The Wave Equation and Solitons
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The Wave Equation and Solitons

The Wave Equation and Solitons

Description:

Published: March 2, 2018 1 0 463
By: Zachary Houghton, Cal Poly Pomona
Category: Math & Physics
Hashtags: #1345163 #appliedmathematics #Calculus #Math #Mathematics #nonlinear #NSF #NSFCREST #PDE #Physics #Problemsolving #SolitaryWaves #solitons #Visualization #waveequation #WaveModeling

The project was team based, funded by NSF-CREST grant #1345163, with me and five other students working in tandem. The main objective of the project was to investigate the phenomena known as solitons, a special kind of solitary wave that is non-dispersive and non-linear. During the course of this four-week winter internship, we looked at three main non-linear yet solvable partial differential equations (PDEs): the Korteweg-Devries equation, the Sine-Gordon equation, and the Non-Linear Schrodinger equation. The wave equation was also considered, in 1, 2, and 3 spatial dimensions. Using the Maple mathematical envirnoment, time-domain simulations of the various types of soliton solutions possible for each PDE were modeled, studied, and the findings were presented to the PI of the grant along with other members of the physics staff. Later on the project was presented at the Meeting of the Minds undergraduate research consortium at Cal State University San Bernardino, in May 2017. This project was extremely challenging, considering all but one of our group members had even taken differential equations. We also had to learn how to typeset in LaTeX, as well as solve partial differential equations and create 2, and 3 dimensional simulations and graphs in Maple. It was really my first experience at dealing with such a difficult subject matter with such little time to spare, but I was able to improve my problem solving skills through practice. This was also one of my first experiences working on a team, which meant we had to divide up the tasks, and pool our resources in order to complete the project on time. I really think the short time span of this project has given me the opportunity to preform under pressure, and I know it will come in handy moving forward. The presentation aspect of the project was also heavily emphasized, and has definitely left me better prepared for future academic presentations.

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