On Miura Transformations Among Second-order Nonlinear Partial Differential Equations

In this paper we classify Miura transformations u v from second-order partial differential equations of the form u xx = F(u ,ux,ut) defined via associated integrable systems of the formThe first equation of the above integrable system gives u = v x?ω( v), and substituting it into the second equation yields a partial differential equation only involvingv . So in the classification we do not need to assume the form of the partial differential equations about v . Our results show that there are four kinds of such nonlinear partial differential equations. The first one is the equation u xx = p (u ) ? p′(u )u x + ut,where p is an arbitrary nonlinear smooth function, and the associated integrable system is whereλis an arbitrary constant. In this case, v satisfies the equation vt = p ( v x ?λ? v )+ vxx;The second one is just the Burgers equation u xx = 2u u x + ut,and the associated integrable system is In this case, v satisfies the nonlinear partial differential equation v t = vxx?vx2 +2e?vvx, which is equivalent to the Burgers equation; The third one is the equation uux2+uut, where p is an arbitrary smooth function andμan arbitrary non-zero constant, and the associated integrable system is whereλis arbitrary constant, In this case, v solves the equationThe last one is the equation u xx =λu x + ue u u x + e uut,whereλis arbitrary constant, and the associated integrable system is In this case, v solves the equation [ln( ) 1] .As applications of the Miura transformations, from some special solutions of the u -equations, by solving the associated integrable systems, we generate solutions of the corresponding v -equations.