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Painlevé-Darboux Transformation in Nonlinear PDEs

  • M. Musette
Conference paper
Part of the Inverse Problems and Theoretical Imaging book series (IPTI)

Abstract

Let us first compare some results concerning invariance properties in the Painlevé 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.

Keywords

Invariant Function Darboux Transformation Linear Differential Operator Expansion Variable Singular Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • M. Musette
    • 1
  1. 1.Dienst Theoretische NatuurkundeVrije Universiteit BrusselBrusselsBelgium

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