV hierarchy with self-consistent sources by inverse scattering method. Phys. A 291, 287–298 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zeng Y., Shao Y., Xue W.: Positon solutions of the KdV equation with self-consistent sources. Theor. Math. Phys. 137, 1622–1631 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Bondarenko N., Freiling G., Urazboev G.: Integration of the matrix KdV equation with self-consistent sources. Chaos Solitons Fractals 49, 21–27 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wu H., Zeng Y., Fan T.: The Boussinesq equation with self-consistent sources. Inverse Probl. 24, 035012 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Khasanov A., Urazboev G.: On the sine-Gordon equation with a self-consistent source. Sib. Adv. Math. 19, 153–166 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhang D., Chen D.-Y.: The N-soliton solutions of the sine-Gordon equation with self-consistent sources. Phys. A 321, 467–481 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang D.: The N-soliton solutions of some soliton equations with self-consistent sources. Chaos Solitons Fractals 18, 31–43 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Dimakis A., Müller-Hoissen F.: Solutions of matrix NLS systems and their discretizations: a unified treatment. Inverse Probl. 26, 095007 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mel’nikov V.: Integration of the nonlinear Schrödinger equation with a self-consistent source. Commun. Math. Phys. 137, 359–381 (1991)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Shao Y., Zeng Y.: The solutions of the NLS equations with self-consistent sources. J. Phys. A Math. Gen. 38, 2441–2467 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Hu J., Wang H.-Y., Tam H.-W.: Source generation of the Davey-Stewartson equation. J. Math. Phys. 49, 013506 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shen S., Jiang L.: The Davey-Stewartson equation with sources derived from nonlinear variable separation method. J. Comput. Appl. Math. 233, 585–589 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Huang Y., Liu X., Yao Y., Zeng Y.: A new extended matrix KP hierarchy and its solutions. Theor. Math. Phys. 167, 590–605 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Boiti M., Leon J., Martina L., Pempinelli F.: Scattering of localized solitons in the plane. Phys. A 132, 432–439 (1988)MathSciNetGoogle Scholar
  48. 48.
    Fokas A., Santini P.: Coherent structures in multidimensions. Phys. Rev. Lett. 63, 1329–1333 (1989)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Hirota R., Hietarinta J.: Multidromion solutions to the Davey-Stewartson equation. Phys. Lett. A 145, 237–244 (1990)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Gilson C., Nimmo J.: A direct method for dromion solutions of the Davey-Stewartson equations and their asymptotic properties. Proc. R. Soc. A Math. Phys. Eng. Sci. 453, 339–357 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Gilson C.: Resonant behaviour in the Davey-Stewartson equation. Phys. Lett. A 161, 423–428 (1992)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Mikhailov A.: Integrability of a two-dimensional generalization of the Toda chain. JETP Lett. 30, 414–418 (1979)ADSGoogle Scholar
  53. 53.
    Hirota R., Ito M., Kako F.: Two-dimensional Toda lattice equations. Prog. Theor. Phys. Suppl. 94, 42–58 (1988)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Hirota R.: The direct method in soliton theory, Cambridge Tracts in Mathematics, vol. 155. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  55. 55.
    Wang H.-Y., Hu X.-B.: Gegenhasi: 2D Toda lattice equation with self-consistent sources: Casoratian type solutions, bilinear Bäcklund transformation and Lax pair. J. Comput. Appl. Math. 202, 133–143 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Hu X.-B., Wang H.-Y.: Construction of dKP and BKP equations with self-consistent sources. Inverse Probl. 22, 1903–1920 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Liu X., Zeng Y., Lin R.: An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources. J. Math. Phys. 49, 093506 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Doliwa A., Lin R.: Discrete KP equation with self-consistent sources. Phys. Lett. A 378, 1925–1931 (2014)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Dimakis A., Müller-Hoissen F.: Functional representations of integrable hierarchies. J. Phys. A Math. Gen. 39, 9169–9186 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Grimshaw R.: The modulation of an internal gravity-wave packet, and the resonance with the mean motion. J. Appl. Math. Phys. 56, 241–266 (1977)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Shul’man E.: On the integrability of equations of Davey-Stewartson type. Theor. Math. Phys. 1984, 720–724 (1984)zbMATHGoogle Scholar
  62. 62.
    Oikawa M., Okamura M., Funakoshi M.: Two-dimensional resonant interaction between long and short waves. J. Phys. Soc. Jpn. 58, 4416–4430 (1989)ADSCrossRefGoogle Scholar
  63. 63.
    Maccari A.: The Kadomtsev-Petviashvili equation as a source of integrable model equations. J. Math. Phys. 37, 6207–6212 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Berkela Y., Sidorenko Y.: The exact solutions of some multicomponent integrable models. Mat. Stud. 17, 47–58 (2002)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Ohta Y., Maruno K., Oikawa M.: Two-component analogue of two-dimensional long wave - short wave resonance interaction equations: a derivation and solutions. J. Phys. A Math. Theor. 40, 7659–7672 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Chen J., Chen Y., Feng B., Maruno K.: Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems. Phys. Lett. A 379, 1510–1519 (2015)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    Strampp W.: Multilinear forms associated with the Yajima–Oikawa system. Phys. Lett. A 218, 16–24 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Oevel W., Carillo S.: Squared eigenfunction symmetries for soliton equations: Part I. J. Math. Anal. Appl. 217, 161–178 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Oleksandr Chvartatskyi
    • 1
    • 2
  • Aristophanes Dimakis
    • 3
  • Folkert Müller-Hoissen
    • 2
  1. 1.Mathematisches InstitutGeorg-August Universität GöttingenGöttingenGermany
  2. 2.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  3. 3.Department of Financial and Management EngineeringUniversity of the AegeanChiosGreece

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