Nonlocal symmetries and negative hierarchies related to bilinear Bácklund transformation

In this paper, nonlocal symmetries defined by bilinear Bácklund transformation for bilinear potential KdV (pKdV) equation are obtained. By introducing an auxiliary variable which just satisfies the Schwartzian form of KdV (SKdV) equation, the nonlocal symmetry is localized and the Levi transformation is presented. Besides, based on three different types of nonlocal symmetries for potential KdV equation, three sets of negative pKdV hierarchies along with their bilinear forms are constructed. An impressive result is that the coefficients of the third type of (bilinear) negative pKdV hierarchy (N > 0) are variable, which are obtained via introducing an arbitrary parameter by considering the translation invariance of the pKdV equation.