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nonlinear partial differential equations

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1: 23.21 Physical Applications
§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. …
2: 9.16 Physical Applications
These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …
3: 22.19 Physical Applications
§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. …
4: Peter A. Clarkson
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. …
5: 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)). …and the nonlinear Schrödinger equations …Here, and in what follows, x , y , and t suffixes indicate partial derivatives. …
6: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
Corresponding numerical factors in this example for other zeros and other values of j are obtained in Gautschi (1984, §4).
§3.8(vii) Systems of Nonlinear Equations
For fixed-point iterations and Newton’s method for solving systems of nonlinear equations, see Gautschi (1997a, Chapter 4, §9) and Ortega and Rheinboldt (1970). …
7: 18.38 Mathematical Applications
Differential Equations
Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. … While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …with H n ( x ) as in §18.3, satisfies the Toda equation
8: Bibliography K
  • A. A. Kapaev (1988) Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).
  • A. V. Kashevarov (2004) The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.
  • J. Koekoek, R. Koekoek, and H. Bavinck (1998) On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.
  • J. J. Kovacic (1986) An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2 (1), pp. 3–43.
  • M. D. Kruskal (1974) The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.