Nonlinear, Theory, Merhods & Apphcaorions. Vol. 5, No. 4, pp. 423-432. 0362-546X/81/040423-10 $02.00/O Printed in Great Britain. 0 1981 Pergamon Press Ltd. BACKLUND TRANSFORMATIONS FOR HEREDITARY SYMMETRIES A. S. FOKAS*and B. FuCHssTErNERt California Institute of Technology, Pasadena, CA 91125, U.S.A. (Received 29 February 1980) Key WOI& urul phruses: Symmetries, conservation laws, solitons, Bicklund-transformation. 1. INTRODUCTION A HEREDITARY symmetry Y(U) is an operator-valued function of u (u is an element of a suitable vector space) having the property that it generates a hierarchy of evolution equations for which Y(u) is a strong symmetry (or recursion operator in the terminology of [I]). Since a strong sym- metry describes more or less completely the symmetries, the conservation laws and the soliton solutions (if they exist) of an evolution equation a method for finding hereditary symmetries is very desirable. The aim of this paper is to show that suitable implicit functions defined by B(u, s) = 0 give rise to transformations between hereditary symmetries, thus (in principle) generating out of a given hereditary symmetry a class of others. Further, a Blcklund transformation B(u, s) = 0 between two evolution equations defines such a transformation between the corresponding hereditary symmetries possessed by those equations. This then explains the fact that in general the whole hierarchy of evolution equations has the same Backlund transformation. For a general discussion and for a convenient characterization of Backlund transformations see [2] and [3] respectively. Hereditary symmetries are discussed in [4]. 2. REVIEW OF BASIC NOTIONS In this paper we only deal with differentiable functions F(u): u E E, -+ zyxwvutsrqponmlkjihgfedcbaZ E, between suitable vector spaces E, and E,. By differentiable we always mean a notion of differentiability such that the derivative F”(u) is a linear map between E, and E, and such that the chain rule holds. In other words, we are dealing with functions which are Hadamard-differentiable [S] (with respect to a suitable topology). If 2: E E, the derivative of F(u) in the direction v is denoted by F”(u)[u] and may be calculated by FU(U)[U] = ; F(u + Et$= (). Let B(u, s) be a function in the two arguments u E E, and s E E, with values in some vector space E,. Then by BU, Bs we denote the partial derivatives with respect to the arguments indicated. This function is called admissible if B(u, s) = 0 gives rise to a one-to-one map between the tangent *Research supported in part by the Saul Kaplun Memorial Fund. TPermanent address Universitat Paderborn, Mathematik, D-479 Paderborn, West Germany. 423