Coverings And Nonlocal Symmetries For Some Differential Equations Of Mathematical Physics

The theory of coverings in differential equations created by I. S. Krasilshchik and A. M. Vinogradov give a rigorous geometric interpretation of nonlocal phenomenon of integrable system. Wahlquist-Estabrook(WE) prolongation structures are special type of nontrivial coverings (WE coverings). These type of coverings have many ap-plications in integrable system. In Chapter2to Chapter5, we study WE coverings, realizations and equivalence classifications of the one-dimensional WE coverings and nonlocal symmetries in the WE coverings of some important differential equations in mathematical physics, including:(1) second and third order AKNS equations;(2) nonlinear Schrodinger(NLS) equation;(3) derivative nonlinear Schrodinger(DNLS) e-quation;(4) modified derivative nonlinear Schrodinger(MDNLS) equation;(5) modified Boussinesq(MB) equation. The bi-Hamiltonian structures are very important in inte-grable systems. In recent years, A.De Sole and V. Kac use the language of Poisson vertex algebra(PVA) to describe Hamiltonian structure. It is very convenient to study some integrable Hamiltonian systems by PVA language. Chapter6of this thesis is based on A.De Sole and V. Kac’s work. From three compatible A-brackets, we obtain two compatible Hamiltonian operators. Base on the two Hamiltonian operators, we construct two integrable Hamilton hierarchies. One of which is Boussinesq hierarchy, and the other one is a new integrable Hamiltonian system.