Korteweg%E2%80%93de%20Vries%20equation
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11: 23.21 Physical Applications
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§23.21(ii) Nonlinear Evolution Equations
►Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). … ►Ellipsoidal coordinates may be defined as the three roots of the equation …12: 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
…13: 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: …14: 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
…15: 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).16: Bibliography R
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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I.
Gauthier-Villars, Paris.
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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II.
Gauthier-Villars, Paris.
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Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome III.
Collection Technique et Scientifique du C. N. E. T.
Gauthier-Villars, Paris.
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Sur quelques classes nouvelles de polynômes orthogonaux.
C. R. Acad. Sci. Paris 188, pp. 1023–1025.
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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent.
Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
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17: 22.19 Physical Applications
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►Classical motion in one dimension is described by Newton’s equation
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§22.19(iii) Nonlinear ODEs and PDEs
►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. These include the time dependent, and time independent, nonlinear Schrödinger equations (NLSE) (Drazin and Johnson (1993, Chapter 2), Ablowitz and Clarkson (1991, pp. 42, 99)), the Korteweg–de Vries (KdV) equation (Kruskal (1974), Li and Olver (2000)), the sine-Gordon equation, and others; see Drazin and Johnson (1993, Chapter 2) for an overview. … …18: Bibliography C
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Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams.
J. Comput. Appl. Math. 115 (1-2), pp. 93–99.
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Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique.
Masson et Cie, Paris (French).
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Accélération de calcul de nombres de Bernoulli.
J. Number Theory 28 (3), pp. 347–362 (French).
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Inverse Acoustic and Electromagnetic Scattering Theory.
2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.
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Exact elliptic compactons in generalized Korteweg-de Vries equations.
Complexity 11 (6), pp. 30–34.
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