Rogue waves in Jacobian elliptic periodic background for the nonlinear Schrödinger equation with a linear external potential

Abstract In this paper, we investigate the nonlinear Schrödinger equation with a linear external potential, which describes the propagation of quasi-monochromatic waves in weakly nonlinear media. By leveraging the periodicity of two Jacobian elliptic functions, namely the dn- and cn-functions, we construct explicit rogue wave solutions within a periodic background. For perturbations in the form of solitary waves, we demonstrate that the resulting rogue wave solutions associated with the dn- and cn-functions exhibit modulational instability. Using the Darboux transformation and the Lax spectral problem of the equation, we derive the analytical expressions for these rogue wave solutions, which can also be reduced to the Peregrine breather wave solutions. Our findings contribute to a deeper understanding of rogue wave dynamics in the context of the nonlinear Schrödinger equation.