Reduction of the symmetries of the self-dual Yang-Mills equation
We know that the self-dual Yang-Mills (SDYM) equation has infinitely many symmetries, and these symmetries constitute an infinite dimensional Lie algebra. This property is commonly shared by almost all the \(1+1\)- dimensional integrable evolution equations (soliton equations), and has become a very important criterion of integrability for the evolution equations. So in a sense the SDYM equation is integrable. In recent years people have found that some typical soliton equations, such as the KdV equation, the nonlinear Schrödinger (NLS) equation and the Toda lattice equation, can be reduced from the SDYM equation. In this note we show that the symmetries and their Lie algebraic structure of the above-mentioned soliton equations can be reduced from those of the SDYM equation. Since the symmetries of the above-mentioned soliton equations correspond to some isospectral and non-isospectral hierarchies of soliton equations, we thus build more relations between the SDYM equation and the soliton equations.
scientific article; zbMATH DE number 558934