Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
Numerical Methods, Software, and Analysis
Numerical Methods, Software, and Analysis
On backtracking failure in newton-GMRES methods with a demonstration for the navier-stokes equations
Journal of Computational Physics
Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms
Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms
Journal of Computational Physics
Iteration methods for stability spectra of solitary waves
Journal of Computational Physics
Journal of Computational Physics
Newton-conjugate-gradient methods for solitary wave computations
Journal of Computational Physics
Simulation of coherent structures in nonlinear Schrödinger-type equations
Journal of Computational Physics
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Nonlinear travelling waves and standing waves can computed by discretizing the appropriate partial differential equations and then solving the resulting system of nonlinear algebraic equations. Here, we show that the ''small denominator'' problem of Kolmogorov-Arnold-Moser (KAM) theory is equally awkward for numerical algorithms. Furthermore, Newton's iteration combined with continuation in a parameter often exhibits ''erratic failure'' even in the absence of bifurcation. Wave resonances can interlock a countable infinity of branches in an extremely complex topology, as will be illustrated through the fifth-degree Korteweg-deVries equation. Continuation can easily jump, unsuspected, from one branch to another. Constraints, sometimes finite and sometimes infinite in number, are usually needed to specify a unique solution. This confluence of numerical difficulties can be overcome only by combining the latest numerical algorithms with a strong understanding of travelling wave physics.