Numerical analysis of nonlinear localized modes in vibrational and magnetic lattices

We numerically investigate the existence of nonlinear, spatially localized modes for various lattice Hamiltonians using Newton-Raphson method to obtain numerically exact solutions. We start by examining the well-known one-dimensional Fermi-Pasta-Ulam lattice with quadratic and quartic potentials and obtain solution branches in both frequency and nonlinear coefficient via continuation. We continue by propagating the solution in time with Runge-Kutta degree 4 (RK4) method. We then turn to two-dimensional ferromagnetic and antiferromagnetic lattices: here intrinsic localized modes were demonstrated in previous research, and more recently the topological magnetic skyrmions has stimulated intense interest. We again apply the Newton-Raphson method to obtain nontrivial solutions for certain spin-lattice Hamiltonians, and then propagate the solution in time with RK4-method.