W-shaped soliton, breather and rogue wave solutions on the elliptic function background in a fifth-order nonlinear Schrödinger equation

In this paper, we investigate a fifth-order nonlinear Schrödinger equation, which can be applied to describe the propagation of ultrashort pulses in optical fibers. We provide the eigenfunctions of the Lax pair associated with the elliptic function seed solutions cn and dn. Using the Darboux transformation method, the W-shaped solitons, breathers, periodic solutions and rogue waves on the elliptic functions cn and dn background are obtained, the corresponding dynamical properties and evolutions are illustrated graphically by choosing proper parameters, the variations for amplitudes and periods of these solutions are analyzed. The relationship between parameters and solutions’ structures is discussed. To the best of our knowledge, the W-shaped solitons on the elliptic function background are presented for the first time. The results in this paper might be useful for us to understand some characteristics and relations of breathers and rogue waves on the elliptic functions cn and dn background in various physical equations with higher-order effects.