Deleted Residuals, the QR-Factored Newton Iteration, and Other Methods for Formally Overdetermined Determinate Discretizations of Nonlinear Eigenproblems for Solitary, Cnoidal, and Shock Waves

Solitary waves, cnoidal waves, and shock waves can be computed by solving a nonlinear eigenvalue problem, which in discretized form is a system of nonlinear algebraic equations. Unfortunately, many such systems are singular because the solution is not unique until one or more additional constraints are imposed. For example, if the waves are translationally invariant and u(X) is a solution, then so also is u(X+Φ) for arbitrary Φ. To obtain a unique solution, one must impose an additional condition to reduce the one-parameter family of solutions by constraining Φ. We describe five methods for coping with such singular systems: (i) reformulation of the problem, (ii) deleting residuals, (iii) Kellerʹs bordered matrix scheme, (iv) QR-factored, overdetermined Newton iteration, and (v) pseudoinverse-Newton iteration. We illustrate these ideas using the cnoidal waves of the Korteweg–de Vries equation, the traveling shocks of the Korteweg–de Vries-Burgers equation, and the weakly nonlocal solitary waves of the nonlinear equatorial beta-plane equations. Finite difference, Fourier and rational Chebyshev pseudospectral methods, and spectrally upgraded finite differences are applied. Reformulation and deleting residuals are the cheapest strategies, but the QR-factored Newton iteration is needed for the shock waves, which lack the symmetry of the other two wave species.