Multiple singular manifold method and extended direct method: Application to the Burgers equation

This paper considers the relationship between the multiple singular manifold method (MSMM) and the extended direct method (EDM) for studying partial differential equations. It is shown that the similarity reductions using EDM can be obtained by MSMM. The prototype example for illustrating the approach is the Burgers equation, which is the simplest evolution equation to embody nonlinearity and dissipation. As a conclusion of the MSMM, we obtain a set of Backlund transformations of the Burgers equation.