CURF: About Me and Dispersive Waves

Hi! My name is Neil MacLachlan, I am a junior majoring in mathematics and statistics. I enjoy biking, hiking, writing, knitting, and climbing in my free time. I plan to pursue a PhD in applied mathematics, and ultimately research and teach at a university. The CURF is an important step along that path since it allows me to focus my time on research, with the potential for publication, or presentation. It also allows me to interact with other aspiring scientists and scholars at Pitt, and through that improve the quality of my own work and the breadth of my understanding. 

I am a nontraditional student. I transferred to Pitt from the Community College of Allegheny County in the Fall of 2021. I am twenty-four and have over ten years of experience working with children as a tutor, after school program teacher, and kindergarten substitute. My background is unique to me and allows me to appreciate diversity of thought and experience throughout scholarship. Through my academic career I hope to facilitate a learning environment and community surrounding mathematics which supports and encourages individuals of every background.  

My research project is on the quantization of dispersive waves in higher dimensions. My mentor is Dr. Ming Chen. In wave dynamics dispersion is a term used to describe how waves of different frequencies move at different speeds: waves, which were originally synchronized, gradually split up as they propagate through space. My work concerns a special case where these dispersive waves resynchronize later in time, which is described as quantization. Some work has already been done concerning this behavior in one dimension using the Korteweg–De Vries (KdV) equation. My work extends the existing theory into the two dimensional case using the Schrödinger equation. Quantization occurs at time steps which are rational, i.e. fraction multiples of the period of wave oscillations. At these time steps the solution to the wave equation is region-wise constant, as you can see in the figure below. At all other time steps, at least in the KdV equation, the solution forms a fractal curve. My research is significant since it serves to increase our understanding of wave dynamics in general. The result of this quantization is mathematically remarkable, and is directly connected to the Talbot effect in optics. Through exploring the behavior of the equations there is potential to uncover unintuitive results which are directly applicable to physical systems. Results from wave dynamics can be applied in numerous areas, to name a few: water waves, optics, acoustics, solid mechanics, and medical imaging. Since my research is focused on the Schrödinger equation, which is directly connected to optics and quantum mechanics, it has potential for applications in microscopy, plasmonics, and x-rays.

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