Various truncations in Painlevé analysis of PDEs (Q2734887)

The author summarizes the most recent stage of the WTC truncation method for integrable partial differential equations (PDEs) \(E(u)= 0\) with only one family of movable singularities. The method consists of five steps: a Darboux transformation \(u= U+D\log\tau\) (where \(D\) is an operator given by the Painlevé test); the scalar Lax pair \(L_1(u,\lambda)\psi= L_2(u, \lambda)\psi= 0\) (where the vanishing \([L_1, L_2]= 0\) is equivalent with the original equation \(E(u)= 0\)); the link \(D\log\tau= f(\psi)\) between functions \(\tau\) and \(\psi\); the ``truncation'' which provides the operators \(L_1\), \(L_2\) in explicit terms and a differential equation for the function \(U\); and the Bäcklund transformation between \(u\) and \(U\). The exposition is rather concise, however, numerous examples and references are stated.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00047].