Fractional-Order Extension, Painlevé Integrability and Generalized Darboux Transformations for Broer–Kaup Equations

Broer and Kaup (BK) equations that include both spatial and temporal variables were independently introduced in the early 1970s. Renowned for describing the propagation of bidirectional long waves in shallow water, the BK equations are an important classical model among nonlinear evolution equations in fluid mechanics and nonlinear science. Although the BK equations have been extensively studied, their Painlevé integrability and generalized Darboux transformations (DTs) have not been reported. Addressing these two issues constitutes the starting point of this work. First, this paper extends the BK equations to their fractional-order counterparts, the fractional-order BK (fBK) equations, and proves their Painlevé integrability. Based on the Painlevé truncation expansions, we derive the Bäcklund transformations for the fBK equations. Subsequently, we reduce the fBK equations to the fractional linear heat conduction equation and the fractional Burgers equation. Meanwhile, starting from the fractional linear heat equation, we construct fractional N-soliton solutions of the fBK equations. On the other hand, starting from the classical DTs based on their Lax pair formulation of the BK equations, we derive the generalized DTs for the BK equations. By applying a limit process to the eigenfunctions, we deduce iteration eigenfunctions within the framework of the generalized DTs. These iteration eigenfunctions, along with two forms of seed solutions of the related Lax pairs, are then used to derive first-, second-, third-, and fourth-order rational-hyperbolic function solutions of the BK equations. The obtained high-order rational-hyperbolic function solutions are new and exhibit novel spatial structures, as they combine the external explicit representations of independent variables with hyperbolic functions to form rational forms.