From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

Han-Kwan, Daniel

doi:10.1007/s00220-013-1825-8

Cited by 42 publications

(39 citation statements)

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“…The derivation of ZK from the Euler-Poisson system with magnetic field in the long wave limit was carried out by Lannes, Linares and Saut in [25]. The ZK equation was also derived by Han-Kwan [19] from the Vlasov-Poisson system in a combined cold ions and long wave limit. Moreover, the following quantities are conserved by the flow of ZK, (1.3) M (u) = u(x, t) 2 dx, and (1.4)…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

Côte

Muñoz

Pilod

et al. 2015

Arch Rational Mech Anal

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Abstract. We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [10]. Our proofs follow the ideas by Martel [30] and Martel and Merle [35], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

Côte

Muñoz

Pilod

et al. 2015

Arch Rational Mech Anal

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show abstract

“…The equation, considered is the present paper, is known as a model of two-dimensional nonlinear waves in dispersive media propagating in one preassigned ( x ) direction with deformations in the transverse ( y ) direction. A rigorous derivation of the ZK model can be found, for example, in [20,22]. From the point of view of solubility and well-posedness the most significant results for ZK equation and its generalizations were obtained for the initial value problem.…”

Section: Introduction Description Of Main Resultsmentioning

confidence: 99%

Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation

Faminskii

2018

Journal of Mathematical Analysis and Applications

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“…where compared to the two-component VPB system (1.1), the dynamical equation of electrons and the ions-electrons collisions have been omitted, and the number density n e = R 3 f e dξ has been replaced by an analogue of the classical Boltzmann relation n e = exp{φ/T e }, or a general function depending on the potential function φ. We remark that the Boltzmann relation has been recently extensively visited in a lot of studies of kinetic and related fluid dynamic equations, for instance, [16,47,48,49,50,72]. Inspired by our previous works [25,26,28], we expect in the paper to further consider the much more physical two-component VPB system, particularly extending the results in [42,44] to the case of perturbations of the non-constant equilibrium state.…”

Section: Literature and Backgroundmentioning

confidence: 85%

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

Duan

Liu²

2017

J Stat Phys

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The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In the paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in [59] that we can be able to develop into the case for the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a nonconstant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system. 10 2.3. Diffusion and heat-conductivity 15 3. Quasineutral Euler equations and rarefaction waves 20 4. Preliminary estimates on two-component collision operator 23 5. Proof of the main result 27 6. A priori estimates on the fluid part 32 6.1. Estimate on zero-order energy 32 6.2. Estimate on first-order dissipation 37 6.3. Estimate on first-order energy 44 7. A priori estimates on the non-fluid part 51 7.1. Estimate on zero-order dissipation 51 7.2. Estimate on high-order energy 55 7.3. Estimate on energy with mixed derivatives 60 References 61 2010 Mathematics Subject Classification. Primary: 35Q20, 76P05; Secondary: 35B35, 35B40.

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From Vlasov–Poisson to Korteweg–de Vries and Zakharov–Kuznetsov

Cited by 42 publications

References 50 publications

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

Initial-boundary value problems in a rectangle for two-dimensional Zakharov–Kuznetsov equation

The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

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