Ott-Antonsen-Ansatz

Question

I'm reading a paper https://doi.org/10.1063/1.2930766 about the Ott-Antonsen-Ansatz that is used to describe the dynamics of global coupled oscillator. There is a computational step from equation (4),(5) to (6),(7) that I don't understand.

I would really appreciate it if someone could explain to me what the author has done in these steps

Answer 1

Let

$$Z=Re^{i\Theta} = \int_{-\pi}^\pi \int_{-\infty}^\infty e^{i\theta} \rho(\theta,\omega,t)g(\omega) d\theta d\omega\tag{1}$$

be the complex meanfield of the Kuramoto model in the thermodynamic limes, where $\rho$ is the oscillator density and $g$ the distribution of the frequency.

The continuity equation for oscillator density will be $\frac{\partial \rho}{\partial \theta} + \frac{\partial }{\partial \theta} [\omega + KR\sin(\Theta - \theta)\rho]\tag{2}$

Make a Fourier series of $\rho$ in $\theta$

$$\rho(\theta,\omega,t)= \frac{g(\omega)}{2\pi} \sum_{l=-\infty}^{\infty} f_l(\omega,t) e^{il\theta}\qquad, f_l = f^*_{-l}, f_0=1$$

Assume that $f_l (\omega,t)$ is given by $f_l (\omega,t) = [\alpha (\omega,t)]]^n \qquad | \alpha(\omega,t)|\leq 1$

Now comes the part that I don't understand, the author writes: Substituting this series expansion into Eqs. (1) and (2), we find the remarkable result that this special form of f represents a solution to Eqs. (1) and (2) if

$$\frac{\partial \alpha}{\partial t} + \frac{K}{2}[Z\alpha^2 -Z^* ] + i\omega \alpha=0$$

$$Z^*=\int_{-\infty}^\infty d\omega \alpha (\omega,t) g(\omega) $$

Answer 2

Please try to compare the terms of $e^{in\theta}$ for specified $n$. You will find your answer. This question has been over two years. So I wish this answer should be useful for later ones.