General high-order lump solutions are derived for the Levi equations based on the Hirota bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. These lump solutions are given in terms of Gram determinants whose matrix elements are connected to Schur polynomials. Thus, our solutions have explicit algebraic expressions. Their dynamic behaviors are analyzed by using density maps. It is shown that when the absolute value of one group of these internal parameters in the lump solutions is very large, lump solutions exhibit obvious geometric structures. Interestingly, we have shown that their initial and middle state solutions possess various exciting geometric patterns, including hexagon, decagon, tetradecagon, etc. and other quasi-structures in addition to the standard triangle, pentagon type patterns. Because the internal parameters are not complex conjugates of each other, the dynamic behaviors of solutions are richer. These results make several contributions to the current literature and have a number of important implications for further analysis of fluid dynamics in non-homogeneous media.
In this paper, we focus on the fourth-order nonlinear Schrödinger equation, which can describe the optical system and the Heisenberg spin system. We consider a continuous wave perturbed by the one-dimensional random rough surface as the initial condition. First, we numerically resolve the eigenvalues under different control parameters utilizing the Fourier collocation method. Then, we simulate the evolution of this equation under the above initial conditions via the symmetrical split-step Fourier method. Moreover, we investigate the “steady” chaotic state by evolving a large number of initial conditions for the same control parameters. We find that the control parameters of the initial condition affect the number and intensity of rogue waves (RWs) in integrable turbulence. In particular, we locate the inflection point where the control parameter affects the velocities of solitons and the inconsistency within the parameter of the contribution to the generation of RWs. We further verify that the collision between breathers, solitons, and breathers and solitons can generate RWs. These results will enable us to understand the turbulent state and the formation mechanism of RWs.
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