Influence of impurities on solitons in the nonlinearLCtransmission line

Abstract

This paper studies the propagation of solitons in the nonlinear LC transmission line (NLCTL) with capacitor impurity. Based on Kirchhoff's laws, the numerical simulation shows that the amplitude of the soliton will be increased or decreased when it is close to the positive or negative impurity. Then, it will be split into reflected and transmitted waves by both the positive and negative impurities. Furthermore, their final amplitude and propagating speeds are almost independent of the impurity polarity. The observations near the impurity can be understood in the physical picture of the linear uncoupled energy absorption. By these results, we find that the impurity-soliton interactions (ISIs) in NLCTLs for both inductance and capacitance impurities, which have been seen to be different before, actually can be unified. They also indicate that the ISIs in NLCTLs essentially can be integrated with those in many other soliton systems, such as the Frenkel-Kontorova model and hydrodynamics. Moreover, the impurity-induced influence on the NLCTL solitons can also be well interpreted in the framework of the nonlinear Schrödinger model with an impurity term derived from the discrete voltage propagation equations by means of the perturbation method.

Turbulence-Induced Relative Velocity of Dust Particles. Iv. The Collision Kernel

“…At steady state, the rms Mach number, M , of the simulated flow is ≃ 0.1, consistent with turbulence conditions in protoplanetary disks. There has been compelling evidence that the behavior of structure functions of all orders in a subsonic turbulent flow with M ∼ < 1 is almost identical to incompressible turbulence (e.g., Porter et al 2002, Padoan et al 2004, Pan & Scannapieco 2011. This suggests that, at M ∼ < 1, the particle collision statistics would be insensitive to the Mach number, and our results are applicable for any subsonic turbulent flow.…”

“…The RDF of different particles has also a much weaker r-dependence than for particles of the same size. As shown in Pan et al (2011), the bidisperse RDF as a function of r becomes flat and approaches a constant at small r (see also Chun et al 2005). Intuitively, the increases of the RDF toward small r would eventually stop at scales below the typical separation between local concentration peaks of two different particle species 5 .…”

“…The monodisperse RDF peaks for St ≃ 1 particles, whose friction time couples to the smallest eddies of the flow. In Pan et al (2011), we showed that the peak of the RDF at St ≃ 1 can be explained by an analysis of the effective compressibility of the particle "flow", which is the largest at St ≃ 1. The RDF of equal-size 10 0 10 1 10 −1 10 0 10 1 10 2 10 −2 10 −1 10 0 10 1 particles with St ∼ < 6 shows a significant dependence on r, increasing as a power law with decreasing r (see Pan et al 2011 and Paper I).…”

“…This is because particles of different sizes tend to cluster at different locations. If one continuously changes the particle size, the positions of maximum clustering intensity would shift spatially (see Pan et al 2011). As the size difference of the two particles increases, the spatial separation between their local concentration peaks becomes larger, leading to a decreases in their "relative" clustering.…”

Dual constant-flux energy cascades to both large scales and small scales

“…It has also been argued that the direction of the cascade can change with physical properties of the flow, as for example for the passive scalar in the compressible case, depending on the dimension of the system and on its degree of compressibility: when the fluid becomes strongly supersonic, specifically in the sense that, in a Helmoltz decomposition, the curl-free part of the flow, u C , dominates the solenoidal div-free part, u S , then the passive scalar should undergo an upscale cascade [73]; it requires a high u C /u S ratio, and is difficult to observe in numerical simulations of the interstellar medium at high Mach number, possibly because shocks dissipate fast, including in the case of pressure-less (infinite Mach number) Burgers turbulence [74].…”