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A few Lie algebras and their applications for generating integrable hierarchies of evolution types. (English) Zbl 1220.37064
Summary: A Lie algebra consisting of $$3 \times 3$$ matrices is introduced, whose induced Lie algebra is obtained as well by using an inverted linear transformation. As for application examples, we obtain a unified integrable model of the integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy as well as their induced integrable hierarchies. These integrable couplings are different from those results obtained before. However, the Hamiltonian structures of the integrable couplings cannot be obtained by using the quadratic-form identity or the variational identity. For solving the problem, we construct a higher-dimensional subalgebra $$R$$ and its reduced algebra $$Q$$ of the Lie algebra $$A_{2}$$ by decomposing the induced Lie algebra and then again making some linear combinations. The subalgebras of the Lie algebras $$R$$ and $$Q$$ do not satisfy the relation ($$G = G_1 \oplus G_2, [G_1,G_2]\subset G_2$$), but we can deduce integrable couplings, which indicates that the above condition is not necessary to generate integrable couplings. As an application example, an expanding integrable model of the AKNS hierarchy is obtained whose Hamiltonian structure is generated by the trace identity. Finally, we give another Lie algebras which can be decomposed into two simple Lie subalgebras for which a nonlinear integrable coupling of the classical Boussinesq-Burgers (CBB) hierarchy is obtained.

##### MSC:
 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
##### Keywords:
integrable couplings; Lie algebra; soliton hierarchy
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##### References:
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