In this paper, the N th-order rogue waves are investigated for an inhomogeneous higher-order nonlinear Schrödinger equation. Based on the Heisenberg ferromagnetic spin system, the higher-order nonlinear Schrödinger equation is generated. The generalized Darboux transformation is constructed by the Darboux matrix. The solutions of the N th-order rogue waves are given in terms of a recursive formula. There are complex nonlinear phenomena in the rogue waves, add the first-order to the fourth-order rogue waves are discussed in some figures obtained by analytical solutions. It is shown that the general N th-order rogue waves contain 2n − 1 free parameters. The free parameters play a crucial role to affect the dynamic distributions of the rogue waves. The results obtained in this paper will be useful to understand the generation mechanism of the rogue wave.

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The rogue waves of the nonlinear Schrödinger equation with time-dependent linear potential function are investigated by using the similarity transformation in this paper. The first-order and second-order rogue waves solutions are obtained and the nonlinear dynamic behaviors of these solutions are discussed in detail. In addition, the amplitudes of the rogue waves under the effect of the gravity field and external magnetic field changing with the time are analyzed by using numerical simulation. The results can be used to study the matter rogue waves in the Bose-Einstein condensates and other fields of nonlinear science. Hindawi Publishing Corporation