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

SIGMA 5 (2009), 109, 34 pages      arXiv:0908.3569      http://dx.doi.org/10.3842/SIGMA.2009.109

### Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy

Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida, Sakyo, Kyoto, 606-8501, Japan

Received August 27, 2009, in final form December 15, 2009; Published online December 19, 2009

Abstract
Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as ''the coupled KP hierarchy'' and ''the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called ''the Pfaff-Toda hierarchy''). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.

Key words: integrable hierarchy; auxiliary linear problem; Fay-like identity; dispersionless limit; spectral curve; quasi-classical deformation.

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References

1. Takasaki K., Differential Fay identities and auxiliary linear problem of integrable hierarchies, arXiv:0710.5356.
2. Jimbo M., Miwa T., Soliton equations and infinite dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
3. Kac V., van de Leur J., The geometry of spinors and the multicomponent BKP and DKP hierarchies, in The Bispectral Problem (Montreal, PQ, 1997), CRM Proc. Lecture Notes, Vol. 14, Amer. Math. Soc., Providence, RI, 1998, 159-202, solv-int/9706006.
4. Hirota R., Ohta Y., Hierarchies of coupled soliton equations. I, J. Phys. Soc. Japan 60 (1991), 798-809.
5. Adler M., Horozov E., van Moerbeke P., The Pfaff lattice and skew-orthogonal polynomials, Internat. Math. Res. Notices 1999 (1999), no. 11, 569-588, solv-int/9903005.
6. Adler M., Shiota T., van Moerbeke P., Pfaff τ-functions, Math. Ann. 322 (2002), 423-476, solv-int/9909010.
7. Adler M., van Moerbeke P., Toda versus Pfaff lattice and related polynomials, Duke Math. J. 112 (2002), 1-58.
8. Kakei S., Orthogonal and symplectic matrix integrals and coupled KP hierarchy, J. Phys. Soc. Japan 99 (1999), 2875-2877, solv-int/9909023.
9. Kakei S., Dressing method and the coupled KP hierarchy, Phys. Lett. A 264 (2000), 449-458, solv-int/9909024.
10. van de Leur J., Matrix integrals and the geometry of spinors, J. Nonlinear Math. Phys. 8 (2001), 288-310, solv-int/9909028.
11. Adler M., van Moerbeke P., Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. of Math. (2) 153 (2001), 149-189, math-ph/0009001.
12. Adler M., Kuznetsov V.B., van Moerbeke P., Rational solutions to the Pfaff lattice and Jack polynomials, Ergodic Theory Dynam. Systems 22 (2002), 1365-1405, nlin.SI/0202037.
13. Isojima S., Willox R., Satsuma J., On various solutions of the coupled KP equation, J. Phys. A: Math. Gen. 35 (2002), 6893-6909.
14. Isojima S., Willox R., Satsuma J., Spider-web solutions of the coupled KP equation, J. Phys. A: Math. Gen. 36 (2003), 9533-9552.
15. Kodama Y., Maruno K.-I., N-soliton solutions to the DKP hierarchy and the Weyl group actions, J. Phys. A: Math. Gen. 39 (2006), 4063-4086, nlin.SI/0602031.
16. Kodama Y., Pierce V.U., Geometry of the Pfaff lattice, arXiv:0705.0510.
17. Willox R., On a coupled Toda lattice and its reductions as derived from the coupled KP hierarchy, Proceedings of the Research Institute for Applied Mechanics, Kyushu University 13ME-S4 (2002), 18-23 (in Japanese).
18. Willox R., On a generalized Tzitzeica equation, Glasgow Math. J. 47 (2005), 221-231.
19. Santini P.M., Nieszporski M., Doliwa A., An integrable generalization of the Toda law to the square lattice, Phys. Rev. E 70 (2004), 056615, 6 pages, nlin.SI/0409050.
20. Hu X.-B., Li C.-X., Nimmo J.J.C., Yu G.-F., An integrable symmetric (2+1)-dimensional Lotka-Volterra equation and a family of its solutions, J. Phys. A: Math. Gen. 38 (2005), 195-204.
21. Gilson C.R., Nimmo J.J.C., The relation between a 2D Lotka-Volterra equation and a 2D Toda lattice, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 169-179.
22. Zabrodin A.V., Dispersionless limit of Hirota equations in some problems of complex analysis, Theoret. and Math. Phys. 129 (2001), 1511-1525, math.CV/0104169.
23. 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.
24. Kostov I.K., Krichever I., Mineev-Weinstein M., Wiegmann P.B., Zabrodin A., τ-function for analytic curve, in Random Matrices and Their Applications, Editors P. Bleher and A. Its, Math. Sci. Res. Inst. Publ., Vol. 40, Cambridge University Press, Cambridge, 2001, 285-299, hep-th/0005259.
25. Boyarsky A., Marshakov A., Ruchayskiy O., Wiegmann P., Zabrodin A., Associativity equations in dispersionless integrable hierarchies, Phys. Lett. B 515 (2001), 483-492, hep-th/0105260.
26. Teo L.-P., Analytic functions and integrable hierarchies - characterization of tau functions, Lett. Math. Phys. 64 (2003), 75-92, hep-th/0305005.
27. Kodama Y., Pierce V.U., Combinatorics of dispersionless integrable systems and universality in random matrix theory, arXiv:0811.0351.
28. Konopelchenko B., Martinez Alonso L., Integrable quasi-classical deformations of algebraic curves, J. Phys. A: Math. Gen. 37 (2004), 7859-7877, nlin.SI/0403052.
29. Kodama Y., Konopelchenko B., Martinez Alonso L., Integrable deformations of algebraic curves, Theoret. and Math. Phys. 144 (2005), 961-967.
30. 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 Scientific Publishing, Singapor, 1983, 39-119.
31. Dickey L., Soliton equations and Hamiltonian systems, Advanced Series in Mathematical Physics, Vol. 12, World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
32. Kac V.G., van de Leur J.W., The n-component KP hierarchy and representation theory, in Important Developments in Soliton Theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, 302-343, hep-th/9308137.
33. Takasaki K., Takebe T., Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743-808, hep-th/9405096.