Unified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

This paper is an attempt to present and discuss at some length the Singular Mani- fold Method. This Method is based upon the Painleve Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Backlund or Darboux Transformations as well as τ -functions, in a unified way. Besides to present the ba- sics of the Method we exemplify this approach by applying it to four equations in (1 + 1)-dimensions. Two of them are related with the other two through Miura trans- formations that are also derived by using the Singular Manifold Method.

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