A general solution: Localized waves on the periodic wave background of the NLS–H equation with spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

Performing the linear superposition principle and the constant change method, a general solution of the Lax pairs of the nonlinear Schrödinger–Hirota (NLS–H) equation with spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics is presented for the first time. Two types of periodic wave solutions of the NLS–H equation are derived by the separated variable approach. Taking the two types of periodic wave solutions as seed solutions, some localized waves for the NLS–H equation on periodic waves background are derived by the general solution of the Lax pairs and the Darboux transformation of the NLS hierarchy. Under specific reduction conditions, left–right periodic wave background and full periodic wave background are generated from the plane wave and the Jacobian elliptic periodic wave as seed solutions, respectively. For the left–right periodic wave background, the amplitudes of the periodic background waves are different on both sides of the breather. For the full periodic wave background, solitons and rogue waves on the five kinds periodic waves background are expressed by the Jacobian elliptic functions cn, dn, nc, sc and sd. Moreover, we obtain that breathers on the Weierstrass elliptic function ℘ periodic wave background are established by the Darboux transformation. In addition, the dynamic behaviors of these composite waves are described graphically.