Volume 157, number 1 PHYSICS LET1’ERS A 15 July 1991 (1 + 1) -dimensional integrable systems as symmetry constraints of (2 + 1) -dimensional systems Boris Konopeichenko Institute of Nuclear Physics, 90 Novosibirsk 630090, USSR Jurij Sidorenko’ and Walter Strampp 2 Fachbereich 17, Mathematik GH-Universität Kassel, Nora-Platiel-Strasse 1, W-3500 Kassel, Germany Received 6 February 1991; accepted for publication 8 May 1991 Communicated by A.P. Fordy We consider linear problems associated with integrable systems in 2 + 1 dimensions. From the bilocal approach we obtain generating functions for symmetries. Constraining the generating functions to special symmetries leads to integrable systems in 1 + I dimensions. 1. Introduction striction of the KdV flow to the pure multisoliton submanifold turns out as constraint of yi~to the symmetry u~. Many finite dimensional integrable Hamiltonian In this paper we propose to constraint symmetry systems can be obtained as certain constraints of generators to arbitrary symmetries and to generalize (1+1)-dimensional integrable soliton equations [1— the approach to problems in 2+1 dimensions. The 3]. A well known example is the restriction of the Kadomtsev—Petviashvili (KP) equation KdV flow to the pure multisoliton submanifold [2]. In this case we impose the constraint u = cw,~ U~ = ~ Uxxx + 3uu~ + ~ 8x ‘u~ (1) on the KdV potential u and eigenfunctions w~. This is one of the most widely studied integrable equation leads to the finite dimensional integrable system in 2 + 1 dimensions. It is connected with the linear ~ ck~)~=,~wI; i= 1, ..., N. The KdV problem hierarchy can be written as u 1 = 8~(R”l ), n~1, where R = ~ 2 + 2U —8; ‘u~ is the so-called squared eigen- Wy = t~U~ + 2uyi, (2) function operator. Let us consider the squared ei- and its adjoint version genfunctions of the KdV linear problem ~ + 2Uyi=A 2yi as a symmetry generator through the W’— _W~x_2UW*. (3) asymptotic expansion From (2) and (3) we obtain the bilocal quantity ~(x, y, 1) W*(x~, y’, I) which is of fundamental value 00 ~ s,,A~” for the whole KP theory. In particular, it is a key ob- ject in the theory of the bilocal recursion operator for the KP equation (see e.g. refs. [4—7]). In this paper where S 0= 1 and S, = R”1. In view of this, the re- we are mainly interested in the fact that the bilocal quantity ~v(x,y, t)W*(x~, y’, t) after projection onto Permanent address: University of Lwow, Lwow, USSR. the diagonal x=x’, y=y’ acts as a generator of syrn- 2 To whom correspondence should be addressed. metries [4—8]. 0375-9601/91/S 03.50 © 1991 — Elsevier Science Publishers B.V. (North-Holland) 17