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

SIGMA 10 (2014), 023, 13 pages      arXiv:1308.4584
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Dispersionless BKP Hierarchy and Quadrant Löwner Equation

Takashi Takebe
Faculty of Mathematics, National Research University - Higher School of Economics, 7 Vavilova Str., Moscow, 117312 Russia

Received August 23, 2013, in final form March 10, 2014; Published online March 14, 2014

We show that N-variable reduction of the dispersionless BKP hierarchy is described by a Löwner type equation for the quadrant.

Key words: dBKP hierarchy; quadrant Löwner equation; N-variable reduction.

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  1. Abate M., Bracci F., Contreras M.D., Díaz-Madrigal S., The evolution of Loewner's differential equations, Eur. Math. Soc. Newsl. 78 (2010), 31-38.
  2. Błaszak M., Classical R-matrices on Poisson algebras and related dispersionless systems, Phys. Lett. A 297 (2002), 191-195.
  3. Bogdanov L.V., Konopelchenko B.G., On dispersionless BKP hierarchy and its reductions, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 64-73, nlin.SI/0411046.
  4. Chen Y.-T., Tu M.-H., A note on the dispersionless BKP hierarchy, J. Phys. A: Math. Gen. 39 (2006), 7641-7655.
  5. Date E., Kashiwara M., Miwa T., Transformation groups for soliton equations. II. Vertex operators and τ functions, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 387-392.
  6. Del Monaco A., Gumenyuk P., Chordal Loewner equation, arXiv:1302.0898.
  7. Duren P.L., Univalent functions, Grundlehren der Mathematischen Wissenschaften, Vol. 259, Springer-Verlag, New York, 1983.
  8. Gibbons J., Tsarev S.P., Reductions of the Benney equations, Phys. Lett. A 211 (1996), 19-24.
  9. Gibbons J., Tsarev S.P., Conformal maps and reductions of the Benney equations, Phys. Lett. A 258 (1999), 263-271.
  10. Kamke E., Differentialgleichungen. Lösungsmethoden und Lösungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Bd. 18, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1959.
  11. Kodama Y., Gibbons J., A method for solving the dispersionless KP hierarchy and its exact solutions. II, Phys. Lett. A 135 (1989), 167-170.
  12. Kufarev P.P., Sobolev V.V., Sporyševa L.V., A certain method of investigation of extremal problems for functions that are univalent in the half-plane, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 200 (1968), 142-164.
  13. Kupershmidt B.A., Deformations of integrable systems, Proc. Roy. Irish Acad. Sect. A 83 (1983), 45-74.
  14. Mañas M., S-functions, reductions and hodograph solutions of the rth dispersionless modified KP and Dym hierarchies, J. Phys. A: Math. Gen. 37 (2004), 11191-11221, nlin.SI/0405028.
  15. Mañas M., Martínez Alonso L., Medina E., Reductions and hodograph solutions of the dispersionless KP hierarchy, J. Phys. A: Math. Gen. 35 (2002), 401-417.
  16. Odesskii A.V., Sokolov V.V., Systems of Gibbons-Tsarev type and integrable 3-dimensional models, arXiv:0906.3509.
  17. Odesskii A.V., Sokolov V.V., Integrable (2+1)-dimensional systems of hydrodynamic type, Theoret. and Math. Phys. 163 (2010), 549-586, arXiv:1009.2778.
  18. Pavlov M.V., The Kupershmidt hydrodynamic chains and lattices, Int. Math. Res. Not. 2006 (2006), Art. ID 46987, 43 pages, nlin.SI/0604049.
  19. Pommerenke C., Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975.
  20. Schramm O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221-288, math.PR/9904022.
  21. Takasaki K., Quasi-classical limit of BKP hierarchy and W-infinity symmetries, Lett. Math. Phys. 28 (1993), 177-185, hep-th/9301090.
  22. Takasaki K., Dispersionless Hirota equations of two-component BKP hierarchy, SIGMA 2 (2006), 057, 22 pages, nlin.SI/0601063.
  23. Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743-808, hep-th/9405096.
  24. Takasaki K., Takebe T., Radial Löwner equation and dispersionless cmKP hierarchy, nlin.SI/0601063.
  25. Takasaki K., Takebe T., Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy, J. Phys. A: Math. Theor. 41 (2008), 475206, 27 pages, arXiv:0808.1444.
  26. Takebe T., Teo L.-P., Zabrodin A., Löwner equations and dispersionless hierarchies, J. Phys. A: Math. Gen. 39 (2006), 11479-11501, math.CV/0605161.
  27. Teo L.-P., Analytic functions and integrable hierarchies - characterization of tau functions, Lett. Math. Phys. 64 (2003), 75-92, hep-th/0305005.
  28. Tsarëv S.P., The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR-Izv. 37 (1991), 397-419.

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