Rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects

Abstract

In this paper, we study the dynamics of rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects. First, we consider two types of double-periodic solutions in terms of the Jacobi elliptic functions. When the elliptic modulus approaches 1, such both double-periodic wave solutions can reduce to the Akhmediev breather solution. Second, with both double-periodic waves as seed solutions, we derive rogue wave solutions from the Darboux transformation method. Further, we demonstrate localized structures of rogue waves formed on double-periodic waves for two sets of different eigenvalues. In addition, we discuss how the higher-order effect affects double-periodic background waves and rogue waves. We expect that our obtained results may help understand rogue waves manifestations on the double-periodic background occurring in hydrodynamics and nonlinear optics with higher-order effects.