Breaking of nonlinear cylindrical plasma oscillations

Gorbunov, L. M.; Frolov, A. A.; Chizhonkov, E. V.; Andreev, N. E.

doi:10.1134/s1063780x10040045

Cited by 30 publications

(38 citation statements)

References 16 publications

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“…we can establish that for P = 0 and E ≈ −∂ γ/∂ ρ the form of the initial distribution of the perturbing electric filed is described as in (1.4) (see details in [12]). Note also that in addition to the formal correspondence of the initial conditions for axiallysymmetric oscillations and the mechanism of wakefield exciting, the formula for E in (1.4) also takes into account the profound internal correlation between these processes.…”

Section: Statement Of the Problem And Scenario Of The Processmentioning

confidence: 84%

“…In this case the inaccurate quantitative data are generated by the twodimensional modelling, because the magnitude of the axial maximum (for given a * , ρ * ) is equal to approximately 11 magnitudes of the background and has been accurately calculated by different methods and within different (but one-dimensional!) models [4,11,12]. Moreover, system (3.1) has particular one-dimensional axial solutions, for which not only the necessary and sufficient existence conditions, but also analytic formulas were obtained in [5].…”

Section: Computation Resultsmentioning

confidence: 99%

“…First we suppose that a * monotonically decreases. Then the evolution of plasma oscillations will be stretched in time and will asymptotically approach the results of the weakly nonlinear model studied in detail in [3,12] (see also [11]). Recall that for ρ * ≪ 1 and sufficiently small a * we have asymptotic formulas for the oscillation amplitude N max = C 1 (a * /ρ * ) 2 , and the time of formation of the first off-axial maximum of elec-Brought to you by | University of Calgary Authenticated Download Date | 5/26/15 6:36 PM tron density θ 4 , the time of oscillation breaking θ br = C 3 (ρ * /a * ) 6 with some constants C i , i = 1, 2, 3.…”

Section: Statement Of the Problem And Scenario Of The Processmentioning

confidence: 99%

“…The relationship between the electric field and the displacement can be obtained from system (2.1) in an explicit form (see [6,12])…”

Section: One-dimensional Numerical Modelling In Lagrange Variablesmentioning

confidence: 99%

“…In the first case, the criterion of oscillation breaking is the infinite value taken by the function describing the density of electrons, in the second case the criterion is the intersection of electron trajectories. In [21] (see also Appendix in [12]), a strict justification of the formation of a density singularity in the medium under the intersection of electron trajectories was presented. Therefore, below we use both criteria in the discussion of oscillation breaking being guided only by the convenience of the presentation.…”

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confidence: 99%

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Numerical simulation of the breaking effect in nonlinear axially-symmetric plasma oscillations

Chizhonkov¹,

Frolov²

2011

Russian Journal of Numerical Analysis and Mathematical Modelling

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The effect of non-axial breaking of relativistic and nonrelativistic cylindrical axiallysymmetric nonlinear plasma oscillations was earlier identified and then numerically and analytically studied by the authors within a one-dimensional hydrodynamic model. In this paper, an explicit method of the second order of accuracy (on smooth solutions) is proposed and implemented in Euler variables for the two-dimensional simulation of the indicated effect. The results of calculations admitting comparison with the one-dimensional model are presented.

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Section: Statement Of the Problem And Scenario Of The Processmentioning

confidence: 84%

Section: Computation Resultsmentioning

confidence: 99%

Section: Statement Of the Problem And Scenario Of The Processmentioning

confidence: 99%

“…The relationship between the electric field and the displacement can be obtained from system (2.1) in an explicit form (see [6,12])…”

Section: One-dimensional Numerical Modelling In Lagrange Variablesmentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical simulation of the breaking effect in nonlinear axially-symmetric plasma oscillations

Chizhonkov¹,

Frolov²

2011

Russian Journal of Numerical Analysis and Mathematical Modelling

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On radially symmetric oscillations of a collisional cold plasma

Rozanova,

Delova

2024

Math Methods in App Sciences

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We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler–Poisson equations with a non‐zero background, corresponding to cold plasma oscillations in multiple spatial dimensions. It is shown that for any arbitrarily small non‐negative constant friction coefficient, there exists a neighborhood of the zero equilibrium in the norm such that the solution of the Cauchy problem with initial data belonging to this neighborhood remains globally smooth in time. Moreover, this solution converges to zero as . This result contrasts with the situation of zero friction, where any small deviation from the zero equilibrium generally leads to a blow‐up. Our method allows us to estimate the lifetime of smooth solutions. Further, we prove that for any initial data, one can find such a coefficient of friction that the respective solution to the Cauchy problem keeps smoothness for all and converges to zero. We also present the results of numerical experiments for physically reasonable situations, which allows us to estimate the value of the friction coefficient, which makes it possible to suppress the formation of singularities of solutions.

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On the properties of multidimensional electrostatic oscillations of an electron plasma

Rozanova

2022

Math Methods in App Sciences

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We consider the classical Cauchy problem for a system of equations describing 3D arbitrary electrostatic oscillations of the cold plasma and introduce an iteration procedure that allows estimating the blow‐up time from below. This procedure is constructive provided one succeeds in obtaining a two‐sided estimate of an additional quantity depending on the solution. We show that this is possible in the case of one and two dimensions, as well as for solutions with zero vorticity. For the particular case of two‐dimensional initial data with radial symmetry, refined sufficient conditions for destruction and preservation of smoothness in the first period of oscillations are obtained. Moreover, we give the example of estimating the blow‐up time for the data for which results of numerics exist and discuss the roughness of our estimate.

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Breaking of nonlinear cylindrical plasma oscillations

Cited by 30 publications

References 16 publications

Numerical simulation of the breaking effect in nonlinear axially-symmetric plasma oscillations

Numerical simulation of the breaking effect in nonlinear axially-symmetric plasma oscillations

On radially symmetric oscillations of a collisional cold plasma

On the properties of multidimensional electrostatic oscillations of an electron plasma

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