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

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.