Vertex operator solutions to the discrete KP-hierarchy.pdf

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Vertex operator solutions to the discrete KP-hierarchy.pdf

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Vertex operator solutions to the discrete KP-hierarchy

a r X i v : s o l v - i n t / 9 9 1 2 0 1 4 v 1 2 3 D e c 1 9 9 9 Vertex operator solutions to the discrete KP-hierarchy? M. Adler? P. van Moerbeke? August 24, 1998 Contents 1 The KP τ-functions, Grassmannians and a residue formula 7 2 The existence of a τ-vector and the discrete KP bilinear iden- tity 13 3 Sequences of τ-functions, flags and the discrete KP equation 18 4 Discrete KP-solutions generated by vertex operators 23 5 Example of vertex generated solutions: the q-KP equation 24 Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite ?The final version appeared in: Comm. Math. Phys., 203, 185–210 (1999) ?Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. E- mail: adler@math.brandeis.edu. The support of a National Science Foundation grant # DMS-9503246 is gratefully acknowledged. ?Department of Mathematics, Universite? de Louvain, 1348 Louvain-la-Neuve, Bel- gium and Brandeis University, Waltham, Mass 02454, USA. E-mail: vanmoer- beke@geom.ucl.ac.be and @math.brandeis.edu. The support of a National Science Founda- tion grant # DMS-9503246, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. 1 Adler-van Moerbeke:Discrete KP August 24, 1998 §0, p.2 sequence obtained by Darboux transforming an arbitrary KP solution recur- sively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space x-variable gets re- placed by a discrete n-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag). The discrete KP hierarchy can thus be viewed as a container for an entire ensemble of vertex or Darbo