The VlasovHMF equation: simulations
The VlasovHMF equation
The full VlasovHMF equation reads
$\begin{aligned} \qquad\partial_t F+p\partial_q F\partial_q \phi[F]\partial_p F=0 \end{aligned}$ $\begin{aligned} \phi[F](q)=\int_{\infty}^\infty\int_{\pi}^\pi V_{\rm{HMF}}(qq')F(q',p',t)\,\mathrm{d}q'\mathrm{d}p'=V_{\rm{HMF}}\star_q \int_{\infty}^\infty F\,\mathrm{d}p \end{aligned}$ $\begin{aligned} \int_{\infty}^\infty\int_{\pi}^\pi F \,\mathrm{d}q\mathrm{d}p =1 \end{aligned}$where $\star_q$ is the convolution in space (this formulation makes easy the Fourier transform). From now on we will omit the integration bounds. For the HMF potential the mean field potential $\phi[F]$ becomes
$\begin{aligned} \phi[F](q)&=M_c[F](t)\cos qM_s[F](t)\sin q=\M\\cos(q\varphi_M) \end{aligned}$ $\begin{aligned} M[F]&=M_c[F]+iM_s[F]=\iint Fe^{i q}\,\mathrm{d}q\mathrm{d}p, \end{aligned}$where $M[F](t)$ is the order parameter (magnetization). In all simulation shown below by symmetry, it will always be real. This equation describes the time evolution of a density of articles $F$ in the phase space of position $q$ (or $x$ or $\theta$ depending on the context) and velocity $p$ (or $v$). The colors represent the magnitude of $F(q,p,t)$ (or $F(x,v,t)$). I show the evolution of the density $F(q,p,t) = F^0(q,p) + P(q,p,t)$ under a small initial perturbation $P(q,p)$ when the initial distribution is a stationary solution $F^0(q,p)$ of the Vlasov equation. I show different cases, whether the initial distribution is homogeneous in space or not, stable, or weakly unstable. The homogeneous case is already well documented theoretically and numerically. My PhD work consisted in studying (both theoretically and numerically) the case when the initial distribution is nonhomogeneous $F^0(q,p)$ (depends on $q$).
The simulations were done thanks to a GPU VlasovHMF solver provided by T. M. Rocha Filho Comput. Phys. Commun. 184, 34 (2013). More details to come on the context of these simulations. In the meantime, you can refer to Physical Review E, 93(4), 042207 or the most recent Phys. Rev. E 102, 052208. For an longer description of the problems (theory and simulation details), please refer to the Part Two, Chapter V and VI, of PhD thesis. These two chapters correspond respectively to the homogeneous and nonhomogeneous Vlasov equation cases.
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I want to illustrate the fundamental differences between the dynamics around homogeneous steady states $F^0(p)$ with associate magnetization $M_0=0$ and the nonhomogeneous states $F^0(q,p)$ with associate magnetization $M_0\neq 0$. In the first case we get a "singular supercritical" bifurcation, while in the latter we get a "transcritical" bifurcation. The quotes mean that the situation is more subtle than that.

Simulations movies
Homogeneous steady states $F^0(p)$
Stable homogeneous steady state of the VlasovHMF equation. It is perturbed with a small perturbation. We observe Landau damping i.e., phase space mixing. The perturbation looks as if it is damped but it actually disappear into the velocity high Fourier modes.
Unstable homogeneous steady state of the VlasovHMF equation. It is perturbed with a small perturbation. This time the perturbation grows and we observe the \"trapping phenomenon\" with a characteristic cateye pattern. There are strong resonances between the particles with a velocity close to the frequency of the perturbation, here close to $p=0$ that leads to the trapping.
NonHomogeneous steady states $F^0(q,p)$
Unstable inhomogeneous steady state $F^0(q,p)$ of the VlasovHMF equation.
Small "positive" perturbation $P(q,p)$ perturbation, meaning that the associate magnetization is positive. The perturbation stops is quickly saturated by the nonlinear terms at $O(\lambda^2)$. We observe small nonlinear oscillations.
To highlight this perturbative dynamic, we remove subtract the reference state and show $FF^{0}$.
Small "negative" perturbation $P(q,p)$ perturbation (sponge scenario), meaning that the associate magnetization is negative. The perturbation grows outside the perturbative regime. Outside the perturbative regime, we observe a limit cycle type behavior.
We show the same simulations for another initial nonhomogeneous state $F^0$.