Research

My research interests are Partial Differential Equations and applications to industry such as fuel cells and wind turbines.

The Hirota Equation

My undergraduate thesis Hirota's Bilinear Method and Maple Animations of Vector Solitons deals with the modified complex Kortweg-DeVries equation (mKdV). An example is shown in the image above. This equation can simulate tsunami waves, a wave which retains its shape and speed as it moves through the ocean, as well as optical fiber signals, plasmas, atmospheric waves, vortex filaments, superconductivity, and gravitational fields with cylindrical symmetry. The Hirota Equation is the nonlinear PDE

u_t + |u|²u_x + u_xxx = 0

For more information about this, you can go here.

The Allen-Cahn Equation

In many nonlinear diffusion equations, there are transition layers in solutions that separate two binary fluids in a medium when the reaction term is very large. The Allen-Cahn equation is a prototype model that describes simple adsorption and desorption in the surface of the medium. The transition layers evolve over time at the scale of the reaction term and will move slowly depending on this term. The Allen-Cahn Equation is the nonlinear PDE

u_t = Δu - w'(u)

where w is the double-well potential w(u) = 1/4 (1-u²)²