Symmetry reductions and exact solutions of two new generalized negative KdV type equations

We give a simple geometric interpretation of the mapping of the negative KdV equation [Formula presented] as proposed by Qiao and Li (2011) [26] and the Fuchssteiner equation [Formula presented] using geometry of projective connection on S¹ or stabilizer set of the Virasoro orbit. We propose a similar connection between [Formula presented] and [Formula presented] with the higher-order negative KdV equations of Fuchssteiner type described as [Formula presented] and [Formula presented] respectively. Then we perform the symmetry analysis of these newly found equations and study one-dimensional optimal classifications of their inequivalent subalgebras. We then obtain several exact solutions for these two equations by using those subalgebras. Further we establish some traveling wave solutions for these two equations. Moreover we construct some new exact solutions of the negative KdV equation which have not been reported in earlier studies. We illustrate the physical significance of some of the obtained solutions by performing numerical simulations and they appeared to be several kinds of soliton solutions.