Explicit Solution For A Sort Of Nonlinear Evolution Equations

As we all kown,Darboux transformation is one of more direct and effective methods to study the explicit solutions of soliton equations.However,it is not easy to construct the Darboux transformation for CH-type equatons.The aim of the present paper is to apply the reciprocal transformation and Darboux transformation to study the smooth soliton solutions in a parametric form to the Novikov equation and the first negative flow of the Novikov hierarchy.First,with the help of reciprocal transformation,we relate the Novikov equation and the first negative flow of the Novikov hierarchy to the first negative flow of Sawada-Kotera hierarchy and Sawada-Kotera equation,respectively.Second,resorting to the Darboux transformation of the first negative flow of Sawada-Kotera hierarchy and Sawada-Kotera equation,reciprocal transformation and the asymptotic behaviours of wave functions,we obtain smooth soliton solutions to the Novikov equation and the first negative flow of the Novikov hierarchy.