The group-theoretical side of N-pseudoreductions is discussed. The resulting equations are shown to be easy transformations of the N-KdV hierarch

Post, G.

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Post, Gerhard F.

University of Twente Research Information

J .  Phqs. A:  Math.  Gen.  20 (1987)  6591-6592. Printed in  the U K  C OM NIENT Note on 'N-pseudoreductions' of the KP hierarchy G Post Faculty of Applied Mathematics, University of Twente, PO Box 217, 7500 A E  Enschede, Netherlands Received 27 J u l y  1987 Abstract. The group-theoretical side of N-pseudoreductions is discussed. The resulting equations are  shown to be easy transformations of the N - K d V  hierarchy. In Hirota ( 1986), so-called N-pseudoreductions of the K P  hierarchy are introduced. Our purpose here is to give a group-theoretical interpretation, in the spirit of Segal and Wilson (1985). 'This leads to a description of the pseudoreductions in terms of the pseudodifferential operator a L =  D + u , D - ' + u , D - ~ +  . . .  D =-; U ,  = U ,  ( X  = t ,  , t z ,  . . . ). ax The K P  hierarchy is given by aL/at,  == [ ( L ' ) , ,  L ]  ( i z  1).  The 7-functions of the K P  hierarchy are elements of the orbit of 1 E C [ [  t , ,  t 2 , .  . . I ]  under the action p of a group G related to gl,. The elements of G are linear operators on a part of C [ [  t , ,  t 2 , .  . . I ] .  Examples of G can be found in Kac (1983) and Segal and Wilson (1985). G is a central extension of some 6, 6 being a group of Z x Z  matrices. Now for N-pseudoreduced T functions, it holds (by definition) Clearly g E p (  G )  with g, - - c,, "1 = 0 [ a:, a t ,  maintains this condition. Projecting G onto d this condition reads The elements from G, for which the projection satisfies ( 2 ) ,  will be our group Gpr, associated to the pseudoreduction ( 1  1. Equation ( 2 )  has only trivial solutions (namely linear combinations of the '1') if the group G is contained in { A  = (u#,) Iu, ,  = 0 if i - j  > M ,  some M } .  0305-4470/87/ 186591 +02$02.50 1987 IOP Publishing Ltd 6591 6592 G Post To find non-trivial solutions one has to pass to extensions as in Segal and Wilson (1985). Moreover, using this Grassmannian approach, one proves the following state- ments: an  element of G determines an operator L ;  in the case co=O, condition (2) yields L" = (LN)+, leading to the Kdv-type hierarchies. In a similar way, one can prove that in the general case N L - C O L  = (L"  - COL)+ =: P. The following equation holds for P :  In particular, since [( L N ) + ,  P ]  = [c,D, PI ,  we find that u2,,,  = u ~ , ~ .  Calculating the N = 2  (modified) and N = 3  pseudoreductions, using the time flow of (3), we find exactly (4.7) and (4.3) of Hirota (1986). By equation (3 )  it is clear that pseudoreductions will yield linear combinations of the canonical (i.e. graded) Kdv-type equations. In this sense pseudoreductions of the KP-hierarchy are equivalent to the usual reductions, and  yield nothing new. References Hirota R 1986 Physica D 18 161 Kac  V G 1983 Infinite dimensional Lie Algebras (Basle:  Birkhauser) Segal G and Wilson G 1985 Publ. Math. IHES no 61 p 5 

Note on 'N-pseudoreductions' of the KP hierarchy

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