A Class Of Third Order Nonlinear Partial Differential Equations With Variable Coefficients Backlund Transformation Classification

In this paper, we classify three-order nonlinear partial differential equations with variable coefficients of the uxxx=F(x,t,u,ux,u,) which admit Backlund transformations defined via integrable systems of form Our discussion is divided into two parts.1. For the case that co is linear in v, we classify completely both the partial differential equations and their associated integrable systems, and the functions F、w and (?) are given by where and q(x,t,u), Q{x,t,u), w1(x,t),01(x,t) and g(t) are arbitrary smooth functions in their corressponding variables. As applications, from a given solution of u-equation, by solving the associated integrable system, we generate some solutions of the corresponding v-equations. 2. For the case that co is nonlinear in v, we prove, when the integrable systems don’t depend explicitly on the variables x and t, that the nonlinear partial differential equations are independ of the variables x and t either. Therefore, by a result in [14], the partial differential equations considered are equivalent to the MKdV or negative MKdV equations.