Autoresonant excitation of space-time quasicrystals in plasma

We demonstrate theoretically and numerically that a warm fluid model of a plasma supports space-time quasicrystalline structures. These structures are highly nonlinear, two-phase, ion acoustic waves that are excited autoresonantly when the plasma is driven by two small amplitude chirped-frequency ponderomotive drives. The waves exhibit density excursions that substantially exceed the equilibrium plasma density. Remarkably, these extremely nonlinear waves persist even when the small amplitude drives are turned off. We derive the weakly nonlinear analytical theory by applying Whitham's averaged variational principle to the Lagrangian formulation of the fluid equations. The resulting system of coupled weakly nonlinear equations is shown to be in good agreement with fully nonlinear simulations of the warm fluid model. The analytical conditions and thresholds required for autoresonant excitation to occur are derived and compared to simulations. The weakly nonlinear theory guides and informs numerical study of how the two-phase quasicrystalline structure “melts” into a single phase traveling wave when one drive is below a threshold. These nonlinear structures may have applications to plasma photonics for extremely intense laser pulses, which are limited by the smallness of density perturbations of linear waves.

Figure 1

The colormap of the electron density $n_e(x,\tau )$ as a function of slow time $\tau =\sqrt{{\alpha }_1}t$ and coordinate $x$. (a) Two-phase autoresonant ion acoustic wave excited by two driving counter propagating traveling waves with $k_1=-0.5$ and $k_2=1$ obtained by solving the fully nonlinear equations (1, 2, 3). (b) Autoresonant single phase ion acoustic wave driven by the first driving component in (a) with $k_1=-0.5$. (c) Autoresonant single phase ion acoustic wave driven by the second driving component in (a) with $k_2=1$. The red-vertical line indicates the termination of the drive at $\tau =12$.

Figure 2

The maximum (over $x$) of the ion fluid velocity $u(x,\tau )$ vs slow time $\tau =\sqrt{{\alpha }_1}t$ for a two-phase autoresonant ion acoustic wave excited by two driving traveling waves with $k_1=-0.5$ and $k_2=1$ obtained by solving the fully nonlinear equations (1, 2, 3) (denoted as “fully nonlinear”, blue line) and the weakly nonlinear reduced dynamical equations (38)–(41) (denoted as “weakly nonlinear”, pink line). The solid-black line represents the absolute value of the phase velocity ${\omega }_{d,1}\left(\tau \right)/\left|k_1\right|$ of the first driving wave and the dashed-black line represents the absolute value of the phase velocity of the second driving wave ${\omega }_{d,2}\left(\tau \right)/\left|k_2\right|$ vs $\tau$. The parameters used in the simulations are the same as in Figs. 1, 3, and 4

Figure 3

The colormap of the electron density $n_e(x,\tau )$ as a function of slow time $\tau =\sqrt{{\alpha }_1}t$ and coordinate $x$ for a two-phase autoresonant ion acoustic wave excited by two driving counter propagating traveling waves with $k_1=-0.5$ and $k_2=1$ obtained by solving the weakly nonlinear reduced dynamical equations (38, 39, 40, 41). The red-vertical line indicates the termination of the drive at $\tau =12$. The parameters used in the simulation are the same as in Figs. 1, 2, and 4.

Figure 4

The effective actions $I_1,I_2$ (top subplot) and the phase mismatches ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2$ (bottom subplot) vs slow time $\tau =\sqrt{{\alpha }_1}t$ obtained by solving the weakly nonlinear reduced dynamical equations (38, 39, 40, 41). The parameters used in the simulation are the same as in Figs. 1, 2, and 3.

Figure 5

The spectrum of the harmonics of the ion fluid velocity $u(x,\tau )$ in $k$ space obtained by solving the fully nonlinear equations (1, 2, 3) at $\tau =3$ (pink line) and $\tau =12$ (blue line) in linear (top subplot) and logarithmic (bottom subplot) scales. The parameters used in the simulations are the same as in Figs. 1, 2, 3, and 4.

Figure 6

The curves $D(k_1,k_2)=0$ for three values of the ion thermal velocity: $u_{th}=0$ (solid-blue line), $u_{th}=0.003$ (dashed-green line), and $u_{th}=0.03$ (dash-dotted-pink line), plotted in the $({\omega }_1/|k_1|,{\omega }_2/|k_2\left|\right)$ plane assuming $k_1{k}_2>0$. $D$ is negative inside the curve $D=0$ and positive outside the curve. For $k_1{k}_2<0, D$ is always positive. The double autoresonance is possible only in the regions where $D>0$.

Figure 7

Melting of a two-phase quasicrystal into a single phase quasicrystal around the threshold in the fully nonlinear model. The colormaps show the electron density $n_e(x,\tau )$ in the $(x,\tau )$ plane for a two-phase ion acoustic wave excited by two driving counter propagating traveling waves with $k_1=-0.5$ and $k_2=1$ obtained by solving the fully nonlinear equations (1, 2, 3) for various values of ${\overline{\varepsilon }}_2$: just above the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.338$, top subplot), just below the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.337$, middle subplot), and below the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.32$, bottom subplot). The parameters used in the simulations are otherwise the same as in Figs. 1, 2, and 3. The red-vertical line indicates the termination of the drive at $\tau =12$.

Figure 8

Melting of a two-phase quasicrystal into a single phase quasicrystal around the threshold in the weakly nonlinear model. The colormaps show the electron density $n_e(x,\tau )$ in the $(x,\tau )$ plane for a two-phase ion acoustic wave excited by two driving counter propagating traveling waves with $k_1=-0.5$ and $k_2=1$ obtained by solving the weakly nonlinear reduced dynamical equations (38, 39, 40, 41) for various values of ${\overline{\varepsilon }}_2$: just above the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.339$, top subplot), just below the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.338$, middle subplot), and below the threshold (${\overline{\varepsilon }}_2/{\overline{\varepsilon }}_1=0.32$, bottom subplot). The parameters used in the simulations are otherwise the same as in Figs. 1, 2, and 3. The red-vertical line indicates the termination of the drive at $\tau =12$.

Figure 9

The effective actions $I_1,I_2$ (top subplots) and the phase mismatches ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2$ (bottom subplots) vs slow time $\tau =\sqrt{{\alpha }_1}t$ obtained by solving the weakly nonlinear reduced dynamical equations (38, 39, 40, 41). The parameters used in the simulations are the same as in Figs. 1, 2, and 3 except for parameter ${\overline{\varepsilon }}_2$, which is equal to ${\overline{\varepsilon }}_2=0.338{\overline{\varepsilon }}_1$ (left side) and ${\overline{\varepsilon }}_2=0.339{\overline{\varepsilon }}_1$ (right side).