Bäcklund transformations for tri-Hamiltonian dual structures of multi-component integrable systems

In this article, the Bäcklund transformation based-approach is explored to obtain Hamiltonian operators of multi-component integrable systems which are governed by compatible tri-Hamiltonian dual structures. The resulting Hamiltonian operators are used not only to derive multi-component bi-Hamiltonian integrable hierarchies and their dual integrable versions but also to serve as a criterion to verify the compatibility for the corresponding (dual) bi-Hamiltonian operators. The approach is illustrated through the construction of two families of integrable bi-Hamiltonian hierarchies associated with certain two-component and three-component dispersive water wave systems and their modifications. The corresponding dual bi-Hamiltonian systems are also derived. Some of them are new integrable systems. Furthermore, we classify the analytic and non-analytic traveling wave solutions to the two-component dual non-linearly dispersive water wave system.