Solitons, breathers and periodic rogue waves for the variable-coefficient seventh-order nonlinear Schrödinger equation

Abstract Through Darboux transformation (DT) method, Several nonlinear wave solutions of seventh-order variable-coefficient nonlinear Schrödinger (vcNLS) equation are obtained, including solitons, breathers and rogue periodic waves. When the coefficients are linear, parabolic and periodic functions, the parabolic, cubic and quasi-periodic solitons and breathers can be constructed. Then we investigate their effects on the solutions, the variation of the coefficients affects the shape of the solutions. On this basis, the interactions between two solitons are studied and the interactions between two types of breathers are constructed. Next, through the approaches of the nonlinearization of spectral problem and DT method, rogue waves on the background of the Jacobi elliptic functions dn and cn for a seventh-order vcNLS equation are constructed. When the coefficients are selected as linear, exponential and periodic functions, the nonlinear dynamics of two kinds of rogue periodic waves are analysed.