Project Report

Under Professor Okado’s mentorship, we wrote a program in Mathematica to study the scattering of solitons associated with the G2(1)-crystal, and proved a conjecture on the scattering rules of these solitions. Solitons are "solitary waves" in clouds, in ocean waves, and in fundamental physics. Using the theory of crystal bases, we found a previously unknown explicit solution (R-matrix) for the Yang-Baxter equation in statistical mechanics. Crystal bases are a powerful tool for studying Lie group symmetries by relating them to associated discrete structures. Using this, and a technique developed by Hatayama, Kuniba, and Takagi, we studied a new model of solitons associated with the Lie group G2(1) by a discrete model known as a cellular automaton. We proved that solitons in this system are parameterized by elements of an A1(1) crystal base and that the scattering rule is the R-matrix for A1(1). The scattering rule in this case is interesting because it leads to negative phase shift of a longer faster soliton colliding with a short slow soliton. This is not observed except in the D4(1) and G2(1) cases. Our paper-Soliton cellular automata associated with G2(1)-crystal base, is soon to appear in J. Math. Phys., and is on the web at www.arxiv.org 1109:2836.

Agency
National Science Foundation (NSF)
Institute
Office of International and Integrative Activities (IIA)
Application #
1107105
Program Officer
Carter Kimsey
Project Start
Project End
Budget Start
2011-06-01
Budget End
2012-05-31
Support Year
Fiscal Year
2011
Total Cost
$5,967
Indirect Cost
Name
Wilson Evan A
Department
Type
DUNS #
City
Durham
State
NC
Country
United States
Zip Code
27713