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**Some evolution hierarchies derived from self-dual Yang-Mills equations.**
*(English)*
Zbl 1247.37075

Summary: We develop in this paper a new method to construct two explicit Lie algebras \(E\) and \(F\). By using a loop algebra \(\overline{E}\) 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 \(g_{N}\). As an application, we apply the loop algebra \(\widetilde{E}\) 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 \(\alpha \) and \(\beta \), can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra \(\widetilde{F}\) 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 \(R^{3}\) based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations.

### MSC:

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |

37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |

22E67 | Loop groups and related constructions, group-theoretic treatment |