Multi-Component Coupled Fokas-Lenells Equations and Theirs Localized Wave Solutions

As the first negative flow of the integrable generalization of the nonlinear Schrodinger equation, the Fokas-Lenells equation has attracted extensive attention in recent years. In this paper, we derive the general structure of the multi-component coupled Fokas-Lenells equations which have Lax representation in matrix form. Then we construct a basic theory of the general form of Lax pairs and Darboux transformations (classical and generalized) for the previously mentioned equation. As applications, we study two examples in detail, both of the four-component and the three-component coupled Fokas-Lenells equations can be reduced to the ubiquitous Fokas-Lenells equation. Furthermore, we apply the basic theory to obtain kinds of localized wave solutions, that is to say we use the classical Darboux transformation to obtain soliton solutions and use the generalized Darboux transformation to obtain soliton-positon solutions, rogue wave solutions and breather solutions. At last these localized wave solutions are illustrated by three-dimensional structure plots and two-dimensional density plots, as well as their dynamic properties are discussed.