Two soliton hierarchies associated with \(\mathrm{SO}(4)\) and the applications of \(\mathrm{SU}(2) \otimes\mathrm{SU}(2) \cong\mathrm{SO}(4)\) (Q2924914)

This paper deals with two soliton hierarchies associated with \(\mathrm{SO}(4)\) and the applications of the isomorphism \(\mathrm{SU}(2)\otimes\mathrm{SU}(2)\cong\mathrm{SO}(4)\). By using the two matrix Lie algebras of the six-dimensional real special orthogonal Lie algebra \(\mathrm{SO}(4)\), the authors generate two hierarchies associated with \(\mathrm{SO}(4)\) and construct two different isospectral problems. The two hierarchies of soliton equations can be generated from zero curvature equations. Finally, based on the property of \(\mathrm{SU}(2)\otimes\mathrm{SU}(2)\cong\mathrm{SO}(4)\), they obtain some relations between the hierarchy of \(\mathrm{SU}(2)\otimes\mathrm{SU}(2)\) and the hierarchy of \(\mathrm{SO}(4)\). Many interesting unsolved problems are mentioned, for example, the relation between the hierarchy of \(\mathrm{SU}(2)\) and the hierarchy of \(\mathrm{SO}(3)\), the hierarchies associated with \(\mathrm{SO}(6)\) and the bi-integrable and tri-integrable couplings of the two integrable hierarchies derived in the paper.