Nonlinear wave collapse, shock, and breather formation in an electron magnetohydrodynamic plasma

Low-frequency nonlinear wave dynamics is investigated in a two-dimensional inhomogeneous electron magnetohydrodynamic (EMHD) plasma in the presence of electron viscosity. In the long-wavelength limit, the dynamics of the wave is found to be governed by a novel nonlinear equation. The result of the moving-frame nonlinear analysis is noteworthy, which shows that this nonlinear equation does have a breather solution and electron viscosity is responsible for the breather. A breather is a nonlinear wave in which energy accumulates in a localized and oscillatory manner. Analytical solution and time-dependent numerical simulation of this novel equation reveal the collapse of a soliton (localized pulse) into a weak noise shelf and formation of shocklike structures.

Figure 1

Phase-space trajectories in the $\varphi \text{−}\psi$ plane of the dimensionless dynamical system (14) with $M=4$. (a) Homoclinic orbit of a small disturbance around the stationary point $(0,0)$. (b) Elliptic orbit of a small disturbance around the stationary point $(M/3,0)$.

Figure 2

Numerical solution of the dynamical system (14) with $M=4$. (a) Formation of a soliton of a dimensionless small disturbance around the stationary point $(0,0)$. (b) Oscillatory solution of a dimensionless small disturbance around the stationary point $(M/3,0)$.

Figure 3

Numerical solution of the dimensionless dynamical system (17) with $\varepsilon =0.1$, $M=4$, and $(0,0,0)$ as the initial condition. (a) Formation of a shocklike structure for $\varphi$ (dimensionless magnetic field fluctuations) with moving frame $\chi$. (b) Projection of the phase portrait trajectory of (17) in the $\varphi \text{−}\psi$ plane.

Figure 4

Breather solutions of the dimensionless dynamical system (17) with $(M/3,0,0)$ as the initial condition. (a) Single breather for $\varepsilon =0.5,M=2$. (b) Multiple breather for $\varepsilon =1,M=4$.

Figure 5

Time-dependent numerical solution of (9) with soliton solution as an initial profile. The physical parameters are $\mu =0.2$, $\alpha =0.1$, and $U=1$. (a) Decreasing soliton amplitude and noise-tail formation at $\tau =40$ and $\tau =80$. (b) Oscillatory shock structure is formed at $\tau =100$.

Figure 6

Time-dependent numerical solution of (9) with Gaussian initial profile $\varphi (\zeta ,0)=exp(-{\zeta }^2/2)$. (a) Decreasing amplitude and formation of noise tails at $\tau =20$ and $\tau =40$. (b) Oscillatory shock structure is formed at $\tau =180$. In the simulation the physical parameters are as in Fig. 5.