KP equation
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1: 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
…2: 31.2 Differential Equations
§31.2 Differential Equations
►§31.2(i) Heun’s Equation
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31.2.1
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§31.2(v) Heun’s Equation Automorphisms
… ►Composite Transformations
…3: 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).4: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
►§15.10(i) Fundamental Solutions
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15.10.1
►This is the hypergeometric differential equation.
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5: 32.2 Differential Equations
§32.2 Differential Equations
►§32.2(i) Introduction
►The six Painlevé equations – are as follows: … ►§32.2(ii) Renormalizations
… ► …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: 21.9 Integrable Equations
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►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)):
…The KP equation has a class of quasi-periodic solutions described by Riemann theta functions, given by
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►Furthermore, the solutions of the KP equation solve the Schottky
problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)).
Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4).
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9: Bibliography D
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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes.
Phys. Rev. E (3) 57 (1), pp. 252–275.
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Formes canoniques des équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
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Sur les équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.
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The KP equation with quasiperiodic initial data.
Phys. D 123 (1-4), pp. 123–152.
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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation.
Math. Phys. Anal. Geom. 3 (1), pp. 49–74.
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