Abstract
We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.
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M. J. Ablowitz and J. Satsuma,J. Math. Phys.,19, 2180–2186 (1978).
H. Airault, H. P. McKean, and J. Moser,Commun. Pure Appl. Math.,30, 95–148 (1977).
F. Calogero,Nuovo Cimento B,43, 177–241 (1978).
D. V. Chudnovsky and G. V. Chudnovsky,Nuovo Cimento B,40, 339–353 (1977).
I. M. Krichever,Funct. Anal. Appl.,12, 59–61, (1978).
M. D. Kruskal, “The Korteweg-de Vries equation and related evolution equations,” in:Nonlinear Wave Motion (A. C. Newell, ed.) (Lect. Appl. Math., Vol. 15), Am. Math. Soc., Providence, RI (1974), pp. 61–83.
V. B. Matveev,Lett. Math. Phys.,3, 503–512 (1979).
T. Shiota,J. Math. Phys.,35, 5844–5849 (1994).
G. Wilson,Invent. Math.,133, 1–41 (1998).
M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold,” in:Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo) (P. Lax and H. Fujita, eds.), North-Holland, Amsterdam (1982), pp. 259–271.
Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro,Progr. Theor. Phys. Suppl.,94, 210–241 (1988).
J. P. Zubelli,Lett. Math. Phys.,24, 41–48 (1992).
G. Wilson,J. Reine Angew. Math.,442, 177–204 (1993).
J. Harnad and A. Kasman, eds.,The Bispectral Problem (Montreal, PQ, 1997), Am. Math. Soc., Providence, RI (1998).
G. Falqui, F. Magri, and M. Pedroni,Commun. Math. Phys.,197, 303–324 (1998).
K. Takasaki,Rev. Math. Phys.,1, 1–46 (1989).
I. V. Cherednik,Funct. Anal. Appl.,12, 195–203 (1979).
G. Wilson,Quart. J. Math. Oxford,32, 491–512 (1981).
E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Transformation groups for soliton equations,” in:Proc. RIMS Symp. Nonlinear Integrable Systems: Classical Theory and Quantum Theory (M. Jimbo and T. Miwa, eds.), World Scientific, Singapore (1983), pp. 39–119.
L. A. Dickey,Soliton Equations and Hamiltonian Systems (Adv. Series in Math Phys, Vol. 12), World Scientific, Singapore (1991).
P. Casati, G. Falqui, F. Magri, and M. Pedroni,J. Math. Phys.,38, 4606–4628 (1997).
M. F. de Groot, T. J. Hollowood, and J. L. Miramontes,Commun. Math. Phys.,145, 57–84 (1992).
G. Falqui, F. Magri, and G. Tondo,Theor. Math. Phys. (forthcoming).
K. Takasaki, “Integrable systems as deformations ofD-modules”, in:Theta Functions—Bowdoin 1987 (L. Ehrenpreis and R. C. Gunning, eds.) (Proc. Symp. Pure Math., Vol. 49, Part 1), Am. Math. Soc., Providence, RI (1989), pp. 143–168.
M. Mulase, “Algebraic theory of the KP equations,” in:Perspectives in Mathematical Physics (R. Penner and S.-T. Yau, eds.), International Press, Boston (1994), pp. 151–217.
F. Magri, M. Pedroni, and J. P. Zubelli,Commun. Math. Phys.,188, 305–325 (1997).
W. T. Reid,Riccati Differential Equations, Acad. Press, New York (1972).
V. G. Kac,Infinite Dimensional Lie Algebras (3rd ed.), Cambridge Univ. Press, Cambridge (1990).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 23–36, January, 1999.
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Falqui, G., Magri, F., Pedroni, M. et al. An elementary approach to the polynomial τ-functions of the KP Hierarchy. Theor Math Phys 122, 17–28 (2000). https://doi.org/10.1007/BF02551166
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DOI: https://doi.org/10.1007/BF02551166