Iteration methods for stability spectra of solitary waves

Three iteration methods are proposed for the computation of eigenvalues and eigenfunctions in the linear stability of solitary waves. These methods are based on iterating certain time evolution equations associated with the linear stability eigenvalue problem. The first method uses the fourth-order Runge–Kutta method to iterate the pre-conditioned linear stability operator, and it usually converges to the most unstable eigenvalue of the solitary wave. The second method uses the Euler method to iterate the “square” of the pre-conditioned linear stability operator. This method is shown to converge to any discrete eigenvalue in the stability spectrum. The third method is obtained by incorporating the mode elimination technique into the second method, which speeds up the convergence considerably. These methods are applied to various examples of physical interest, and are found to be efficient, easy to implement, and low in memory usage.