Modeling relativistic soliton interactions in overdense plasmas: A perturbed nonlinear Schrödinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrödinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrödinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude.

Figure 1

Simulation of a propagating soliton with $u_e=0.9$, $u_p=0.1$, and $\epsilon =0.141$, using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 2

Estimate of the relative error introduced by neglecting higher-order terms in pNLS Eq. (46). We plot $\max \left[a(x,T)-a(x,0)\right]/a(0,0)$ for solutions of Eq. (46) up to $T/{\epsilon }^2=2\times {10}^4$, with initial condition Eq. (80) with $x_1=0$. Dots correspond to $\omega =(0.999,0.99,0.98,0.97,0.96,0.95)$. The solid line is the best fit to the data, showing that $\mathrm{rel}.\mathrm{error}\sim {\epsilon }^{4.3}$.

Figure 3

Comparison of simulations of two-soliton interaction with frequencies ${\omega }_1={\omega }_2=0.999$, $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.0896$ (corresponding to $\epsilon \simeq 0.0447$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$) and initial distance $d_0=100$ using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 4

Comparison of results of quasiparticle theory for $\omega =0.999$ (red, dashed line) and $\omega =0.99$ (blue, solid line) with numerical results from simulations of the fluid model (circles and cross signs, respectively) for the first collision time $t_{\mathrm{coll}}$ for solitons of equal amplitude and no initial phase difference, as a function of initial separation $d_0$.

Figure 5

Comparison of simulations of two-soliton interaction with frequency ${\omega }_1={\omega }_2=0.999$, amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.0896$ (corresponding to $\epsilon \simeq 0.0447$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), initial distance $d_0=100$, and phase difference ${\Psi }_0=0.1\pi$, using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 6

Comparison of simulations of two-soliton interaction with ${\omega }_1=0.999$, $R_0^{\left(1\right)}\simeq 0.0896$ and ${\omega }_2=0.998$, $R_0^{\left(2\right)}\simeq 0.127$ (corresponding to $\epsilon \simeq 0.0447$, $k_1^{\left(1\right)}=1$, $k_1^{\left(2\right)}\simeq 1.4139$) and initial distance $d_0=100$ using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 7

Comparison of simulations of soliton interaction with frequencies ${\omega }_1={\omega }_2=0.99$, amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.2879$ ($\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), and initial distance $d_0=60$, using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 8

Two snapshots from cold-fluid model simulations of soliton interaction with frequencies ${\omega }_1={\omega }_2=0.99$, amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.2879$ ($\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), and initial distance $d_0=60$ [see also Fig. 7]. Dashed (blue) line corresponds to $t=0$; solid (green) line corresponds to $t\simeq 10745$, i.e., at maximum separation after the third collision. The inset corresponds to the area in the gray box.

Figure 9

Comparison of simulations of two-soliton interaction with frequency ${\omega }_1={\omega }_2=0.99$ (amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.2879$, $\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), initial distance $d_0=60$, and phase difference ${\Psi }_0=0.1\pi$, using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 10

Comparison of simulations of two-soliton interaction with frequency ${\omega }_1={\omega }_2=0.99$, amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.2879$ ($\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), initial distance $d_0=60$, and phase difference ${\Psi }_0={10}^{-6}$, using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 11

Total energy $E_{\mathrm{tot}}$ in the computational domain (red, dot-dashed line) and energy $E_L$ (blue, solid line) and $E_R$ (green, dashed line) in the left and right half of the domain, respectively, corresponding to cold-fluid model simulation described in the legend of Fig. 10.

Figure 12

Cold fluid model simulations of two-soliton interaction with frequency ${\omega }_1={\omega }_2=0.99$, amplitude $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.2879$ ($\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=k_1^{\left(2\right)}=1$), initial distance $d_0=60$, and phase difference using (a) ${\Psi }_0={10}^{-9}$, (b) ${\Psi }_0={10}^{-7}$, (c) ${\Psi }_0={10}^{-5}$.

Figure 13

Comparison of simulations of two-soliton interaction with frequencies ${\omega }_1=0.99$, ${\omega }_2=0.98$, amplitudes $R_0^{\left(1\right)}\simeq 0.2879$, $R_0^{\left(2\right)}\simeq 0.4144$ (corresponding to $\epsilon \simeq 0.1411$, $k_1^{\left(1\right)}=1$, $k_1^{\left(2\right)}\simeq 1.4107$), initial distance $d_0=30$, and relative phase ${\Psi }_0=0$ using (a) the fluid model Eq. (1), (b) pNLS Eq. (46), and (c) NLS Eq. (24).

Figure 14

Solitary wave interaction using the cubic-quintic NLS equation approximation Eq. (85) for two solitons with (a) frequencies ${\omega }_1={\omega }_2=0.99$, amplitudes $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.288$, and initial distance $d_0=60$, as described in the legend of Fig. 7, (b) frequencies ${\omega }_1={\omega }_2=0.99$, amplitudes $R_0^{\left(1\right)}=R_0^{\left(2\right)}\simeq 0.288$, initial distance $d_0=60$, and phase difference ${\Psi }_0={10}^{-6}$, as described in the legend of Fig. 9, and (c) frequencies ${\omega }_1=0.99$, ${\omega }_2=0.98$ ($R_0^{\left(1\right)}\simeq 0.288$ and $R_0^{\left(2\right)}\simeq 0.414$), and initial distance $d_0=30$, as described in the legend of Fig. 13.