Breather and rogue wave solutions for the variable coefficient nonlinear Schrödinger equation on Jacobian elliptic function periodic backgrounds

This study concentrates on exact solutions on the Jacobian elliptic function periodic background to the variable-coefficient nonlinear Schr & ouml;dinger (vcNLS) equation. Through constructing the new eigenvalue solution for Lax pair and using the known Darboux transformation (DT) of vcNLS equation, the breather and rogue wave (RW) structures on Jacobian elliptic function backgrounds are revealed. By changing the variable coefficients in the equation, some novel localized wave structures are discussed graphically. The results presented in this letter will provide a valuable theoretical support for solving localized waves on the complicated seed background of variable coefficient nonlinear equations.