Painlevé-Darboux Transformation in Nonlinear PDEs

Abstract

Let us first compare some results concerning invariance properties in the Painleve analysis of a partial differential equation $$ E\left( {x,t,u,Du} \right) = 0 $$ (1) which depends polynomially on its solution u and its partial derivatives Du with respect to x and t. Considering a series expansion for u in the neighbourhood of the singular manifold ϕ(x,t) = 0, we are looking for, as a solution of (1), the formal expression $$ u = \sum\limits_{j = 0}^\infty {{u_j}} {x^{j + p}}, \left( {p constant, {u_0} \ne 0} \right) $$ (2) where the expansion variable x goes to zero as ϕ in the following way: $$ x = \frac{{\alpha \varphi }}{{\beta \varphi + \gamma }} , \frac{\alpha }{\gamma } \ne 0 $$ (3) and the coefficients α, β, γ and uj are only functions of the derivatives Dϕ of ϕ.