Inverse cascade based on nonlinear Schrödinger equation analysis with nonlinear feedback control

This paper focuses on the wave inverse cascade instability analysis with self-regulating feedback control for a fixed external potential field and a highly localized finite-amplitude initial pulse. The wave inverse cascade instability analysis is carried out by solving the corresponding two-dimensional generalized nonlinear Schrödinger equation. The wave field firstly suffers from the modulation instability, followed by collapse into turbulence containing the shortest-wavelength modes in the system. This is followed by inverse cascade of the shortest wavelength modes back to the longer-wavelength ones, until a statistical stationary turbulent state is reached. It is found that the inverse cascade is limited to the shorter-wavelength modes with the wavenumber $\left |k\right |≥ 100$. This shows that the viscous damping $p_i$ acts like a control switch to the inverse cascade, and the feedback control can also regulate the intensity of the inverse cascade mode.