Self-adjustment and disintegration threshold of Langmuir solitons in inhomogeneous plasmas

Dynamics of Langmuir solitons in the presence of a background density gradient is investigated numerically, including cases with steep gradients to the extent the solitons can disintegrate. The disintegration threshold is explained by regarding the electric field part of the soliton as a point mass moving along the self-generated potential well corresponding to the density cavity. On the other hand, it is demonstrated that the Langmuir solitons are robust when the density gradient is below the threshold. During the acceleration phase toward low density regions, Langmuir solitons adjust themselves to balance the electric field pressure and the negative plasma pressure by expelling the imbalanced portion as density cavities at the sound velocity. When the density gradient is below the disintegration threshold, the electric field part of the soliton bounces back and forth within the potential well suggesting the solitons have internal structures.

Figure 1

Time evolution of (a) electric field soliton and (b) density cavity at $L=5\times {10}^3$. (c) Time versus the positions of the electric field peak ($X_E$) and the position of density minimum ($X_N$).

Figure 2

Time evolution of (a) electric field soliton, and (b) density cavity at $L=500$. Local density minima of the emitted cavities are located at $X=-5.36$ (blue curve), $X=-3.56$ (green curve), and $X=-1.76$ (red curve), which are separated by a distance of 1.80.

Figure 3

An expansion of the boxed region signified by the dashed lines in Fig. 1.

Figure 4

Time evolution of (a) electric field soliton, and (b) density cavity at $L=50$. Disintegration of the Langmuir solitons caused by a quick escape of the electric field soliton.

Figure 5

Disintegration of solitons at a high density gradient. (a) Conceptual figure of a soliton being regarded as a particle falling down toward the right under potential $V\left(X\right)$ given by a superposition of a linearly varying function and a cavity. (b) Spread of the electric field soliton compared with the solution of a linear Schrödinger equation. Here, LS stands for linear Schrödinger. (c) The center of gravity $X_{EC}$ of the electric field soliton versus time for the two cases. The solid curve is obtained from the solution from the Zakharov equations, and the dashed curve is obtained from the solution of the linear Schrödinger equation.

Figure 6

Values of the solitons' disintegration threshold $L$ versus the soliton shape parameter $K_0$. Here we set $K_1=0$, and thus the initial velocity of the solitons is zero. The solid black curve is given by an analytical estimate $L=3MW/{\left(2K_0\right)}^2$ (see the Appendix). Numerically measured values are given by red circles.

Figure 7

Demonstration of the mismatch between the velocity of the peak position of the electric field soliton ($V_E$) and the velocity of the density minimum position ($V_N$) for the $L=5\times {10}^3$ case of Fig. 1. The green dashed line is a tangent line to the soliton velocity in the very initial phase. If a static density limit ($N=-{\left|E\right|}^2$) is taken, the velocity of the electric field soliton should obey the green dashed line all the way. In practice, they cannot due to a dragging effect by the density cavity, however.

Figure 8

The acceleration of the center of gravity of the electric field soliton ($A_{EC}$) for the $L=500$ case of Fig. 2. The solid curve is obtained from simulation by the Zakharov equation, whereas the dashed curve is from the static density limit.