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In this paper, the Hirota bilinear method is applied to investigate the exact solutions of the (3+1)-dimensional Jimbo-Miwa (JM) equation, including solitons, breathers and lumps, which satisfy specific Wronskian conditions. Their dynamic behaviors and the effects of free parameters on the propagation direction and velocity are analyzed through three-dimensional images and the corresponding contour plots. Especially, based on the 2Mth-order Wronskian determinant solutions, the determinant expression of arbitrary Mth-order lump solutions is constructed by employing elementary transformation and long wave limit. The experimental results show that the interaction between multiple lumps is a completely elastic collision. These results may be helpful to understand the propagation processes of nonlinear waves in some nonlinear physical systems, such as fluid mechanics, nonlinear optics and so on.

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The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.