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Register a new userThe existence of stable BGK waves
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The 1D Vlasov-Poisson system is the simplest kinetic model for describing an electrostatic collisonless plasma, and the BGK waves are its famous exact steady solutions. They play an important role on the long time dynamics of a collisionless plasma as potential final states or attractors, thanks to many numerical simulations and observations. Despite their importance, the existence of stable BGK waves has been an open problem since their discovery in 1958. In this paper, first linearly stable BGK waves are constructed near homogeneous states.
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Consider Vlasov-Poisson system with a fixed ion background and periodic condition on the space variables, in any dimension dgeq2. First, we show that for general homogeneous equilibrium and any periodic x-box, within any small neighborhood in the Sobolev space W_{x,v}^{s,p} (p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary traveling speed. This implies that nonlinear Landau damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and in any period box. The BGK waves constructed are one dimensional, that is, depending only on one space variable. Higher dimensional BGK waves are shown to not exist. Second, for homogeneous equilibria satisfying Penroses linear stability condition, we prove that there exist no nontrivial invariant structures in the (1+|v|^{2})^{b}-weighted H_{x,v}^{s} (b>((d-1)/4), s>(3/2)) neighborhood. Since arbitrarilly small BGK waves can also be constructed near any homogeneous equilibria in such weighted H_{x,v}^{s} (s<(3/2)) norm, this shows that s=(3/2) is the critical regularity for the existence of nontrivial invariant structures near stable homogeneous equilibria. These generalize our previous results in the one dimensional case.
Consider 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space W^{s,p} (p>1,s<1+(1/p)) of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria and any spatial period. Indeed, in W^{s,p} (s<1+(1/p)) neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penroses linear stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some W^{s,p} (p>1,s>1+(1/p)) neighborhood. Furthermore, when p=2,we prove that there exist no nontrivial invariant structures in the H^{s} (s>(3/2)) neighborhood of stable homogeneous states. These results suggest the long time dynamics in the W^{s,p} (s>1+(1/p)) and particularly, in the H^{s} (s>(3/2)) neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for initial perturbations in very rough spaces, for linearly stable homogeneous state. This suggests that the contrasting dynamics in W^{s,p} spaces with the critical power s=1+(1/p) is a trully nonlinear phenomena which can not be traced back to the linear level.
This paper reexamines the physical roles of trapped and passing electrons in electron Bernstein-Greene-Kruskal (BGK) solitary waves, also called the BGK phase space electron holes (EH). It is shown that the charge density variation in the vicinity of the solitary potential is a net balance of the negative charge from trapped electrons and positive charge due to the decrease of the passing electron density. A BGK EH consists of electron density enhancements as well as a density depletion, instead of only the density depletion as previously thought. The shielding of the positive core is not a thermal screening by the ambient plasma, but achieved by trapped electrons oscillating inside the potential energy trough. The total charge of a BGK EH is therefore zero. Two separated EHs do not interact and the concept of negative mass is not needed. These features are independent of the strength of the nonlinearity. BGK EHs do not require thermal screening, and their size is thus not restricted to be greater than the Debye length $lambda_D$. Our analysis predicts that BGK EHs smaller than $lambda_D$ can exist. A width($delta$)-amplitude($psi$) relation of an inequality form is obtained for BGK EHs in general. For empty-centered EHs with potential amplitude $gg 1$, we show that the width-amplitude relation of the form $deltaproptosqrt{psi}$ is common to bell-shaped potentials. For $psill 1$, the width approaches zero faster than $sqrt{psi}$.
In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainerman-Sideris [1996_CPAM], and global existence was built by Agemi [2000_Invent. Math.] and Sideris [1996_Invent. Math., 2000_Ann. Math.] independently. Here we establish the corresponding almost global and global existence theory in the inhomogeneous case.
The nature of the plasma wave modes around the ion kinetic scales in highly Alfvenic slow solar wind turbulence is investigated using data from the NASAs Parker Solar Probe taken in the inner heliosphere, at 0.18 Astronomical Unit (AU) from the sun. The joint distribution of the normalized reduced magnetic helicity ${sigma}_m ({theta}_{RB}, {tau})$ is obtained, where ${theta}_{RB}$ is the angle between the local mean magnetic field and the radial direction and ${tau}$ is the temporal scale. Two populations around ion scales are identified: the first population has ${sigma}_m ({theta}_{RB}, {tau}) < 0$ for frequencies (in the spacecraft frame) ranging from 2.1 to 26 Hz for $60^{circ} < {theta}_{RB} < 130^{circ}$, corresponding to kinetic Alfven waves (KAWs), and the second population has ${sigma}_m ({theta}_{RB}, {tau}) > 0$ in the frequency range [1.4, 4.9] Hz for ${theta}_{RB} > 150^{circ}$, corresponding to Alfven ion Cyclotron Waves (ACWs). This demonstrates for the first time the co-existence of KAWs and ACWs in the slow solar wind in the inner heliosphere, which contrasts with previous observations in the slow solar wind at 1 AU. This discrepancy between 0.18 and 1 AU could be explained, either by i) a dissipation of ACWs via cyclotron resonance during their outward journey, or by ii) the high Alfvenicity of the slow solar wind at 0.18 AU that may be favorable for the excitation of ACWs.
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