Painlevé equations
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1: 32.12 Asymptotic Approximations for Complex Variables
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§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
4: Mark J. Ablowitz
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►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.
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5: 32.4 Isomonodromy Problems
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§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
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►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).
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8: Alexander A. Its
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►Books by Its are The Isomonodromic Deformation Method in the Theory of Painlevé Equations (with V.
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9: 32.10 Special Function Solutions
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