Abstract
The Darboux transformation has been discovered several times in history (cf. references in [EK] and in [G3]): the reason for this is its deep geometric significance. In this note we explore some geometric aspects of the transformation and their relevance to the difficult problems of describing explicitly higher-rank commutative algebras of ordinary differential operators and KP flows.
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References
M. Adler and P. van Moerbeke, Limits of operators and the geometry of Grassmannians, in preparation.
M. Adler and J. Moser, On a class of polynomials connected with the Korteweg de Vries equations, Comm. Math. Phys. 61 (1978), 1–30.
J.L. Burchnall and T.W. Chaundy, a) Commutative ordinary differential operators, Proc. London Math. Soc. 211 (1923), 420–440; b) —, Proc. Royal Soc. London (A) 118 (1928), 557–583; c) — II. The identity Pn = Q m, ibid. 134 (1932), 471–485.
P. Dehornoy, Opérateurs différentiels et courbes elliptiques, Compositio Math. 43 (1981), 71–99.
J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240.
V.G. Drinfeld and V.V. Sokolov, On equations related to the Korteweg de Vries equation, Soviet Math. Dokl. 32 (1985), 361–365.
F. Ehlers and H. Knörrer, An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg-de Vries equation, Comment. Math. Helvetici 57 (1982), 1–10.
N.M. Ercolani and H.P. McKean, a) A quick proof of Fay’s secant identities, in Analysis, et cetera, Moser’s Birthday Volume, eds. P.H. Rabinowitz and E. Zhender, Academic Press Inc., Boston 1990, pp. 301–307; b) Geometry of KdV (4): Abel sums, Jacobi variety, and theta functions in the scattering case, Invent. Math. 99 (1990), 483–544.
F.A. Grünbaum, Commuting pairs of linear ordinary differential operators of orders four and six, Physica 31D (1988), 424–433.
F.A. Grünbaum, The Kadomtsev-Petviashvili equation: an elementary approach to the “rank 2” solutions of Krichever and Novikov, Phys. Lett. A 139 (1989), 146–150.
F.A. Grünbaum, Darboux’s method and some “rank two” explicit solutions of the KP equation, in Nonlinear evolution equations: integrability and spectral methods, eds. A Degasperis, A.P. Fordy and M. Lakshmanan, Manchester University Press, Manchester, UK, 1990, 271–277.
M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Pub. RIMS Kyoto Univ. 19 (1983), 943–1001.
I.M. Krichever, Commutative rings of ordinary differential operators, Functional Anal. Appl. 12 (1978), 175–185.
I.M. Krichever and S.P. Novikov, Holomorphic fiberings and nonlinear equations, Finite zone solutions of rank 2, Sov. Math. Dokl. 20 (1979), 650–654.
G.A. Latham, Solutions of the Kadomtsev-Petviashvili equation associated to higher rank commuting ordinary differential operators, PhD Thesis, Berkeley, 1989.
E. Previato and G. Wilson, Vector bundles over curves and solutions of the KP equations, Proc. Sympos. Pure Math. 49 (1989), 553–569.
G.B. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. I.H.E.S. 61 (1985), 5–65.
S.I. Svinolupov, V. Sokolov and R. Yamilov, On Bäcklund transformations for integrable evolution equations, Soviet Math. Dokl. 28 (1983), 165–168.
G. Wilson, Algebraic curves and soliton equations, in Geometry Today, eds. E. Arbarello et al., Birkhäuser, Boston, 1985, 303–329.
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Latham, G., Previato, E. (1994). Higher Rank Darboux Transformations. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds) Singular Limits of Dispersive Waves. NATO ASI Series, vol 320. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2474-8_9
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