A mode elimination technique to improve convergence of iteration methods for finding solitary waves

We extend the key idea behind the generalized Petviashvili method of [T.I. Lakoba, J. Yang, A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity, J. Comput. Phys., this issue, doi:10.1016/j.jcp.2007.06.009] by proposing a novel technique based on a similar idea. This technique systematically eliminates from the iteratively obtained solution a mode that is “responsible” either for the divergence or the slow convergence of the iterations. We demonstrate, theoretically and with examples, that this mode elimination technique can be used both to obtain some nonfundamental solitary waves and to considerably accelerate convergence of various iteration methods. As a collateral result, we compare the linearized iteration operators for the generalized Petviashvili method and the well-known imaginary-time evolution method and explain how their different structures account for the differences in the convergence rates of these two methods.