Date of Award
In this project, I used various numerical techniques to investigate the existence of nonlinear, spatially localized modes for vibrational and magnetic lattices. I started by using Newton-Raphson algorithm to solve the well-known 1-D Fermi-Pasta-Ulam (FPU) lattice for nonlinear, stable and localized solutions, also known as intrinsic localized modes (ILMs). Using this approach, I found lattice ILMs at the nonlinear frequency just above the minimum nonlinear threshold, then input them into Newton-Raphson to obtain more solutions at higher frequencies via continuation. I then evolved these solutions in time with Runge-Kutta (RK4) algorithm to evaluate their stability.
Afterwards, I turned to the 1-D ferromagnetic and antiferromagnetic lattices, and tried a similar approach to the FPU lattice. Due to the huge in- crease in complexity of the magnetic lattices and possibly other complications, the Newton-Raphson algorithm failed to converge even when backtracking line search and trust-region constraint variations were implemented. Thus I used the shooting method to solve for ILMs in the lattice, and successfully found ILMs in the antiferromagnetic lattice. Afterwards, I also explored both the formation of nontrivial localized modes in 1-D and 2-D ferromagnetic lattice from small perturbations from the uniform mode and their long term stability. ii
Le, Hieu, "Numerical Analysis of Nonlinear Localized Modes in Virbrational and Magnetic Lattices" (2019). Dickinson College Honors Theses. Paper 324.