nonlinear

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11—20 of 32 matching pages

11: 21.9 Integrable Equations

… ►

21.9.1

4u_{t}=6uu_{x}+u_{xxx},

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Symbols:: $u(x,y,t)$ : elevation
Permalink:: http://dlmf.nist.gov/21.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.9 and Ch.21

►and the nonlinear Schrödinger equations …

12: 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. 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. …

13: 32.15 Orthogonal Polynomials

… ►Then

u_{n}(z)=(a_{n}(z))^{2}

satisfies the nonlinear recurrence relation …

14: Bibliography O

… ►

A. B. Olde Daalhuis (2005a) Hyperasymptotics for nonlinear ODEs. I. A Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2060), pp. 2503–2520.

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External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

►

A. B. Olde Daalhuis (2005b) Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.

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External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §2.11(v), §32.12(i).
Encodings:: BibTeX

… ►

J. M. Ortega and W. C. Rheinboldt (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:: Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania
External Links:: MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))
Referenced by:: §3.8(vii).
Encodings:: BibTeX

…

15: Bibliography W

… ►

E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.

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External Links:: Document
Referenced by:: §2.11(vi), §3.9(v), §3.9(vi).
Encodings:: BibTeX

►

E. J. Weniger (1996) Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Computers in Physics 10 (5), pp. 496–503.

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External Links:: Document
Referenced by:: §2.11(vi), §2.11(vi).
Encodings:: BibTeX

… ►

G. B. Whitham (1974) Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Notes:: Reprinted in 1999.
External Links:: ISBN 0471359424, MathReview, ZentralBlatt
Referenced by:: §9.16.
Encodings:: BibTeX

…

16: Bibliography

… ►

M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-38730-2, Document, Link, MathReview (Walter Oevel), ZentralBlatt
Referenced by:: §22.19(ii), §22.19(iii), §32.11(ii), §32.5, §9.16, Profile Mark J. Ablowitz, Profile Peter A. Clarkson.
Encodings:: BibTeX

… ►

V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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External Links:: ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt
Referenced by:: §32.2(v).
Encodings:: BibTeX

… ►

S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

…

17: Bibliography I

… ►

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

… ►

A. R. Its and A. A. Kapaev (2003) Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.

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External Links:: ISSN 0951-7715, Document, MathReview (Youmin Lu), ZentralBlatt
Referenced by:: §32.12(ii).
Encodings:: BibTeX

…

18: Bibliography D

… ►

B. Deconinck and H. Segur (1998) The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Notes:: Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)
External Links:: ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt
Referenced by:: §21.9.
Encodings:: BibTeX

… ►

K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.

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External Links:: ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §3.7(v).
Encodings:: BibTeX

… ►

J. M. Dixon, J. A. Tuszyński, and P. A. Clarkson (1997) From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.

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Referenced by:: Profile Peter A. Clarkson.
Encodings:: BibTeX

…

19: Bibliography C

… ►

L. D. Carr, C. W. Clark, and W. P. Reinhardt (2000) Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

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External Links:: Document
Referenced by:: §22.19(iii).
Encodings:: BibTeX

… ►

P. A. Clarkson and E. L. Mansfield (2003) The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16 (3), pp. R1–R26.

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External Links:: ISSN 0951-7715, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

… ►

P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:: MathReview
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

P. A. Clarkson (2006) Painlevé Equations—Nonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.

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External Links:: ISBN 3-540-31062-2, MathReview (Marta Mazzocco), ZentralBlatt
Referenced by:: Ch.32.
Encodings:: BibTeX

…

20: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which

F

is a rational function of

w

and

\ifrac{\mathrm{d}w}{\mathrm{d}z}

, and is locally analytic in

z

, that is, analytic except for isolated singularities in

\mathbb{C}

. … ►When

\beta=0

this is a nonlinear harmonic oscillator. …