Some Evolution Hierarchies Derived from Self-dual Yang—Mills Equations

Abstract

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra Ē of the Lie algebra E and the reduced self-dual Yang—Mills equations, we obtain an expanding integrable model of the Giachetti—Johnson (GJ) hierarchy whose Hamiltonian structure can also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra gN. As an application, we apply the loop algebra Ẽ of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup—Newell (KN) hierarchy which, consisting of two arbitrary parameters α and β, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra F̃ of the Lie algebra F to obtain an expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R3 based on the self-dual Yang—Mills equations, which include Poisson structures, irregular lines, and the reduced equations.