integrable equations

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1: 21.9 Integrable Equations

§21.9 Integrable Equations

►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►

2: 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). …

3: 33.23 Methods of Computation

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§33.23(iii) Integration of Defining Differential Equations

►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). …

4: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. …

5: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …

6: 9.17 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. …

7: Bibliography B

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A. I. Bobenko (1991) Constant mean curvature surfaces and integrable equations. Uspekhi Mat. Nauk 46 (4(280)), pp. 3–42, 192 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 46(1991), no. 4, pp. 1–45
External Links:: ISSN 0042-1316, MathReview (P. E. Markov), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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8: 28.34 Methods of Computation

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(a)

Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

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(b)

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

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9: 14.32 Methods of Computation

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Numerical integration (§3.7) of the defining differential equations (14.2.2), (14.20.1), and (14.21.1).

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10: 15.19 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. …