Nonlinear Partial Differential Equations B (?) Backlund Transform

With the development of science, non-linear phenomena appear in the natu-ral sciences, engineering technology and many other areas, then the correspond-ing non-linear models are complicated. Non-linear equations, describing the above models, become an important research topic. There are many methods to solve non-linear partial differential equations, one of which is Backlund trans-formation. On one hand, this approach could construct a new solution from the known solution, and obtain Multi-soliton solutions of the original equation by repeated application. On the other hand, the solution of another equation could be deduced from the one of known equation via the approach. Thus Backlund transformation is an effective method to solve partial differential equations.In this paper, we find the integrable systems of Burgers equation and uχχχ= auuχ+but, then deduce all Auto-Backlund transformation of those equations under the above integrable systems. The conclusions obtained are dif-ferent from the ones under other methods, and they give all the Auto-Backlund transformation under the integrable system of equations. In the end, the Miura transformation between the equations is obtained by the integrable system of partial differential equations. These transformations are foundation for getting more new exact solutions of non-linear partial differential equations.This article is arranged as follows:Chapter 1 introduces the Backlund transformation, and gives two methods to find the Auto-Backlund transformation by some examples:WTC method and extended homogeneous balance method.Chapter 2 finds the integrable systems of Burgers equation and uχχχ= auuχ+but, under which we derive all of the Auto-Backlund transformation of the equations.Chapter 3 deduces a class of Miura transformations between simple non-linear equations by their integrable systems.