Breather and rogue wave solutions to integrable equations in the Ablowitz–Kaup–Newell–Segur scheme on elliptic function backgrounds

The nonlinear evolution equations in the mixed Ablowitz–Kaup–Newell–Segur hierarchy are studied, and a unified method for solving all equations in the whole hierarchy on elliptic function backgrounds is given. With the aid of Lenard recursive sequence, we derive the mixed hierarchy containing positive and negative flows. A Riemann surface KN of genus N is defined by a Lax matrix associated with the Nth stationary equation, from which the relations between the elliptic variables and the potentials are established. By means of meromorphic functions, we discuss the Dubrovin equations and the corresponding spectral problems on one-genus Riemann surface to derive the elliptic function solutions. Resorting to parameterizing the algebraic curve, the Jacobi elliptic function solutions to two classical models, the AB systems and Lakshmanan–Porsezian–Daniel (LPD) equation are derived. On this basis and using the Darboux transformation, we finally obtain some special solutions for the AB system and LPD equation such as Ma breathers, Akhmediev breathers and rogue waves on Jacobi elliptic function backgrounds.