Why Newton's method is hard for travelling waves: small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure

DOI10.1016/j.matcom.2006.10.001zbMath1107.65351OpenAlexW2021848228MaRDI QIDQ871062

No author found.

Publication date: 15 March 2007

Published in: Mathematics and Computers in Simulation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.matcom.2006.10.001

zbMATH Keywords

algorithmsnumerical examplesbifurcationKAM theorynonlinear equationssmall denominatorsnonlinear travelling wavesKolmogorov-Arnold-Moser theoryfifth-order Korteweg-de Vries equation

Mathematics Subject Classification ID

Nonlinear parabolic equations (35K55) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Computational methods for bifurcation problems in dynamical systems (37M20) Numerical bifurcation problems (65P30) Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems (37K50)

Related Items (15)

A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS ⋮ Revisiting the Thomas-Fermi equation: accelerating rational Chebyshev series through coordinate transformations ⋮ Newton-conjugate-gradient methods for solitary wave computations ⋮ Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems ⋮ Numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems ⋮ Efficient computation of capillary-gravity generalised solitary waves ⋮ Numerical and perturbative computations of solitary waves of the Benjamin-Ono equation with higher order nonlinearity using Christov rational basis functions ⋮ Simulation of coherent structures in nonlinear Schrödinger-type equations ⋮ Iteration methods for stability spectra of solitary waves ⋮ Numerical approximation of solitary waves of the Benjamin equation ⋮ A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity ⋮ Accelerated Imaginary‐time Evolution Methods for the Computation of Solitary Waves ⋮ A plethora of generalised solitary gravity–capillary water waves ⋮ Strongly nonlinear perturbation theory for solitary waves and bions ⋮ A Numerical Method for Computing Time‐Periodic Solutions in Dissipative Wave Systems

Uses Software

• NewtonLib

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Deleted residuals, the QR-factored Newton iteration, and other methods for formally overdetermined determinate discretizations of nonlinear eigenproblems for solitary, cnoidal, and shock waves
• On backtracking failure in Newton-GMRES methods with a demonstration for the Navier-Stokes equations
• Solitons from sine waves: Analytical and numerical methods for non- integrable solitary and cnoidal waves
• Modeling nonlinear resonance: A modification to the Stokes' perturbation expansion
• Fourier pseudospectral method with Kepler mapping for travelling waves with discontinuous slope: application to corner waves of the Ostrovsky-Hunter equation and equatorial Kelvin waves in the four-mode approximation
• Solving Nonlinear Equations with Newton's Method
• Numerical Methods for Bifurcations of Dynamical Equilibria
• Rational Chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions

This page was built for publication: Why Newton's method is hard for travelling waves: small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure