Abstract
This paper employs a synthesis of the Darboux transformation (DT) and nonlinearization to construct rogue waves for the complex short pulse (cSP) equation on the Jacobian elliptic functions \(\mathrm{dn}\) and \(\mathrm{cn}\) background. We first derive two seed solutions for the associated complex short pulse (acSP) equation, expressed in terms of the Jacobian elliptic functions \(\mathrm{dn}\) and \(\mathrm{cn}\) , respectively. Suitable eigenvalues and their corresponding periodic and nonperiodic eigenfunctions are then determined. By applying the DT to these seed solutions, rogue wave solutions for the cSP equation on the Jacobian elliptic dn function and cn function background are obtained. As illustrative examples, specific rogue wave profiles on the elliptic function background and their dynamical features are presented graphically.
