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

SIGMA 9 (2013), 035, 16 pages      arXiv:1212.1952      https://doi.org/10.3842/SIGMA.2013.035

On Addition Formulae of KP, mKP and BKP Hierarchies

Yoko Shigyo
Department of Mathematics, Tsuda College, Kodaira, Tokyo, 187-8577, Japan

Received December 12, 2012, in final form April 04, 2013; Published online April 23, 2013

Abstract
In this paper we study the addition formulae of the KP, the mKP and the BKP hierarchies. We prove that the total hierarchies are equivalent to the simplest equations of their addition formulae. In the case of the KP and the mKP hierarchies those results had previously been proved by Noumi, Takasaki and Takebe by way of wave functions. Here we give alternative and direct proofs for the case of the KP and mKP hierarchies. Our method can equally be applied to the BKP hierarchy.

Key words: KP hierarchy; modified KP hierarchy; BKP hierarchy.

pdf (369 kb)   tex (18 kb)

References

  1. Bobenko A.I., Suris Yu.B., Discrete differential geometry. Integrable structure, Graduate Studies in Mathematics, Vol. 98, Amer. Math. Soc., Providence, RI, 2008.
  2. Buchstaber V.M., Enolski V.Z., Leykin D.V., Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math and Math. Phys. 10 (1997), no. 2, 1-125, solv-int/9603005.
  3. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. I, J. Phys. Soc. Japan 51 (1982), 4116-4124.
  4. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan 51 (1982), 4125-4131.
  5. Date E., Jimbo M., Miwa T., Method for generating discrete soliton equations. V, J. Phys. Soc. Japan 52 (1982), 766-771.
  6. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39-119.
  7. Eilbeck J.C., Enolski V.Z., Gibbons J., Sigma, tau and Abelian functions of algebraic curves, J. Phys. A: Math. Theor. 43 (2010), 455216, 20 pages, arXiv:1006.5219.
  8. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin, 1973.
  9. Hirota R., Generalizations of determinant identities by Pfaffian, in Mathematical Theories and Applications of Nonlinear Waves and Nonlinear Dynamics, Research Institute for Applied Mechanics, Kyushu University, 2004, 148-156.
  10. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
  11. Inoguchi J., Kajiwara K., Matsuura N., Ohta Y., Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves, arXiv:1108.1328.
  12. Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
  13. 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.
  14. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  15. Miwa T., On Hirota's difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 9-12.
  16. Nakayashiki A., Sigma function as a tau function, Int. Math. Res. Not. 2010 (2010), no. 3, 373-394, arXiv:0904.0846.
  17. Noumi M., Takebe T., Algebraic analysis of integrable hierarchies, in preparation.
  18. Ohta Y., Bilinear theory of solitons with Pfaffian labels, Sūurikaisekikenkyūusho Kōokyūuroku (1993), no. 822, 197-205.
  19. Raina A.K., Fay's trisecant identity and conformal field theory, Comm. Math. Phys. 122 (1989), 625-641.
  20. Sato M., Sato Y., Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, in Nonlinear PDE in Applied Science, North-Holland Math. Stud., Vol. 81, Editors H. Fujita, P. Lax, G. Strang, Tokyo, 1982, 259-271.
  21. Takasaki K., Differential Fay identities and auxiliary linear problem of integrable hierarchies, in Exploring New Structures and Natural Constructions in Mathematical Physics, Adv. Stud. Pure Math., Vol. 61, Math. Soc. Japan, Tokyo, 2011, 387-441, arXiv:0710.5356.
  22. Takasaki K., Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2 (2006), 057, 22 pages, nlin.SI/0604003.
  23. Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743-808, hep-th/9405096.
  24. Teo L.P., Fay-like identities of the Toda lattice hierarchy and its dispersionless limit, Rev. Math. Phys. 18 (2006), 1055-1073, nlin.SI/0606059.

Previous article  Next article   Contents of Volume 9 (2013)