Bäcklund transformations of multi-component Boussinesq and Degasperis-Procesi equations

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Backlund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Backlund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Backlund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis-Procesi (DP) equation, and construct a Lax pair of it. In addition, the Backlund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a Zn-DP equation. Considering the superposition principle, we obtain the solution of an associated Zn-DP equation by using Backlund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena.