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… CONTINUE READING