Virasoro Symmetry Algebra of Dirac Soliton Hierarchy

A hierarchy of first-degree time-dependent symmetries is proposed for Dirac soliton hierarchy and their commutator relations with time-dependent symmetries are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is elucidated and a Lax operator algebra associated with Virasoro symmetry algebra is given. The main purpose of the present letter is to construct a hierarchy of first-degree time-dependent symmetries for Dirac soliton hierarchy [1]. This kind of symmetries was first proposed in the famous Olver’s paper [2]. They include ones corresponding to Galilean invariance and scalar invariance and they are generally related to the first-degree master symmetries [3]. Afterwards, some theory to describe timedependent symmetries were developed for various classes of soliton equations [4] [5]. Moreover it is found that for some nonlinear systems, there exist W∞ symmetry algebras involving arbitrary functions of some independent variables, for example, time variable t [6]. However, for systems of evolution equations, the kind of symmetries involving arbitrary functions of time variable t doesn’t exist [7]. These systems may possess polynomial time-dependent symmetries, which relate to master symmetries of any degree. Usually only first-degree time-dependent symmetries can be found for soliton equations in 1+1 dimensions. Recently, in terms of Lax operators, a simple but systematic scheme for generating first-degree time-dependent symmetries in 1 + 1 dimensions has been established in [8]. Here we would like to discuss the case of Dirac soliton hierarchy through that trick. Dirac soliton hierarchy reads as [1] [9] utm = (