A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

Inverse Scattering and the Geroch Group

We study the integrability of gravity-matter systems in D=2 spatial dimensions with matter related to a symmetric space G/K using the well-known linear systems of Belinski-Zakharov (BZ) and Breitenlohner-Maison (BM). The linear system of BM makes the group structure of the Geroch group manifest and we analyse the relation of this group structure to the inverse scattering method of the BZ approach in general. Concrete solution generating methods are exhibited in the BM approach in the so-called soliton transformation sector where the analysis becomes purely algebraic. As a novel example we construct the Kerr-NUT solution by solving the appropriate purely algebraic Riemann-Hilbert problem in the BM approach.Comment: 30 pages. v2: Typos correcte

Numerical studies on the Zakharov system

Master'sMASTER OF SCIENC

Spurious four-wave mixing processes in generalized nonlinear Schrödinger equations

Numerical solutions of a nonlinear Schödinger equation, e.g., for pulses in optical fibers, may suffer from the spurious four-wave mixing processes. We study how these nonphysical resonances appear in solutions of a much more stiff generalized nonlinear Schödinger equation with an arbitrary dispersion operator and determine the necessary restrictions on temporal and spatial resolution of a numerical scheme. The restrictions are especially important to meet when an envelope equation is applied in a wide spectral window, e.g., to describe supercontinuum generation, in which case the appearance of the numerical instabilities can occur unnoticed