Nonlinear Partial Differential Equation Of Second Order Variable Coefficient Between The Classification Of Their Transformation

In this paper we classify Miura transformations uâ†'v from second-order partial differential equations of the form uxx=F{x,t,u,ux,ut) defined via associated integrable systems of the form where the functions F, w, ζ, η depend explicitly on the independent variables x and t. From the first equation of the integrable system we get u=vx-w(x,t,v). Substituting it into the second equation of the integrable system yields partial differential equations satisfied by v. So we don’t need to assume the form of v-equation in our classification.Our results show that such a partial differential equation and its associated integrable system are equivalent to one of the following four cases.1. The partial differential equation is and the associated integrable system is where f(x,t),r(x,t),a(x,t,u)(≠0)are arbitrary smooth functions in their corresponding variables, and the smooth functions g,h satisfy gt=hx. In this case, v satisfies the following partial differential equation2. The partial differential equation is and the associated integrable system is where the functions w,ζ,, p are given by p(x,t,u)=w,+wvζ-ζx-ζv(w+u), c1(x,t),c2(x,t),c3(x,t)(≠0),λ(x,t)are arbitrary smooth fucntions in their corresponding variables, and the function G is given by In this case, v satisfies the following partial differential equation3. The partial differential equation is and the associated integrable system is where A(x,t),r(x,t),β(x,t,u)(≠0) are arbitrary smooth functions in their corresponding variables, and smooth functions μ,h satisfy μt=hx. In this case, v satisfies the following partial differential equation 4. The partial differential equation is and the associated integrable system is where and c1(x,t)(≠0),c2(x,t),λ(x,t) are arbitrary smooth functions in corresponding variables. In this case, v satisfies the following partial differential equationAs applications of the above Miura transformations, we finally give some examples. By choosing some trivial solutions of the u-equation and then substituting them into the associated integrable systems, we generate nontrivial solutions of the corresponding v-equation.