Hamiltonian hierarchies in 2+1 dimensional classical integrable models
Honours Project.
Description of the Project
A classical integrable model in n+1 dimensional space-time is by definition a classical field theory which has infinitely many conservation laws. Integrals of corresponding currents over an n-dimensional sub-space are time-conserved quantities called "integrals of motion" or "hamiltonians''. Whereas the way of investigating non-integrable theories is via perturbative approximation, the advantage of integrable theories is that results may be obtained exactly.
Contrary to the conformal world n=1, only few integrable models are known for n=2. Two of these are of special interest: the so called 3-wave system [1] and its twin, "the other" 3-wave system [2]. The hierarchy of hamiltonians for the second model is not known yet - their construction, along with a review of classical integrability, is the topic of this project.
An advantage of the project is that it involves the gradual study of the modern theory of exact integrability.
Further reading:
[1] D. J. Kaup, "The inverse scattering solution for the full three dimensional three-wave resonant interaction", Physica 1D (1980) 45-67.
[2] V. V. Mangazeev and S. M. Sergeev, "The continuous limit of the triple tau-function model", Theoretical and Mathematical Physics 129 (2001) 317-326.
