Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations

Trogdon, Thomas; Olver, Sheehan; Deconinck, Bernard

doi:10.1016/j.physd.2012.02.016

Cited by 53 publications

(93 citation statements)

References 26 publications

(56 reference statements)

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“…The reflection coefficient is obtained using the method described in [110]. We use the notation u(n, x, t) to denote the approximate solution obtained with n collocation points per contour.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Trogdon¹,

Olver²

2015

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101

197

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The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painlevé transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are reduced to the study of a nonlinear special function in particular limits. The universal object that these functions share is a Riemann-Hilbert representation: the nonlinear special function can be recovered from the solution of a Riemann-Hilbert problem (RHP). A RHP consists of finding a piecewiseanalytic function in the complex plane when the behavior of its discontinuities is specified. In this dissertation, the applied theory of Riemann-Hilbert problems, using both Hölder and Lebesgue spaces, is reviewed. The numerical solution of RHPs is discussed. Furthermore, the uniform approximation theory for the numerical solution of RHPs is presented, proving that in certain cases the convergence of the numerical method is uniform with respect to a parameter. This theory shares close relation to the method of nonlinear steepest descent for RHPs.The inverse scattering transform for the Korteweg-de Vries and Nonlinear Schrödinger equation is made effective by solving the associated RHPs numerically. This technique is extended to solve the Painlevé II equation numerically. Similar Riemann-Hilbert techniques are used to compute the so-called finite-genus solutions of the Korteweg-de Vries equation. This involves ideas from Riemann surface theory. Finally, the methodology is applied to compute orthogonal polynomials with exponential weights. This allows for the computation of statistical quantities stemming from random matrix ensembles. ACKNOWLEDGMENTS

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Section: Numerical Resultsmentioning

confidence: 99%

“…We assume x = −12c 2 t 1/3 for some positive constant c. This deformation is found in [110]. We rewrite θ:…”

Section: Uniform Approximation Of Solutions Of the Modified Kdv Equationmentioning

confidence: 99%

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Trogdon¹,

Olver²

2015

Self Cite

101

197

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“…We assume x = −12c 2 t 1/3 for some positive constant c. This deformation is found in [15]. We rewrite θ:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The method of nonlinear steepest descent was adapted by the authors in [15] for numerical purposes. We developed a method to reliably solve the Cauchy initial-value problem for KdV and modified KdV for all values of x and t. A benefit of the approach was that, unlike the standard method of nonlinear steepest descent, we did not require the knowledge of difficult-to-derive local parametrices.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Olver

Trogdon

2013

Comm Pure Appl Math

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The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for RiemannHilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann-Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlevé II equation and the modified Korteweg-de Vries equations to explicitly demonstrate the practical validity of the theory.

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“…In Ref. [20], ehe authors obtained an efficient numerical method to study the asymptotic solution of Equation (1). The authors studied compact solitary waves of the mKdV equation by using the phase portrait theory [21].…”

Section: Introductionmentioning

confidence: 99%

Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method

Dong¹,

Fen²

2018

OALib

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Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations

Cited by 53 publications

References 26 publications

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method

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