Dynamics study of integrable turbulence with fourth-order nonlinear Schrödinger equation.

Abstract

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.