Advertisement

Selecta Mathematica

, 25:43 | Cite as

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

  • Simonetta AbendaEmail author
  • Petr G. Grinevich
Article

Abstract

We associate real and regular algebraic–geometric data to each multi-line soliton solution of Kadomtsev–Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non-negative part of real Grassmannians \(Gr^{\mathrm{TNN}}(k,n)\). In [3] we were able to construct real algebraic–geometric data for soliton data in the main cell \(Gr^{\mathrm{TP}} (k,n)\) only. Here we do not just extend that construction to all points in \(Gr^{\mathrm{TNN}}(k,n)\), but we also considerably simplify it, since both the reducible rational \(\texttt {M}\)-curve \(\Gamma \) and the real regular KP divisor on \(\Gamma \) are directly related to the parametrization of positroid cells in \(Gr^{\mathrm{TNN}}(k,n)\) via the Le-networks introduced in [63]. In particular, the direct relation of our construction to the Le-networks guarantees that the genus of the underlying smooth \(\texttt {M}\)-curve is minimal and it coincides with the dimension of the positroid cell in \(Gr^{\mathrm{TNN}}(k,n)\) to which the soliton data belong to. Finally, we apply our construction to soliton data in \(Gr^{\mathrm{TP}}(2,4)\) and we compare it with that in [3].

Keywords

Total positivity Totally non-negative Grassmannians KP hierarchy Real solitons M-curves Le-diagrams Planar bicolored networks in the disk Baker–Akhiezer function 

Mathematics Subject Classification

37K40 37K20 14H50 14H70 

Notes

Acknowledgements

The authors would like to express their gratitude to the referee for careful reading of the manuscript and useful remarks.

References

  1. 1.
    Abenda, S.: On a family of KP multi-line solitons associated to rational degenerations of real hyperelliptic curves and to the finite non-periodic Toda hierarchy. J. Geom. Phys. 119, 112–138 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abenda, S.: On some properties of KP-II soliton divisors in \(Gr^{{\rm TP}}(2,4)\). Ric. Mat. 68(1), 75–90 (2019)MathSciNetGoogle Scholar
  3. 3.
    Abenda, S., Grinevich, P.G.: Rational degenerations of M-curves, totally positive Grassmannians and KP-solitons. Commun. Math. Phys. 361(3), 1029–1081 (2018)zbMATHGoogle Scholar
  4. 4.
    Abenda, S., Grinevich, P.G.: KP theory, plabic networks in the disk and rational degenerations of M-curves. arXiv:1801.00208
  5. 5.
    Abenda, S., Grinevich, P.G.: Real soliton lattices of the Kadomtsev–Petviashvili II equation and desingularization of spectral curves: the \(Gr^{{\rm TP}}(2,4)\) case. Proceedings of the Steklov Institute of Mathematics, vol. 302(1), pp. 1–15 (2018)Google Scholar
  6. 6.
    Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of Algebraic Curves. Volume II. With a Contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, vol. 268. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  7. 7.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Scattering amplitudes and the positive Grassmannian. arXiv:1212.5605
  8. 8.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Grassmannian Geometry of Scattering Amplitudes. Cambridge University Press, Cambridge (2016)zbMATHGoogle Scholar
  9. 9.
    Atiyah, M., Dunajski, M., Mason, L.J.: Twistor theory at fifty: from contour integrals to twistor strings. Proc. A 473, 20170530 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Biondini, G., Chakravarty, S.: Soliton solutions of the Kadomtsev–Petviashvili II equation. J. Math. Phys. 47, 033514 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Boiti, M., Pempinelli, F., Pogrebkov, A.K., Prinari, B.: Towards an inverse scattering theory for non-decaying potentials of the heat equation. Inverse Probl. 17, 937–957 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Buchstaber, V., Glutsyuk, A.: Total positivity, Grassmannian and modified Bessel functions. arXiv:1708.02154
  14. 14.
    Buchstaber, V.M., Glutsyuk, A.A.: On determinants of modified Bessel functions and entire solutions of double confluent Heun equations. Nonlinearity 29, 3857–3870 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Buchstaber, V.M., Terzic, S.: Topology and geometry of the canonical action of \(T^4\) on the complex Grassmannian \(G_{4,2}\) and the complex projective space \({\mathbb{CP}}^5\). Mosc. Math. J. 16(2), 237–273 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Buchstaber, V.M., Terzic, S.: Toric topology of the complex Grassmann manifolds. arXiv:1802.06449v2 (2018)
  17. 17.
    Chakravarty, S., Kodama, Y.: Classification of the line-solitons of KPII. J. Phys. A Math. Theor. 41, 275209 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chakravarty, S., Kodama, Y.: Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 123, 83–151 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Advanced Series in Mathematical Physics, vol. 26, 2nd edn. World Scientific Publishing Co., Inc., River Edge, NJ (2003)Google Scholar
  20. 20.
    Dimakis, A., Müller-Hoissen, F.: KP line solitons and Tamari lattices. J. Phys. A 44(2), 025203 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dryuma, V.S.: Analytic solution of the two-dimensional Korteweg–de Vries (KdV) equation. JETP Lett. 19(12), 387–388 (1973)Google Scholar
  22. 22.
    Dubrovin, B.A.: Theta functions and non-linear equations. Russ. Math. Surv. 36(2), 11–92 (1981)zbMATHGoogle Scholar
  23. 23.
    Dubrovin, B.A., Malanyuk, T.M., Krichever, I.M., Makhankov, V.G.: Exact solutions of the time-dependent Schrödinger equation with self-consistent potentials. Soviet J. Particles Nuclei 19(3), 252–269 (1988)MathSciNetGoogle Scholar
  24. 24.
    Dubrovin, B.A., Krichever, I.M., Novikov, S.P.: Integrable systems I. Dynamical Systems, IV. In: Arnol’d, V.I., Novikov, S.P.: (eds.) Encyclopaedia of Mathematical Sciences, vol. 4, pp. 177–332. Springer, Berlin (2001)Google Scholar
  25. 25.
    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)zbMATHGoogle Scholar
  26. 26.
    Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. I.H.E.S. 103, 1–211 (2006)zbMATHGoogle Scholar
  27. 27.
    Fomin, S., Zelevinsky, A.: Double Bruhat cells and total positivity. J. Am. Math. Soc. 12, 335–380 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Freeman, N.C., Nimmo, J.J.C.: Soliton solutions of the Korteweg de Vries and the Kadomtsev–Petviashvili equations: the Wronskian technique. Proc. R. Soc. Lond. A 389, 319–329 (1983)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Gantmacher, F.R., Krein, M.G.: Sur les matrices oscillatoires. C. R. Acad. Sci. Paris 201, 577–579 (1935)zbMATHGoogle Scholar
  31. 31.
    Gantmacher, F.R., Krein, M.G.: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (Russian). Gostekhizdat, Moscow-Leningrad (1941) [second edition (1950); revised English edition from AMS Chelsea Publ (2002)]Google Scholar
  32. 32.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, vol. 167. American Mathematical Society, Providence, RI (2010)zbMATHGoogle Scholar
  33. 33.
    Gel’fand, I.M., Goresky, R.M., MacPherson, R.D., Serganova, V.V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Gel’fand, I.M., Serganova, V.V.: Combinatorial geometries and torus strata on homogeneous compact manifolds. Russ. Math. Surv. 42(2), 133–168 (1987)zbMATHGoogle Scholar
  35. 35.
    Goncharov, A.B., Kenyon, R.: Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér. (4) 46(5), 747–813 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, Hoboken (1978)zbMATHGoogle Scholar
  37. 37.
    Gudkov, D.A.: The topology of real projective algebraic varieties. Russ. Math. Surv. 29, 1–79 (1974)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Harnack, A.: Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann. 10, 189–199 (1876)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Hirota, R.: The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, vol. 155. Cambridge University Press, Cambridge (2004)Google Scholar
  40. 40.
    Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry. Oberwolfach Seminars, vol. 35, 2nd edn. Birkhäuser Verlag, Basel (2009)zbMATHGoogle Scholar
  41. 41.
    Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)zbMATHGoogle Scholar
  42. 42.
    Karlin, S.: Total Positivity, vol. 1. Stanford University Press, Stanford (1968)zbMATHGoogle Scholar
  43. 43.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. (2) 163(3), 1019–1056 (2006)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Kodama, Y.: Young diagrams and N-soliton solutions of the KP equation. J. Phys. A Math. Gen. 37, 11169–11190 (2004)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Kodama, Y.: KP solitons in shallow water. J. Phys. A Math. Theor. 43, 434004 (2010)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Kodama, Y., Williams, L.K.: The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244, 979–1032 (2013)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Kodama, Y., Williams, L.K.: KP solitons and total positivity for the Grassmannian. Invent. Math. 198, 637–699 (2014)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Krichever, I.M.: An algebraic-geometric construction of the Zakharov–Shabat equations and their periodic solutions. Sov. Math. Dokl. 17, 394–397 (1976)zbMATHGoogle Scholar
  49. 49.
    Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funct. Anal. Appl. 11(1), 12–26 (1977)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Krichever, I.M.: Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model. Funct. Anal. Appl. 20(3), 203–214 (1986)zbMATHGoogle Scholar
  51. 51.
    Krichever, I.M.: Spectral theory of two-dimensional periodic operators and its applications. Russ. Math. Surv. 44(8), 146–225 (1989)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Krichever, I.M.: The \(\tau \)-function of the universal Whitham hierarchy, matrix models and topological field theories. Commun. Pure Appl. Math. 47, 437–475 (1994)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Lusztig, G.: Total positivity in reductive groups, Lie Theory and Geometry: in honor of B. Kostant, Progress in Mathematics, vol. 123, pp. 531–568. Birkhäuser, Boston (1994)Google Scholar
  54. 54.
    Lusztig, G.: Total positivity in partial flag manifolds. Represent. Theory 2, 70–78 (1998)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Malanyuk, T.M.: A class of exact solutions of the Kadomtsev–Petviashvili equation. Russ. Math. Surv. 46(3), 225–227 (1991)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytope varieties. In: Viro, O.Y. (ed.) Topology and Geometry-Rohlin Seminar. Lecture Notes in Mathematics, vol. 1346, pp. 527–543. Springer, Berlin (1988)Google Scholar
  57. 57.
    Marsh, R., Rietsch, K.: Parametrizations of flag varieties. Represent. Theory 8, 212–242 (2004)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Matveev, V.B.: Some comments on the rational solutions of the Zakharov–Schabat equations. Lett. Math. Phys. 3, 503–512 (1979)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Miwa, T., Jimbo, M., Date, E.: Solitons. Differential Equations, Symmetries and Infinite-Dimensional Algebras. Cambridge Tracts in Mathematics, vol. 135. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  60. 60.
    Natanzon, S.M.: Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobians of real curves. Russ. Math. Surv. 54(6), 1091–1147 (1999)zbMATHGoogle Scholar
  61. 61.
    Novikov, S.P.: The periodic problem for the Korteweg–de vries equation. Funct. Anal. Appl. 8(3), 236–246 (1974)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Pinkus, A.: Totally Positive Matrices, Cambridge Tracts in Mathematics, vol. 181. Cambridge University Press, Cambridge (2010)Google Scholar
  63. 63.
    Postnikov, A.: Total positivity, Grassmannians, and networks. arXiv:math/0609764 [math.CO]
  64. 64.
    Postnikov, A., Speyer, D., Williams, L.: Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Algebraic Combin. 30(2), 173–191 (2009)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Rietsch, K.: An algebraic cell decomposition of the nonnegative part of a flag variety. J. Algebra 213(1), 144–154 (1999)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Sato, M.: Soliton equations as dynamical systems on a infinite dimensional Grassmann manifolds. RIMS Kokyuroku 439, 30–46 (1981)Google Scholar
  67. 67.
    Schoenberg, I.: Über variationsvermindende lineare Transformationen. Math. Zeit. 32, 321–328 (1930)zbMATHGoogle Scholar
  68. 68.
    Taimanov, I.A.: Singular spectral curves in finite-gap integration. Russ. Math. Surv. 66(1), 107–144 (2011)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Talaska, K.: A formula for Plücker coordinates associated with a planar network. IMRN 2008 (2008), Article ID rnn081.  https://doi.org/10.1093/imrn/rnn081
  70. 70.
    Viro, O.Y.: Real plane algebraic curves: constructions with controlled topology. Leningrad Math. J. 1(5), 1059–1134 (1990)MathSciNetzbMATHGoogle Scholar
  71. 71.
    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(3), 226–235 (1974)zbMATHGoogle Scholar
  72. 72.
    Zarmi, Y.: Vertex dynamics in multi-soliton solutions of Kadomtsev–Petviashvili II equation. Nonlinearity 27, 1499–1523 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.L.D. Landau Institute for Theoretical Physics, RASChernogolovkaRussia
  3. 3.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  4. 4.Moscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations