Novel Algorithms

The study of the nonlinear dynamics of a higher dimensional system represented by one-dimensional and two-dimensional partial differential equations requires fast, efficient algorithms for generating long time-series data sets (for computing suitable thermodynamic averages). The algorithms remove severe time-step restrictions encountered by earlier workers and therefore can be solved on low-end computational platforms, which are readily accessible. One equation that has been solved using such techniques is the Kuramoto-Shivashinsky equation. The accompanying diagram shows the space-time evolution of a scalar field represented by the KS equation. Another equation is the 3D Ginzburg-Landau equation. The stability of 3D scrolls, rings, and spirals has been studied in collaboration with Prof. Edward Ott, Michael Gabbay, and Keeyeol Nam. More recently we have applied these techniques to develop an efficient algorithm for coupled nonlinear Schrodinger equations representing modulation and Raman scattering of laser beams propagating in an optical fiber. This work is in collaboration with Prof. Raj Roy and graduate student Bhaskar Khubchandani.