Painlevé-Darboux Transformation in Nonlinear PDEs

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


Let us first compare some results concerning invariance properties in the Painlevé analysis of a partial differential equation
$$ E\left( {x,t,u,Du} \right) = 0 $$
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
$$ u = \sum\limits_{j = 0}^\infty {{u_j}} {x^{j + p}}, \left( {p constant, {u_0} \ne 0} \right) $$
where the expansion variable x goes to zero as ϕ in the following way:
$$ x = \frac{{\alpha \varphi }}{{\beta \varphi + \gamma }} , \frac{\alpha }{\gamma } \ne 0 $$
and the coefficients α, β, γ and uj are only functions of the derivatives Dϕ of ϕ.


Manifold Soliton 


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  1. [1]
    Weiss J., Tabor M. and Carnevale G., J. Math. Phys. 24, 522 (1983).CrossRefMATHADSMathSciNetGoogle Scholar
  2. [2]
    Conte R., P.L.A. 140, 383 (1989).CrossRefMathSciNetGoogle Scholar
  3. [3]
    Conte R. and Musette M., J. Phys. A: Math. Gen. 22, 169 (1989).CrossRefMATHADSMathSciNetGoogle Scholar
  4. [4]
    Levi D., Rep. on Math. Phys. 23, 41 (1986) Levi D. and Ragnisco A., Inverse Problems 4, 815 (1989).CrossRefMATHADSGoogle Scholar
  5. [5]
    Musette M., “Painlevé analysis for integrable and nonintegrable p.d.e.’s” presented at the VI International Workshop “Solitons and applications”, 25-27 august 1989, Dubna, USSR - VUB/TENA/89/10.Google Scholar
  6. [6]
    Wahlquist H.D. and Estabrook F.B., J. Math. Phys. J.6, 1 (1975) Estabrook F.B. and Wahlquist H.D., J. Math. Phys. 17, 1293 (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

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

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