Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 024, 43 pages      arXiv:1912.06768      https://doi.org/10.3842/SIGMA.2021.024

Space Curves and Solitons of the KP Hierarchy. I. The $l$-th Generalized KdV Hierarchy

Yuji Kodama and Yuancheng Xie
Department of Mathematics, The Ohio State University, Columbus OH, 43210, USA

Received October 14, 2020, in final form March 02, 2021; Published online March 16, 2021

Abstract
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.

Key words: space curve; soliton solution; KP hierarchy; Sato Grassmannian; numerical semigroup.

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References

  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 (2017), 112-138, arXiv:1605.00995.
  2. Abenda S., Grinevich P.G., Rational degenerations of tt M-curves, totally positive Grassmannians and KP2-solitons, Comm. Math. Phys. 361 (2018), 1029-1081, arXiv:1506.00563.
  3. Belokolos E.D., Bobenko A.I., Enolskii V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
  4. Buchstaber V.M., Enolskii V.Z., Leikin D.V., Hyperelliptic Kleinian functions and applications, in Solitons, Geometry, and Topology: on the Crossroad, Amer. Math. Soc. Transl. Ser. 2, Vol. 179, Amer. Math. Soc., Providence, RI, 1997, 1-33.
  5. Buchstaber V.M., Enolskii V.Z., Leykin D.V., Rational analogues of abelian functions, Funct. Anal. Appl. 33 (1999), 83-94.
  6. Buchweitz R.-O., Greuel G.-M., The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980), 241-281.
  7. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators, Proc. London Math. Soc. 21 (1923), 420-440.
  8. Greuel G.-M., On deformation of curves and a formula of Deligne, in Algebraic Geometry (La Rábida, 1981), Lecture Notes in Math., Vol. 961, Springer, Berlin, 1982, 141-168.
  9. Kodama Y., KP solitons and the Grassmannians: combinatorics and geometry of two-dimensional wave patterns, SpringerBriefs in Mathematical Physics, Vol. 22, Springer, Singapore, 2017.
  10. Kodama Y., Williams L., The Deodhar decomposition of the Grassmannian and the regularity of KP solitons, Adv. Math. 244 (2013), 979-1032, arXiv:1204.6446.
  11. Kodama Y., Williams L., KP solitons and total positivity for the Grassmannian, Invent. Math. 198 (2014), 637-699, arXiv:1106.0023.
  12. Komeda J., Matsutani S., Previato E., The sigma function for Weierstrass semigoups $\langle3, 7, 8\rangle$ and $\langle6, 13, 14, 15, 16\rangle$, Internat. J. Math. 24 (2013), 1350085, 58 pages, arXiv:1303.0451.
  13. Komeda J., Matsutani S., Previato E., The Riemann constant for a non-symmetric Weierstrass semigroup, Arch. Math. (Basel) 107 (2016), 499-509, arXiv:1604.02627.
  14. Komeda J., Matsutani S., Previato E., The sigma function for trigonal cyclic curves, Lett. Math. Phys. 109 (2019),, arXiv:1712.00694.
  15. Krichever I.M., Integration of nonlinear equations by the methods of algebraic geometry, Funct. Anal. Appl. 11 (1977), 12-26.
  16. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  17. Matsutani S., Komeda J., Sigma functions for a space curve of type $(3,4,5)$, J. Geom. Symmetry Phys. 30 (2013), 75-91, arXiv:1112.4137.
  18. Mulase M., Cohomological structure in soliton equations and Jacobian varieties, J. Differential Geom. 19 (1984), 403-430.
  19. Mulase M., Geometric classification of commutative algebras of ordinary differential operators, in Differential Geometric Methods in Theoretical Physics (Davis, CA, 1988), Editors L.L. Chau, W. Nahm, NATO Adv. Sci. Inst. Ser. B Phys., Vol. 245, Plenum, New York, 1990, 13-27.
  20. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progress in Mathematics, Vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984.
  21. Nakayashiki A., On algebraic expressions of sigma functions for $(n,s)$ curves, Asian J. Math. 14 (2010), 175-211, arXiv:0803.2083.
  22. Nakayashiki A., Degeneration of trigonal curves and solutions of the KP-hierarchy, Nonlinearity 31 (2018), 3567-3590, arXiv:1708.03440.
  23. Nakayashiki A., On reducible degeneration of hyperelliptic curves and soliton solutions, SIGMA 15 (2019), 009, 18 pages, arXiv:1808.06748.
  24. Pinkham H.C., Deformations of algebraic varieties with $G_{m}$ action, Astérisque 20 (1974), i+131 pages.
  25. Rosales J.C., García-Sánchez P.A., Numerical semigroups, Developments in Mathematics, Vol. 20, Springer, New York, 2009.
  26. Sato M., Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds, in Random Systems and Dynamical Systems (Kyoto, 1981), RIMS Kokyuroku, Vol. 439, Kyoto, 1981, 30-46.
  27. Sato M., Noumi M., Soliton equation and universal Grassmann manifold, Sophia University Kokyuroku in Mathematics 18 (1984), 1-131.
  28. Schur I., Über vertauschbare lineare Differentialausdrücke, Stizungsber. der Berliner Math. Gesel 4 (1905), 2-8.
  29. Segal G., Wilson G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5-65.
  30. Takasaki K., Geometry of universal Grassmann manifold from algebraic point of view, Rev. Math. Phys. 1 (1989), 1-46.
  31. Xie Y., Algebraic curves and flag varieties in solutions of the KP hierarchy and the full Kostant-Toda hierarchy, PhD Thesis, Ohio State University, 2021, in preparation.

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