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1: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
►§15.10(i) Fundamental Solutions
… ►This is the hypergeometric differential equation. … ► … ►The connection formulas for the principal branches of Kummer’s solutions are: …2: 30.2 Differential Equations
§30.2 Differential Equations
►§30.2(i) Spheroidal Differential Equation
… ► … ►The Liouville normal form of equation (30.2.1) is … ►§30.2(iii) Special Cases
…3: 31.2 Differential Equations
§31.2 Differential Equations
►§31.2(i) Heun’s Equation
… ►§31.2(ii) Normal Form of Heun’s Equation
… ►§31.2(v) Heun’s Equation Automorphisms
… ►Composite Transformations
…4: 29.2 Differential Equations
§29.2 Differential Equations
►§29.2(i) Lamé’s Equation
… ►§29.2(ii) Other Forms
… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).5: 32.2 Differential Equations
§32.2 Differential Equations
… ►The six Painlevé equations – are as follows: … ►be a nonlinear second-order differential equation in which is a rational function of and , and is locally analytic in , that is, analytic except for isolated singularities in . … ►When this is a nonlinear harmonic oscillator. …6: 28.2 Definitions and Basic Properties
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§28.2(i) Mathieu’s Equation
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28.2.1
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►This is the characteristic equation of Mathieu’s equation (28.2.1).
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§28.2(iv) Floquet Solutions
… ► …7: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
►When is replaced by , (28.2.1) becomes the modified Mathieu’s equation: ►
28.20.1
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28.20.2
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►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
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8: Peter A. Clarkson
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►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations.
… Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M.
…His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M.
…He is also coauthor of the book From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics (with J.
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9: Mark J. Ablowitz
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►Ablowitz is an applied mathematician who is interested in solutions of nonlinear wave equations.
Certain nonlinear equations are special; e.
…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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10: 29.19 Physical Applications
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