On symmetries of KdV-like evolution equations

It is well known that provided scalar (1+1)-dimensional evolution equation possesses the infinitedimensional commutative Lie algebra of time-independent non-classical symmetries, it is either linearizable or integrable via inverse scattering transform [1, 2]. The standard way to prove the existence of such algebra is to construct the recursion operator [2]. But Fuchssteiner [3] suggested an alternative way to do that: if the evolution equation possesses time-independent mastersymmetry and several time-independent symmetries, the required commutative algebra may be generated by the repeated commuting of mastersymmetry with time-independent symmetries. In its turn, to possess the mastersymmetry, the equation in question must have (at least one) polynomial in time t symmetry. This fact was one of the main reasons of growing interest to the study of whole algebra of time-dependent symmetries of evolution equations [10, 11, 12]. However, the complete description of this algebra even for the simplest case of scalar (1+1)dimensional evolution equation is an extremely difficult task, which hardly may be carried out without any a priori conjectures (say, that all the symmetries are polynomial in time t). To the best of author’s knowledge, in the class of nonlinear evolution equations the complete algebras of time-dependent symmetries were found only for KdV equation by Magadeev and Sokolov [5] and for KdV and Burgers equations by Vinogradov et al. [6]. In [6] there were also proved two no-go theorems, which show, when the symmetries of third order KdV-like and second order Burgers-like equations are exhausted by Lie ones. Orlov and Winternitz [7] and Orlov and Shul’man [8] have constructed the rich sets of symmetries of (2+1)-dimensional KP hierarchy and the symmetries of (1+1)-dimensional integrable systems, using the technique of ”infinitesimal dressing”. Unfortunately, it is by no means clear how to pick out from the whole set of non-local symmetries, found in [7, 8], local symmetries and whether all the symmetries of the systems in question are given by that construction. Surprisingly enough, for the scalar (1+1)-dimensional KdV-like non-linearizable evolution equation with time-independent coefficients it is possible to establish the general form of time dependence of its symmetries. Namely, as we show below, any symmetry of such equation as a function of t is a linear combination of products of the exponents by polynomials (i. e. of quasipolynomials).