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

SIGMA 15 (2019), 009, 18 pages      arXiv:1808.06748      https://doi.org/10.3842/SIGMA.2019.009

On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

Atsushi Nakayashiki
Department of Mathematics, Tsuda University, 2-1-1, Tsuda-Machi, Kodaira, Tokyo, Japan

Received August 27, 2018, in final form January 29, 2019; Published online February 08, 2019

Abstract
In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.

Key words: hyperelliptic curve; soliton solution; KP hierarchy; Sato Grassmannian.

pdf (398 kb)   tex (22 kb)

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 M-curves, totally positive Grassmannians and KP2-solitons, Comm. Math. Phys. 361 (2018), 1029-1081, arXiv:1506.00563.
  3. Abenda S., Grinevich P.G., Real soliton lattices of the KP-II and desingularization of spectral curves: the ${\rm Gr}^{{\rm TP}}(2,4)$ case, Proc. Steklov Inst. Math. 302 (2018), 1-15, arXiv:1803.10968.
  4. Abenda S., Grinevich P.G., KP theory, plane-bipartite networks in the disc and rational degenerations of M-curvess, arXiv:1801.00208.
  5. Abenda S., Grinevich P.G., Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons, arXiv:1805.05641.
  6. Chakravarty S., Kodama Y., Classification of the line-soliton solutions of KPII, J. Phys. A: Math. Theor. 41 (2008), 275209, 33 pages, arXiv:0710.1456.
  7. Chakravarty S., Kodama Y., Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math. 123 (2009), 83-151, arXiv:0902.4433.
  8. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), Editors M. Jimbo, T. Miwa, World Sci. Publishing, Singapore, 1983, 39-119.
  9. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Math., Vol. 352, Springer-Verlag, Berlin - New York, 1973.
  10. Freeman N.C., Nimmo J.J.C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A 95 (1983), 1-3.
  11. Kawamoto N., Namikawa Y., Tsuchiya A., Yamada Y., Geometric realization of conformal field theory on Riemann surfaces, Comm. Math. Phys. 116 (1988), 247-308.
  12. Kodama Y., KP solitons and the Grassmannians. Combinatorics and geometry of two-dimensional wave patterns, SpringerBriefs in Mathematical Physics, Vol. 22, Springer, Singapore, 2017.
  13. Kodama Y., Williams L.K., KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. USA 108 (2011), 8984-8989, arXiv:1105.4170.
  14. Krichever I.M., Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surveys 32 (1977), no. 6, 185-213.
  15. Mulase M., Algebraic theory of the KP equations, in Perspectives in Mathematical Physics, Editors R. Penner, S.T. Yau, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, 151-217.
  16. Nakayashiki A., Sigma function as a tau function, Int. Math. Res. Not. 2010 (2010), 373-394, arXiv:0904.0846.
  17. Nakayashiki A., Tau function approach to theta functions, Int. Math. Res. Not. 2016 (2016), 5202-5248, arXiv:1504.01186.
  18. Nakayashiki A., Degeneration of trigonal curves and solutions of the KP-hierarchy, Nonlinearity 31 (2018), 3567-3590, arXiv:1708.03440.
  19. Sato M., Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science, Editors P.D. Lax, H. Fujita, G. Strang, Lecture Notes in Numerical and Applied Analysis, North-Holland, Amsterdam, Kinokuniya, Tokyo, 1982, 259-271.
  20. Sato M., Lecture note of M. Sato (1984-1985 spring), note taken by T. Umeda, Lecture Note on Mathematical Sciences, Vol. 5, RIMS, Kyoto University, 1989 (in Japanese).
  21. Sato M., Noumi M., Soliton equations and universal Grassmann manifold, Mathematical Lecture Note, Vol. 18, Sophia University, Tokyo, 1984 (in Japanese).
  22. Satsuma J., Exact solutions of a nonlinear diffusion equation, J. Phys. Soc. Japan 50 (1981), 1423-1424.
  23. Segal G., Wilson G., Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5-65.

Previous article  Next article   Contents of Volume 15 (2019)