Real Soliton Lattices of the Kadomtsev Petviashvili II Equation and Desingularization of Spectral Curves: The GrTP(2, 4) Case

  • Simonetta AbendaEmail author
  • Petr G. Grinevich


We apply the general construction developed in our previous papers to the first nontrivial case of GrTP(2, 4). In particular, we construct finite-gap real quasi-periodic solutions of the KP-II equation in the form of a soliton lattice corresponding to a smooth M-curve of genus 4 which is a desingularization of a reducible rational M-curve for soliton data in GrTP(2, 4).


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  1. 1.
    S. Abenda, “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).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Abenda, “On some properties of KP-II soliton divisors in GrTP(2, 4),” Ric. Mat., doi: 10.1007/s11587-018-0381-0 (2018).Google Scholar
  3. 3.
    S. Abenda and P. G. Grinevich, “Rational degenerations of M-curves, totally positive Grassmannians and KP2-solitons,” Commun. Math. Phys. 361 (3), 1029–1081 (2018); arXiv: 1506.00563 [nlin.SI].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. Abenda and P. G. Grinevich, “KP theory, plane-bipartite networks in the disk, and rational degenerations of M-curves,” arXiv: 1801.00208 [math-ph].Google Scholar
  5. 5.
    S. Abenda and P. G. Grinevich, “Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons,” arXiv: 1805.05641 [math-ph].Google Scholar
  6. 6.
    E. Arbarello, M. Cornalba, and P. A. Griffiths, Geometry of Algebraic Curves (Springer, Heidelberg, 2011), Vol. II, with a contribution by J. D. Harris, Grundl. Math. Wiss. 268.CrossRefzbMATHGoogle Scholar
  7. 7.
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes (Cambridge Univ. Press, Cambridge, 2016).CrossRefzbMATHGoogle Scholar
  8. 8.
    E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations (Springer, Berlin, 1994), Springer Ser. Nonlinear Dyn.zbMATHGoogle Scholar
  9. 9.
    A. I. Bobenko and L. A. Bordag, “Periodic multiphase solutions of the Kadomtsev–Petviashvili equation,” J. Phys. A.: Math. Gen. 22 (9), 1259–1274 (1989).CrossRefzbMATHGoogle Scholar
  10. 10.
    M. Boiti, F. Pempinelli, A. K. Pogrebkov, and B. Prinari, “Towards an inverse scattering theory for non-decaying potentials of the heat equation,” Inverse Probl. 17 (4), 937–957 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    V. M. Buchstaber and A. A. Glutsyuk, “Total positivity, Grassmannian and modified Bessel functions,” arXiv: 1708.02154 [math.DS].Google Scholar
  12. 12.
    V. M. Buchstaber and S. Terzić, “Topology and geometry of the canonical action of T 4 on the complex Grassmannian G4,2 and the complex projective space CP5,” Moscow Math. J. 16 (2), 237–273 (2016).MathSciNetzbMATHGoogle Scholar
  13. 13.
    V.M Buchstaber and S. Terzić, “Toric topology of the complex Grassmann manifolds,” arXiv: 1802.06449 [math.AT].Google Scholar
  14. 14.
    S. Chakravarty and Y. Kodama, “Soliton solutions of the KP equation and application to shallow water waves,” Stud. Appl. Math. 123 (1), 83–151 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Computational Approach to Riemann Surfaces, Ed. by A. I. Bobenko and C. Klein (Springer, Heidelberg, 2011), Lect. Notes Math. 2013.zbMATHGoogle Scholar
  16. 16.
    B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Math. Comput. 73 (247), 1417–1442 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    B. Deconinck and M. van Hoeij, “Computing Riemann matrices of algebraic curves,” Physica D 152–153, 28–46 (2001).Google Scholar
  18. 18.
    B. Deconinck and H. Segur, “The KP equation with quasiperiodic initial data,” Physica D 123 (1–4), 123–152 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    V. S. Dryuma, “Analytic solution of the two-dimensional Korteweg–de Vries (KdV) equation,” JETP Lett. 19 (12), 387–388 (1974) [transl. from Pis’ma Zh. Eksp. Teor. Fiz. 19 (12), 753–755 (1974)].Google Scholar
  20. 20.
    B. A. Dubrovin, “Theta functions and non-linear equations,” Russ. Math. Surv. 36 (2), 11–92 (1981) [transl. from Usp. Mat. Nauk 36 (2), 11–80 (1981)].CrossRefzbMATHGoogle Scholar
  21. 21.
    B. A. Dubrovin, R. Flickinger, and H. Segur, “Three-phase solutions of the Kadomtsev–Petviashvili equation,” Stud. Appl. Math. 99 (2), 137–203 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems. I,” in Dynamical Systems–4 (VINITI, Moscow, 1985), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 4, pp. 179–284; Engl. transl. in Dynamical Systems IV: Symplectic Geometry and Its Applications, 2nd ed. (Springer, Berlin, 2001), Encycl. Math. Sci. 4, pp. 177–332.Google Scholar
  23. 23.
    B. A. Dubrovin and S. M. Natanzon, “Real theta-function solutions of the Kadomtsev–Petviashvili equation,” Math. USSR, Izv. 32 (2), 269–288 (1989) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 52 (2), 267–286 (1988)].MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    S. Fomin and A. Zelevinsky, “Cluster algebras. I: Foundations,” J. Am. Math. Soc. 15 (2), 497–529 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J. Frauendiener and C. Klein, “Computational approach to compact Riemann surfaces,” Nonlinearity 30 (1), 138–172 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    F. R. Gantmacher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, rev. ed. (Gostekhizdat, Moscow, 1950; AMS Chelsea Publ., Providence, RI, 2002).zbMATHGoogle Scholar
  27. 27.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York, 1978).zbMATHGoogle Scholar
  28. 28.
    D. A. Gudkov, “The topology of real projective algebraic varieties,” Russ. Math. Surv. 29 (4), 1–79 (1974) [transl. from Usp. Mat. Nauk 29 (4), 3–79 (1974)].MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Harnack, “Ueber die Vieltheiligkeit der ebenen algebraischen Curven,” Math. Ann. 10, 189–198 (1876).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. R. Its and V. B. Matveev, “On a class of solutions of the Korteweg–de Vries equation,” in Problems of Mathematical Physics, Issue 8: Differential Equations. Spectral Theory. Wave Propagation Theory (Leningr. Univ., Leningrad, 1976), pp. 70–92 [in Russian].Google Scholar
  31. 31.
    B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersing media,” Sov. Phys., Dokl. 15, 539–541 (1970) [transl. from Dokl. Akad. Nauk SSSR 192 (4), 753–756 (1970)].zbMATHGoogle Scholar
  32. 32.
    C. Kalla and C. Klein, “On the numerical evaluation of algebro-geometric solutions to integrable equations,” Nonlinearity 25 (3), 569–596 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    C. Kalla and C. Klein, “Computation of the topological type of a real Riemann surface,” Math. Comput. 83 (288), 1823–1846 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    S. Karlin, Total Positivity (Stanford Univ. Press, Stanford, CA, 1968), Vol. 1.zbMATHGoogle Scholar
  35. 35.
    Y. Kodama and L. Williams, “The Deodhar decomposition of the Grassmannian and the regularity of KP solitons,” Adv. Math. 244, 979–1032 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Y. Kodama and L. Williams, “KP solitons and total positivity for the Grassmannian,” Invent. Math. 198 (3), 637–699 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    I. M. Kričever, “Algebro-geometric construction of the Zaharov–Šabat equations and their periodic solutions,” Sov. Math., Dokl. 17, 394–397 (1976) [transl. from Dokl. Akad. Nauk SSSR 227 (2), 291–294 (1976)].Google Scholar
  38. 38.
    I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funct. Anal. Appl. 11 (1), 12–26 (1977) [transl. from Funkts. Anal. Prilozh. 11 (1), 15–31 (1977)].MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    I. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model,” Funct. Anal. Appl. 20 (3), 203–214 (1986) [transl. from Funkts. Anal. Prilozh. 20 (3), 42–54 (1986)].CrossRefzbMATHGoogle Scholar
  40. 40.
    G. Lusztig, “Total positivity in reductive groups,” in Lie Theory and Geometry: In Honor of B. Kostant (Birkhäuser, Boston, 1994), Prog. Math. 123, pp. 531–568.Google Scholar
  41. 41.
    G. Lusztig, “Total positivity in partial flag manifolds,” Represent. Theory 2, 70–78 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    T. M. Malanyuk, “A class of exact solutions of the Kadomtsev–Petviashvili equation,” Russ. Math. Surv. 46 (3), 225–227 (1991) [transl. from Usp. Mat. Nauk 46 (3), 193–194 (1991)].MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    V. B. Matveev, “Some comments on the rational solutions of the Zakharov–Schabat equations,” Lett. Math. Phys. 3, 503–512 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    D. Mumford, Tata Lectures on Theta. I, II, 2nd ed. (Birkhäuser, Basel, 2007), Modern Birkhäuser Classics.CrossRefGoogle Scholar
  45. 45.
    S. M. Natanzon, “Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobians of real curves,” Russ. Math. Surv. 54 (6), 1091–1147 (1999) [transl. from Usp. Mat. Nauk 54 (6), 3–60 (1999)].CrossRefzbMATHGoogle Scholar
  46. 46.
    S. P. Novikov, “The periodic problem for the Korteweg–de Vries equation,” Funct. Anal. Appl. 8 (3), 236–246 (1974) [transl. from Funkts. Anal. Prilozh. 8 (3), 54–66 (1974)].MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    A. R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform (Academic, Amsterdam, 2010).zbMATHGoogle Scholar
  48. 48.
    A. Pinkus, Totally Positive Matrices (Cambridge Univ. Press, Cambridge, 2010), Cambridge Tracts Math. 181.zbMATHGoogle Scholar
  49. 49.
    A. Postnikov, “Total positivity, Grassmannians, and networks,” arXiv:math/0609764 [math.CO].Google Scholar
  50. 50.
    M. Sato, “Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds,” in Random Systems and Dynamical Systems: Proc. Symp., Kyoto, 1981 (Kyoto Univ., Kyoto, 1981), RIMS Kokyuroku 439, pp. 30–46.Google Scholar
  51. 51.
    G. Springer, Introduction to Riemann Surfaces (Addison-Wesley, Reading, MA, 1957).zbMATHGoogle Scholar
  52. 52.
    C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simul. 127, 263–272 (2016).MathSciNetCrossRefGoogle Scholar
  53. 53.
    I. A. Taimanov, “Singular spectral curves in finite-gap integration,” Russ. Math. Surv. 66 (1), 107–144 (2011) [transl. from Usp. Mat. Nauk 66 (1), 111–150 (2011)].MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    O. Ya. Viro, “Real algebraic plane curves: Constructions with controlled topology,” Leningr. Math. J. 1 (5), 1059–1134 (1990) [transl. from Algebra Anal. 1 (5), 1–73 (1989)].zbMATHGoogle Scholar
  55. 55.
    V. E. Zakharov and A. B. Shabat, “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) [transl. from Funkts. Anal. Prilozh. 8 (3), 43–53 (1974)].CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBologna (BO)Italy
  2. 2.L.D. Landau Institute for Theoretical Physics of Russian Academy of SciencesChernogolovkaRussia
  3. 3.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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