nonlinear%20evolution%20equations
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21: Alexander I. Bobenko
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►Bobenko’s books are Algebro-geometric Approach to Nonlinear Integrable Problems (with E.
… Matveev), published by Springer in 1994, Painlevé Equations in the Differential Geometry of
Surfaces (with U.
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22: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent.
Phys. D 3 (1-2), pp. 165–184.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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23: Bibliography O
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Studies on the Painlevé equations. I. Sixth Painlevé equation
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Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
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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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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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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Iterative Solution of Nonlinear Equations in Several Variables.
Academic Press, New York.
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24: 22.19 Physical Applications
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►The subsequent time evolution is always oscillatory with period and modulus :
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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. … …25: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…26: Bibliography I
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Ordinary Differential Equations.
Longmans, Green and Co., London.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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A First Course in the Numerical Analysis of Differential Equations.
Cambridge Texts in Applied Mathematics, No. 15, Cambridge University Press, Cambridge.
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On the asymptotic analysis of the Painlevé equations via the isomonodromy method.
Nonlinearity 7 (5), pp. 1291–1325.
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Quasi-linear Stokes phenomenon for the second Painlevé transcendent.
Nonlinearity 16 (1), pp. 363–386.
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27: Bibliography D
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The KP equation with quasiperiodic initial data.
Phys. D 123 (1-4), pp. 123–152.
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Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations.
CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
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From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics.
Oxford University Press, Oxford.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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28: 21.9 Integrable Equations
§21.9 Integrable Equations
… ►Typical examples of such equations are the Korteweg–de Vries equation …and the nonlinear Schrödinger equations … ► … ►29: Bibliography W
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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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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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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Linear and Nonlinear Waves.
John Wiley & Sons, New York.
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30: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Algebro-geometric Approach to Nonlinear Integrable Problems.
Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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