Some Soliton Hierarchies Associated with Lie Algebras $$\mathfrak {sp}(4)$$ and $$\mathfrak {so}(5)$$

Abstract Based on the symplectic Lie algebra $$\mathfrak {sp}(4)$$ sp ( 4 ) , we obtain two integrable hierarchies of $$\mathfrak {sp}(4)$$ sp ( 4 ) , and by using the trace identity, we give their Hamiltonian structures. Then, we use $$2\times 2$$ 2 × 2 Kronecker product, and construct integrable coupling systems of one soliton equation. Next, we consider two bases of Lie algebra $$\mathfrak {so}(5)$$ so ( 5 ) , and we get the corresponding two integrable hierarchies. Finally, we discuss the relation between the integrable hierarchies of two different bases associated with Lie algebra $$\mathfrak {so}(5)$$ so ( 5 ) .