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Painlevé equations

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1: 32.12 Asymptotic Approximations for Complex Variables
§32.12(i) First Painlevé Equation
§32.12(ii) Second Painlevé Equation
§32.12(iii) Third Painlevé Equation
2: 32.13 Reductions of Partial Differential Equations
§32.13 Reductions of Partial Differential Equations
§32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations
The sine-Gordon equation
§32.13(iii) Boussinesq Equation
The Boussinesq equation
3: 32.1 Special Notation
The functions treated in this chapter are the solutions of the Painlevé equations P I P VI .
4: Mark J. Ablowitz
ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …
5: 32.4 Isomonodromy Problems
§32.4(ii) First Painlevé Equation
§32.4(iii) Second Painlevé Equation
§32.4(iv) Third Painlevé Equation
§32.4(v) Other Painlevé Equations
6: 32.5 Integral Equations
§32.5 Integral Equations
7: 32.17 Methods of Computation
The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …
8: Alexander A. Its
9: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … …
10: Bibliography Q
  • H. Qin and Y. Lu (2008) A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.