Rational Degenerations of \({{\mathtt{M}}}\)-Curves, Totally Positive Grassmannians and KP2-Solitons

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Abstract

We establish a new connection between the theory of totally positive Grassmannians and the theory of \({{\mathtt{M}}}\)-curves using the finite-gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev–Petviashvili 2 equation [see (1)], which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian \({Gr^{\textsc{tp}} (N,M)}\) a reducible curve which is a rational degeneration of an \({{\mathtt{M}}}\)-curve of minimal genus \({g=N(M-N)}\), and we reconstruct the real algebraic-geometric data á la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth \({{\mathtt{M}}}\)-curves. In our approach, we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection \({Gr^{\textsc{tp}} (r+1,M-N+r+1)\mapsto Gr^{\textsc{tp}} (r,M-N+r)}\).

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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)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abenda, S., Grinevich, P.G.: KP theory, plane-bipartite networks in the disk and rational degenerations of \({{\mathtt{M}}}\)-curves. arXiv:1801.00208
  3. 3.
    Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Grundlehren der Mathematischen Wissenschaften 268, Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo F., Goncharov A.B., Postnikov A., Trnka J.: Scattering Amplitudes and the Positive Grassmannian. arXiv:1212.5605
  5. 5.
    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)Google Scholar
  6. 6.
    Biondini G., Kodama Y.: On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy. J. Phys. A 36(42), 10519–10536 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Boiti M., Pempinelli F., Pogrebkov A.K.: Properties of the solitonic potentials of the heat operator. Theoret. Math. Phys. 168(1), 865–874 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cavaretta A.S. Jr., Dahmen W.A., Micchelli C.A., Smith P.W.: A factorization theorem for banded matrices. Linear Algebra Appl. 39, 229–245 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chakravarty S., Kodama Y.: Classification of the line-soliton solutions of KPII. J. Phys. A 41(27), 275209,33 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chakravarty S., Kodama Y.: Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 123(1), 83–151 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    de Boor C., Pinkus A.: The approximation of a totally positive band matrix by a strictly banded totally positive one. Linear Algebra Appl. 42, 81–98 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dickey, L. A.: Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics, 26, World Scientific Publishing Co., Inc., River Edge, NJ (2003)Google Scholar
  14. 14.
    Dimakis A., Müller-Hoissen F.: KP line solitons and Tamari lattices. J. Phys. A 44(2), 025203, 49 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dubrovin B.A., Krichever I.M., Novikov S.P.: Dynamical Systems, IV, Encyclopaedia Math. Sci. Springer, Berlin (2001)Google Scholar
  16. 16.
    Dubrovin B.A., Natanzon S.M.: Real theta-function solutions of the Kadomtsev–Petviashvili equation. Izv. Akad. Nauk SSSR Ser. Mat. 52(2), 267–286 (1998)MathSciNetMATHGoogle Scholar
  17. 17.
    Fomin S., Zelevinsky A.: Double Bruhat cells and total positivity. J. Am. Math. Soc. 12(2), 335–380 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fomin S., Zelevinsky A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gantmacher F.R., Krein M.G.: Sur les matrices oscillatoires. C. R. Acad. Sci. Paris 201, 577–579 (1935)MATHGoogle Scholar
  20. 20.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mathematical Surveys and Monographs, 167, American Mathematical Society, Providence, RI (2010)Google Scholar
  21. 21.
    Goncharov A.B., Kenyon R.: Dimers and cluster integrable systems. Ann. Sci. École Norm. Super. (4) 46(5), 747–813 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, Hoboken (1978)MATHGoogle Scholar
  23. 23.
    Gross B.H., Harris J.: Real algebraic curves. Ann. Sci. École Norm. Super. série 4 14(2), 157–182 (1981)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Harnack A.: Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann. 10, 189–199 (1876)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hirota R. (2004) The direct method in soliton theory. Cambridge Tracts in Mathematics, 155, Cambridge University Press, Cambridge (2004)Google Scholar
  26. 26.
    Kadomtsev B.B., Petviashvili V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)ADSMATHGoogle Scholar
  27. 27.
    Kenyon R., Okounkov A., Sheffield S.: Dimers and amoebae. Ann. Math. (2) 163(3), 1019–1056 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kodama Y., Williams L.K.: KP solitons, total positivity, and cluster algebras. Proc. Natl. Acad. Sci. USA 108(22), 8984–8989 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kodama Y., Williams L.: The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244, 979–1032 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kodama Y., Williams L.: KP solitons and total positivity for the Grassmannian. Invent. Math. 198(3), 637–699 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Krichever I.M.: An algebraic-geometric construction of the Zakharov–Shabat equations and their periodic solutions. (Russian) Dokl.Akad. Nauk SSSR 227(2), 291–294 (1976)MathSciNetGoogle Scholar
  32. 32.
    Krichever I.M.: Integration of nonlinear equations by the methods of algebraic geometry. (Russian) Funkcional. Fubk. Anal. i Pril. 11(1), 15–31,96 (1977)Google Scholar
  33. 33.
    Krichever I.M.: The spectral theory of nonstationary Schroedinger operator. The nonstationary Peierls model. (Russian). Funk. Anal. i pril. 20(3), 42–54 (1986)Google Scholar
  34. 34.
    Krichever, I.M., Vaninsky, K.L.: The Periodic and Open Toda Lattice. AMS/IP Stud. Adv. Math., 33, Amer. Math. Soc.,pp. 139-158. Providence, RI (2002)Google Scholar
  35. 35.
    Lam T.: Dimers, webs, and positroids. J. Lond. Math. Soc. (2) 92(3), 633–656 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Lusztig, G.: Total positivity in reductive groups. Lie theory and geometry, 531-568, Progr. Math., 123, Birkhäuser Boston, Boston, MA, (1994)Google Scholar
  37. 37.
    Lusztig G.: Total positivity in partial flag manifolds. Represent. Theory 2, 70–78 (1998)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lusztig G.: A survey of total positivity. Milan J. Math. 76, 125–134 (2008)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Malanyuk T.M.: A class of exact solutions of the Kadomtsev–Petviashvili equation. Russian Math. Surveys 46(3), 225–227 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Marsh R.J., Rietsch K.: Parametrizations of flag varieties. Represent. Theory 8, 212–242 (2004)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Matveev V.B.: Some comments on the rational solutions of the Zakharov–Schabat equations. Lett. Math. Phys. 3, 503–512 (1979)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Matveev V.B., Salle M.A.: Darboux Transformations and Solitons. Springer Series in Nonlinear Dynamics. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  43. 43.
    Miwa, T., Jimbo, M., Date, E.: Solitons. Differential Equations, Symmetries and Infinite-Dimensional Algebras. Cambridge Tracts in Mathematics, 135, Cambridge University Press, Cambridge (2000)Google Scholar
  44. 44.
    Natanzon S.M.: Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobians of real curves. Rus. Math. Surv. 54(6), 1091–1147 (1999)CrossRefMATHGoogle Scholar
  45. 45.
    Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: “Theory of solitons. The inverse scattering method”, Contemporary Soviet Mathematics. Consultants Bureau [Plenum], New York (1984)Google Scholar
  46. 46.
    Petrowsky I.: On the Topology of Real Plane Algebraic Curves. Annals of Mathematics Second Series 39(1), 189–209 (1938)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Pinkus, A.: Totally positive matrices. Cambridge Tracts in Mathematics, 181, Cambridge University Press, Cambridge (2010). ISBN: 978-0-521-19408-2Google Scholar
  48. 48.
    Postnikov, A.: Total positivity, Grassmannians, and networks. arXiv:math/0609764 [math.CO]
  49. 49.
    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
  50. 50.
    Schoenberg J.: Über variationsvermindende lineare Transformationen. Mathematishe Zeitschrift 32, 321–328 (1930)CrossRefMATHGoogle Scholar
  51. 51.
    Talaska K.: Combinatorial formulas for Open image in new window-coordinates in a totally nonnegative Grassmannian. J. Combin. Theory Ser. A 118(1), 58–66 (2011)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Veselov A.P., Novikov S.P.: Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations. Soviet Math. Dokl. 30, 588–591 (1984)MATHGoogle Scholar
  53. 53.
    Veselov A.P., Novikov S.P.: Finite-zone, two-dimensional Schrödinger operators. Potential operators. Soviet Math. Dokl. 30, 705–708 (1984)MATHGoogle Scholar
  54. 54.
    Viro O. Ya.: Real plane algebraic curves: constructions with controlled topology. Leningrad Math. J. 1(5), 1059-1134 (1990)Google Scholar
  55. 55.
    Whitney A.M.: A reduction theorem for totally positive matrices. J. Analyse Math. 2, 88–92 (1952)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    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)CrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di BolognaBolognaItaly
  2. 2.L.D.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  3. 3.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  4. 4.Moscow Institute of Physics and TechnologyMoscow RegionRussia

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