Skip to main content
SpringerLink
Log in

On some properties of KP-II soliton divisors in \(Gr^\mathrm{TP}(2,4)\)

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

We construct the pole divisor of the wavefunction for real regular bounded multi-soliton KP-II solutions represented by points in \(Gr^\mathrm{TP} (2,4)\) on the reducible rational \(\mathtt M\)-curve \(\varGamma ({\mathcal {N}}_T)\) recently introduced in Abenda and Grinevich (KP theory, plane-bipartite networks in the disk and rational degenerations of \({\mathtt M}\)-curves, 2018. arXiv:1801.00208) and we give evidence that the asymptotic behavior of its zero divisor in the real (x, y)-plane at fixed time t is compatible with the behavior of the soliton solution classified in Chakravarty and Kodama (Stud Appl Math 123:83–151, 2009).

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Abenda, S., Grinevich, P.G.: Rational degenerations of \(M\)-curves, totally positive Grassmannians and KP-solitons. arXiv:1506.00563, to appear in Commun. Math. Phys. (2018)

  2. Abenda, S., Grinevich, P.G.: KP theory, plane-bipartite networks in the disk and rational degenerations of \(\mathtt{M}\)-curves (2018). arXiv:1801.00208

  3. Biondini, G., Chakravarty, S.: Soliton solutions of the Kadomtsev–Petviashvili II equation. J. Math. Phys. 47, 033514 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chakravarty, S., Kodama, Y.: Classification of the line-solitons of KPII. J. Phys. A Math. Theor. 41, 275209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chakravarty, S., Kodama, Y.: Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 123, 83–151 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Integrable systems. In: Arnol’d, V.I., Novikov, S.P. (eds.) Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, vol. 4, pp. 177–332. Springer, Berlin (2001)

    Google Scholar 

  7. Dubrovin, B.A., Natanzon, S.M.: Real theta-function solutions of the Kadomtsev–Petviashvili equation. Izv. Akad. Nauk SSSR Ser. Mat. 52, 267–286 (1988)

    MATH  Google Scholar 

  8. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gross, B.H., Harris, J.: Real algebraic curves. Ann. Sci. École Norm. Sup. Série 4(14), 157–182 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)

    MATH  Google Scholar 

  11. Kodama, Y., Williams, L.K.: The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244, 979–1032 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kodama, Y., Williams, L.K.: KP solitons and total positivity for the Grassmannian. Invent. Math. 198, 637–699 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krichever, I.M.: An algebraic–geometric construction of the Zakharov–Shabat equations and their periodic solutions. Dokl. Akad. Nauk SSSR 227, 291–294 (1976)

    MathSciNet  Google Scholar 

  14. Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funkc. Anal. i Priloz. 11, 15–31 (1977). 96

    MathSciNet  MATH  Google Scholar 

  15. Krichever, I.M.: Spectral theory of nonstationary Schroedinger operators and non stationary Peierls model. Funkc. Anal. i Priloz. 20, 42–54 (1986)

    Google Scholar 

  16. Malanyuk, T.M.: A class of exact solutions of the Kadomtsev–Petviashvili equation. Russ. Math. Surv. 46, 225–227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Pinkus, A.: Totally Positive Matrices, xii+182, Cambridge Tracts in Mathematics, vol. 181. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  19. Postnikov, A.: Total positivity, Grassmannians, and networks. arXiv:math/0609764

  20. Sato, M.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. In: Lax, P., Fujita, H. (eds.) Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo), pp. 259–271. North-Holland, Amsterdam (1982)

    Google Scholar 

  21. Zakharov, V.E., Shabat, A.B.: 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)

    Article  MATH  Google Scholar 

  22. Zarmi, Y.: Vertex dynamics in multi-soliton solutions of Kadomtsev–Petviashvili II equation. Nonlinearity 27, 1499–1523 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We warmly thank P.G. Grinevich for useful discussions.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Bologna University, P.zza di Porta San Donato 5, 40126, Bologna, BO, Italy

    Simonetta Abenda

Authors
  1. Simonetta Abenda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Simonetta Abenda.

Additional information

We dedicate our paper to Tommaso A. Ruggeri.

This research has been partially supported by GNFM-INDAM and RFO University of Bologna. The results contained in the present paper have been partially presented in Wascom 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abenda, S. On some properties of KP-II soliton divisors in \(Gr^\mathrm{TP}(2,4)\). Ricerche mat 68, 75–90 (2019). https://doi.org/10.1007/s11587-018-0381-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-018-0381-0

Keywords

Mathematics Subject Classification

Search

Navigation