Statistics and Structures of Strong Turbulence in a Complex Ginzburg-Landau Equation
One-dimensional complex Ginzburg-Landau equation with a quintic nonlinearity (QCGL) is studied numerically to reveal the asymptotic property of its strong turbulence. In the inviscid limit, the QCGL equation tends to the nonlinear Schrödinger (NLS) equation which has a singular solution self-similarly blowing up in a finite time. The probability distribution function (PDF) of fluctuation amplitudes is found to have an algebraic tail with exponent close to -8. This power law is described as the multiplication of the PDF of the amplitude of a singular solution of the NLS equation and that of maximum heights of bursts. The former is shown to have a -7 power law in terms of the scaling property of the NLS singular solution. The latter is found to have a -1 power law by numerical simulation.