A Few Kinds of Loop Algebras and Some Applications

In this paper, we search for some approaches for generating (1+1)-dimensional, (2+1)-dimensional and (3+1)-dimensional integrable equations by making use of various Lie algebras and the corresponding loop algebras under the frame of the Tu scheme. The well-known modified KdV equation, the heat conduction equation, the nonlinear Schrödinger equation, the (2+1)-dimensional cylindrical dissipative Zaboloskaya–Khokhlov equation and the (3+1)-dimensional heavenly equation are obtained, respectively. In addition, some new isospectral integrable hierarchies and their nonisospectral integrable hierarchies are singled out. All the Lie algebras and their loop algebras presented in the paper can be extensively applied to investigate other integrable hierarchies of evolution equations.