Skip to main content
SpringerLink
Log in

Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed.

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

Similar content being viewed by others

References

  1. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Nonlinear integrable systems: classical theory and quantum theory,” in: M. Jimbo and T. Miwa (editors),Proceedings of the RIMS Symposium “Nonlinear Integrable Systems-Classical and Quantum Theory”, World Scientific, Singapore (1983), pp. 39–119.

    Google Scholar 

  2. Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, “An elementary introduction to Sato theory,”Progress Theor. Phys. Suppl.,94, 210–241 (1988).

    MathSciNet  Google Scholar 

  3. M. Sato, M. Jimbo, and T. Miwa,Holonomic Quantum Fields [Russian translation], Mir, Moscow (1983).

    Google Scholar 

  4. L. A. Dikii, “Solition equations and Hamiltonian systems,”Adv. Ser. Math. Phys.,12, 1–310 (1991).

    Google Scholar 

  5. V. E. Zakharov and A. B. Shabat, “Scheme of integration of nonlinear equations of mathematical physics by the inverse scattering method,”Funkts. Anal. Prilozhen.,8, No. 3, 43–53 (1974).

    MathSciNet  Google Scholar 

  6. L. A. Takhtadzhan and L. D. Faddeev,Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).

    Google Scholar 

  7. 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).

    Google Scholar 

  8. B. B. Kadomtsev and V. I. Petviashvili, “On stability of solitary waves in weakly dispersive media,”Dokl. Akad. Nauk SSSR,192, No. 4, 753–756 (1970).

    Google Scholar 

  9. Yu. Sidorenko and W. Strampp, “Symmetry constraints of the KP-hierarchy,”Inverse Probl.,7, L37-L43 (1991).

    Article  MathSciNet  Google Scholar 

  10. B. Konopelchenko, Yu. Sidorenko, and W. Strampp, “(1+1)-Dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems,”Phys. Lett. A,157, 17–21 (1991).

    Article  MathSciNet  Google Scholar 

  11. Yu. Sidorenko, “KP-hierarchy and (1+1)-dimensional multicomponent integrable systems,”Ukr. Mat. Zh.,45, No. 1, 91–104 (1993).

    MathSciNet  Google Scholar 

  12. Yu. Sidorenko and W. Strampp, “Multicomponent integrable reductions in the Kadomtsev-Petviashvili hierarchy,”J. Math. Phys.,34, No. 4, 1429–1446 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  13. W. Oevel, Yu. Sidorenko, and W. Strampp, “Hamiltonian structures of the Melnikov system and its reductions,”Inverse Probl.,9, 737–747 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. G. Samoilenko, “Geometric-differential structure and spectral properties of nonlinear completely integrable dynamic systems of Melnikov type,”Ukr. Mat. Zh.,42, No. 5, 655–659 (1990).

    Google Scholar 

  15. W. Oevel and W. Strampp, “Constrained KP-hierarchy and bi-Hamiltonian structures,”Commun. Math. Phys.,157, 51–81 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Konopelchenko and W. Strampp, “New reductions of the Kadomtsev-Petviashvili and two-dimensional Toda hierarchies via symmetry constraints,”J. Math. Phys.,33, No. 11, 3676–3684 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  17. Yi. Cheng and Li Yi-shen, “Constraints of the (2+1)-dimensional integrable solition systems,”J. Phys. A: Math. Gen.,25, 419–431 (1992).

    Article  MATH  Google Scholar 

  18. A. Kundu, W. Strampp, and W. Oevel, “Gauge transformations of constrained KP flows: new integrable hierarchies,”J. Math. Phys.,36, No. 6, 2972–2984 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  19. R. K. Bullough and P. J. Caudrey (editors),Solitons, Springer-Verlag, Berlin (1980).

    MATH  Google Scholar 

  20. Yu. A. Mitropol'skii, N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, and V. G. Samoilenko,Integrable Dynamical Systems: Spectral and Geometric-Differential Aspects [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  21. A. M. Samoilenko, A. K. Prikarpatskii, and O. Ya. Tymchyshyn, “Geometric analysis of the Poincaré-Melnikov transversal decomposition of separatrix manifolds of slowly perturbed nonlinear dynamic systems,”Ukr. Mat. Zh.,45, No. 12, 1668–1682 (1993).

    Google Scholar 

  22. G. V. Samoilenko, “Integrability of nonlinear dynamic systems and geometric-differential structures,”Ukr. Mat. Zh.,45, No. 2, 419–427 (1993).

    Google Scholar 

  23. R. I. Andrushkiw, A. K. Prikarpatskii, V. G. Samoilenko, Yu. A. Mitropols'kii, and N. N. Prytula, “Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems. I,”J. Math. Phys.,35, No. 4, 1763–1777 (1994).

    MATH  MathSciNet  Google Scholar 

  24. R. I. Andrushkiw, A. K. Prikarpatskii, and V. G. Samoilenko, “Algebraic structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems. II. The reduction via the Dirac and canonical quantization procedure,”J. Math. Phys.,35, No. 8, 4088–4115 (1994).

    MATH  MathSciNet  Google Scholar 

  25. S. V. Manakov, “A remark on the integrability of the Euler equation of dynamics of ann-dimensional solid body,”Funkts. Anal. Prilozhen.,10, No. 4, 93–94 (1976).

    MATH  MathSciNet  Google Scholar 

  26. V. K. Mel'nikov, “Some new nonlinear evolution equations integrated by the method of the inverse problem,”Mat. Sb.,121, No. 4, 469–498 (1983).

    MathSciNet  Google Scholar 

  27. V. K. Mel'nikov,On Equations Integrable by the Inverse Scattering Method, Preprint No. P2-85-958, Joint Institute for Nuclear Research, Dubna (1985).

    Google Scholar 

  28. Xu Bing,A Unified Approach to Recursion Operators of the Reduced (1+1)-Dimensional Systems, Preprint, Hefei University, Hefei (1992).

    Google Scholar 

  29. Yi. Cheng, W. Strampp, and Y. J. Zhang, “Bilinear Bäcklund transformation for the KP- andk-constrained KP-hierarchy,”Phys. Lett. A.,182, 71–76 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  30. Yi. Cheng and Y. J. Zhang, “Bilinear equations for the constrained KP-hierarchy,”Inverse Probl.,10, L11-L17 (1994).

    Article  MathSciNet  Google Scholar 

  31. Yi. Cheng and Y. J. Zhang, “Solutions for the vectork-constrained KP-hierarchy”J. Math. Phys.,35, No. 11, 5869–5884 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  32. Chen Dengyan, Zhu Ningcheng, and Zhu Min, “The potential constraints of the Kadomtsev-Petviashvili system and the corresponding Hamiltonian equations,”J. Math. Phys.,35, No. 9, 4725–4738 (1994).

    Article  MathSciNet  Google Scholar 

  33. W. Oevel and W. Strampp, “Wronskian solutions of the constrained KP-hierarchy,”J. Math. Phys.,37, No. 12, 6213–6219 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  34. I. Loris, and R. Willox, “Bilinear form and solutions of thek-constrained KP-hierarchy,”Inverse Probl.,13, L411-L420 (1997).

    Article  MathSciNet  Google Scholar 

  35. I. Loris and R. Willox, “On the solutions ofc KP-equations: Grammians,”J. Math. Phys.,38, No. 10, 5190–5197 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  36. L. L. Chau and J. C. Shaw, “Solving thec KP-hierarchy by gauge transformations,”J. Math. Phys.,38, No. 8, 4128–4136 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  37. J. C. Shaw and M. N. Tu, “Miura and auto-Bäcklund transformations for thec KP- andcm KP-hierarchy,”J. Math. Phys.,38, No. 11, 5756–5772 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  38. H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, “Constrained KP-models as integrable matrix hierarchies,”J. Math. Phys.,38, No. 3, 1559–1568 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  39. H. Aratyn, E. Nissimov, and S. Pacheva, “Virasoro symmetry of constrained KP-hierarchy,”Phys. Lett. A,228, 164–175 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  40. W. Oevel and W. Shief, “Darboux theorems and the KP-hierarchy,” in: P. A. Clarkson (editor),Applications of Analysis and Geometric Methods to Nonlinear Differential Equations, Kluwer, Dordrecht (1993), pp. 193–206.

    Google Scholar 

  41. J. J. Nimmo, “Darboux transformations from reductions of the KP-hierarchy,” in: V. G. Makhankov, A. R. Bishop, and D. D. Holm (editors)Nonlinear Evolution Equations and Dynamical Systems, World Scientific, Singapore (1995), pp. 168–177.

    Google Scholar 

  42. Yu. M. Sidorenko, “On generalization of the τ-function for the Kadomtsev-Petviashvili hierarchy,”Visn. Kiev. Univ., Ser. Mat. Mekh., 40–49 (1998).

  43. A. Davey and K. Stewartson, “On three-dimensional packets of surface waves,”Proc. Royal Soc. London A,338, 101–110 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  44. V. E. Zakharov, “Integrable systems in multidimensional spaces,”Lect. Notes Phys.,153, 190–216 (1983).

    Article  Google Scholar 

  45. A. S. Fokas, “On the simplest integrable equation in 2+1,”Inverse Problems,10, L19-L22 (1994).

    Article  MathSciNet  Google Scholar 

  46. P. P. Kulish and V. D. Lipovskii, “On a Hamiltonian interpretation of the method of the inverse problem for the Davey-Stewartson equation”,Zap. Nauch. Sem. LOMI Akad. Nauk SSSR,161, 54–71 (1987).

    MATH  Google Scholar 

  47. J. J. Nimmo, “Darboux transformation for a two-dimensional Zakharov-Shabat/AKNS spectral problem,”Inverse Probl.,8, 219–243 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  48. P. A. Klarkson and S. Hood, “New symmetry reductions and exact solutions of the Davey-Stewartson system. I. Reductions to ordinary differential equations,”J. Math. Phys.,35, No. 1, 255–282 (1994).

    Article  MathSciNet  Google Scholar 

  49. V. K. Shivamoggi and D. K. Rollins, “The Painlevé formulations and exact solutions of the nonlinear evolution equations for modulated gravity wave trains,”J. Math. Phys.,35, No. 9, 4779–4798 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  50. R. E. Kates and D. J. Kaup, “Two-dimensional nonlinear Schrödinger equations and their properties,”Phys. D.,75, 458–470 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  51. S. Chakravarty, S. L. Kent, and E. T. Newman, “Some reductions of the self-dual Yang-Mills equations to integrable systems in 2+1 dimensions,”J. Math. Phys.,36, No. 2, 763–772 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  52. A. V. Mikhailov and R. I. Yamilov, “On integrable two-dimensional generalizations of nonlinear Schrödinger-type equations,”Phys. Lett. A,230, 295–300 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  53. V. G. Samoilenko and Yu. M. Sidorenko, “Hierarchy of the Burgers matrix equations and integrable reductions in the Davey-Stewartson system,”Ukr. Mat. Zh.,50, No. 2, 252–264 (1998).

    MathSciNet  Google Scholar 

  54. M. D. Pochynaiko, “Higher spatially two-dimensional nonlinear Schrödinger equations,” in:Spectral Theory of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986), pp. 103–106.

    Google Scholar 

  55. L. P. Nizhnik,Inverse Scattering Problems for Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).

    MATH  Google Scholar 

  56. K. Imai and K. Nozaki, “Darboux covariant (2+1)-dimensional soliton equations associated with a\(\overline {su} \)(2) linear system,”Phys. D.,75, 451–457 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  57. V. E. Zakharov, “Collapse of the Langmuir waves,”Zh. Éksp. Teor. Fiz.,62, No. 5, 1745–1759 (1972).

    Google Scholar 

  58. N. Yajima and M. Oikawa, “Formation and interaction of Sonic-Langmuir solitons: inverse scattering method,”Progress Theoret. Phys.,56, No. 6, 1719–1739 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  59. A. Maccari, “The Kadomtsev-Petviashvili equation as a source of integrable model equations,”J. Math. Phys.,37, No. 12, 6207–6212 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  60. K. Porsezian, “Painlevé analysis of a new higher-dimensional soliton equation,”J. Math. Phys.,38, No. 9, 4675–4679 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  61. A. Maccari, “Universal and integrable nonlinear evolution systems of equations in (2+1)-dimension,”J. Math. Phys.,38, No. 8, 4151–4164 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  62. L. P. Nizhnik, “Integration of many-dimensional nonlinear equations by the method of the inverse problem,”Dokl. Akad. Nauk SSSR,254, No. 2, 332–335 (1980).

    MathSciNet  Google Scholar 

  63. L. P. Nizhnik, “Integration of many-dimensional nonlinear equations by the method of the inverse problem,”Usp. Mat. Nauk,36, No. 4, 228 (1981).

    Google Scholar 

  64. A. G. Reiman and M. A. Semenov-Tyan'-Shanskii, “Hamiltonian structure of equations of the Kadomtsev-Petviashvili type,”Differents. Geom., Li Gruppy, Mekh.: Zap. Nauchn. Sem. LOMI Akad. Nauk SSSR,133, 212–226 (1984).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. M. Samoilenko

  2. Shevchenko National University, Kiev

    V. G. Samoilenko & Yu. M. Sidorenko

Authors
  1. A. M. Samoilenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Samoilenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu. M. Sidorenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 78–97, January, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samoilenko, A.M., Samoilenko, V.G. & Sidorenko, Y.M. 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). https://doi.org/10.1007/BF02591917

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02591917

Keywords

Search

Navigation