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Published: 8 June 2024

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1

Després, Bruno. "Symmetrization of Vlasov--Poisson Equations." SIAM Journal on Mathematical Analysis 46, no. 4 (January 2014): 2554–80. http://dx.doi.org/10.1137/130927942.

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2

JIN, SHI, XIAOMEI LIAO, and XU YANG. "THE VLASOV–POISSON EQUATIONS AS THE SEMICLASSICAL LIMIT OF THE SCHRÖDINGER–POISSON EQUATIONS: A NUMERICAL STUDY." Journal of Hyperbolic Differential Equations 05, no. 03 (September 2008): 569–87. http://dx.doi.org/10.1142/s021989160800160x.

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In this paper, we numerically study the semiclassical limit of the Schrödinger–Poisson equations as a selection principle for the weak solution of the Vlasov–Poisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker–Planck equation. We also numerically justify the multivalued solution given by a moment system of the Vlasov–Poisson equations as the semiclassical limit of the Schrödinger–Poisson equations.
3

Larsson, Jonas. "An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov—Poisson system." Journal of Plasma Physics 48, no. 1 (August 1992): 13–35. http://dx.doi.org/10.1017/s0022377800016342.

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A new action principle determining the dynamics of the Vlasov–Poisson system is presented (the Vlasov–Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler–Lagrange equations contain not only the Vlasov–Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley–Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov–Poisson system.
4

Vedenyapin, V. V., T. V. Salnikova, and S. Ya Stepanov. "Vlasov-Poisson-Poisson equations, critical mass and kordylewski clouds." Доклады Академии наук 485, no. 3 (May 21, 2019): 276–80. http://dx.doi.org/10.31857/s0869-56524853276-280.

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A derivation of the Vlasov-Poisson-Poisson equation is proposed for studying stationary solutions of a system of gravitating charged particles in vicinity of triangular libration points (Kordylevsky cloud). Stationary solutions are sought as functions of integrals, which leads to elliptic nonlinear equations for the potentials of the gravitational and electrostatic fields. This gives a critical mass: for bodies with large masses dominates gravitation forces, and for bodies with smaller masses - electrostatic forces.
5

Vedenyapin, V. V., T. V. Salnikova, and S. Ya Stepanov. "Vlasov–Poisson–Poisson Equations, Critical Mass, and Kordylewski Clouds." Doklady Mathematics 99, no. 2 (March 2019): 221–24. http://dx.doi.org/10.1134/s1064562419020212.

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6

Scovel, Clint, and Alan Weinstein. "Finite dimensional lie-poisson approximations to vlasov-poisson equations." Communications on Pure and Applied Mathematics 47, no. 5 (May 1994): 683–709. http://dx.doi.org/10.1002/cpa.3160470505.

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7

Vedenyapin, Victor Valentinovich, and Dmitry Aleksandrovich Kogtenev. "On Derivation and Properties of Vlasov-type equations." Keldysh Institute Preprints, no. 20 (2023): 1–18. http://dx.doi.org/10.20948/prepr-2023-20.

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Derivation of the gravity and electrodynamics equations in the Vlasov-Maxwell-Einstein form is considered. Properties of Vlasov-Poisson equation and its application to construction of periodic solutions – Bernstein-Greene-Kruskal waves – are proposed.
8

Tyranowski, Tomasz M. "Stochastic variational principles for the collisional Vlasov–Maxwell and Vlasov–Poisson equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2252 (August 2021): 20210167. http://dx.doi.org/10.1098/rspa.2021.0167.

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In this work, we recast the collisional Vlasov–Maxwell and Vlasov–Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov–Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.
9

Karimov, A. R., and H. Ralph Lewis. "Nonlinear solutions of the Vlasov–Poisson equations." Physics of Plasmas 6, no. 3 (March 1999): 759–61. http://dx.doi.org/10.1063/1.873313.

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10

Larsson, Jonas. "Hermitian structure for the linearized Vlasov-Poisson and Vlasov-Maxwell equations." Physical Review Letters 66, no. 11 (March 18, 1991): 1466–68. http://dx.doi.org/10.1103/physrevlett.66.1466.

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11

Belyaeva, Yu O., and A. L. Skubachevskii. "On classical solutions to the first mixed problem for the Vlasov-Poisson system in an infinite cylinder." Доклады Академии наук 484, no. 6 (May 18, 2019): 663–66. http://dx.doi.org/10.31857/s0869-56524846663-666.

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The first mixed problem for the Vlasov-Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a high-temperature two-component plasma under an external magnetic field. For an arbitrary electric field potential and a sufficiently strong external magnetic field, it is shown that the characteristics of the Vlasov equations do not reach the boundary of the cylinder. It is proved that the Vlasov-Poisson system with ion and electron distribution density functions supported at some distance from the cylinder boundary has a unique classical solution.
12

SALORT, DELPHINE. "TRANSPORT EQUATIONS WITH UNBOUNDED FORCE FIELDS AND APPLICATION TO THE VLASOV–POISSON EQUATION." Mathematical Models and Methods in Applied Sciences 19, no. 02 (February 2009): 199–228. http://dx.doi.org/10.1142/s0218202509003401.

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The aim of this paper is to give new dispersive tools for certain kinetic equations. As an application, we study the three-dimensional Vlasov–Poisson equation for initial data having strictly less than six moments in [Formula: see text] where the nonlinear term E is a priori unbounded. We prove via new dispersive effects that in fact the force field E is smooth in space at the cost of a localization in a ball and an averaging in time. We deduce new conditions to bound the density ρ in L∞ and to have existence and uniqueness of global weak solutions of the Vlasov–Poisson equation with bounded density for initial data strictly less than six moments in [Formula: see text]. The proof is based on a new approach which consists in establishing a priori dispersion estimates (moment effects) on the one hand for linear transport equations with unbounded force fields and on the other hand along the trajectories of the Vlasov–Poisson equation.
13

EL-HANBALY, A. M., and A. ELGARAYHI. "Exact solutions of the collisional Vlasov equation." Journal of Plasma Physics 59, no. 1 (January 1998): 169–77. http://dx.doi.org/10.1017/s0022377897006132.

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The symmetry group of the Vlasov–Fokker–Planck equation (VFPE) is constructed. The effects of the Poisson equation on this group is studied, and different types of similarity solutions of the whole system of equations (VFPE+Poisson equation) are obtained.
14

Kaup, D. J., and Gary E. Thomas. "Linearized Vlasov–Poisson equations for the planar magnetron." Physics of Fluids B: Plasma Physics 4, no. 8 (August 1992): 2640–44. http://dx.doi.org/10.1063/1.860180.

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15

Nocera, L., and L. J. Palumbo. "Sectionally analytic solutions of the Vlasov–Poisson equations." Journal of Physics A: Mathematical and Theoretical 45, no. 10 (February 23, 2012): 105501. http://dx.doi.org/10.1088/1751-8113/45/10/105501.

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16

Pokhozhaev, S. I. "On stationary solutions of the Vlasov-Poisson equations." Differential Equations 46, no. 4 (April 2010): 530–37. http://dx.doi.org/10.1134/s0012266110040087.

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17

DOLBEAULT, J., and G. REIN. "TIME-DEPENDENT RESCALINGS AND LYAPUNOV FUNCTIONALS FOR THE VLASOV–POISSON AND EULER–POISSON SYSTEMS, AND FOR RELATED MODELS OF KINETIC EQUATIONS, FLUID DYNAMICS AND QUANTUM PHYSICS." Mathematical Models and Methods in Applied Sciences 11, no. 03 (April 2001): 407–32. http://dx.doi.org/10.1142/s021820250100091x.

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We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger–Poisson system, nonlinear Schrödinger equation).
18

Carrillo, José A., Young-Pil Choi, and Yingping Peng. "Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system." Kinetic and Related Models 15, no. 3 (2022): 355. http://dx.doi.org/10.3934/krm.2021052.

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<p style='text-indent:20px;'>We provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.</p>
19

Vedenyapin, V. V., A. A. Bay, and A. G. Petrov. "ON DERIVATION OF EQUATIONS OF GRAVITATION FROM THE PRINCIPLE OF LEAST ACTION, RELATIVISTIC MILNE-MCCREE SOLUTIONS AND LAGRANGE POINTS." Доклады Российской академии наук. Математика, информатика, процессы управления 514, no. 1 (November 1, 2023): 69–73. http://dx.doi.org/10.31857/s2686954323600532.

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We suggest the derivation of gravitation equations in the framework of Vlasov-Poisson relativistic equations with Lambda-term from the classical least action and use Hamilton-Jacobi consequence for cosmological solutions and investigate Lagrange points.
20

Teichmann, J. "Linear Vlasov stability in one-dimensional double layers." Laser and Particle Beams 5, no. 2 (May 1987): 287–93. http://dx.doi.org/10.1017/s0263034600002779.

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Analytical study of the linear stability of one-dimensional double layers in nonmagnetized plasmas based on the solution of the Vlasov–Poisson system is presented. Electromagnetic effects are not included. A self-consistent equilibrium electrostatic potential Φ0(z) that monotonically increases from a low level at z = − ∞ to a high level at z = + ∞ is assumed. We model this potential as a piecewise continuous function of z and we assume that Φ0(z) has constant values for − ∞ z ≤ 0 and L ≤ z < ∞, L being the thickness of the double layer. The BGK states for the Vlasov–Poisson system provide an explicit expression for the velocity distribution of the reflected electrons required for the particular double layer configuration. The stability of the double layers is studied via the linearized Vlasov and Poisson equations using the WKB approximation.
21

Wang, Yanli, and Shudao Zhang. "Solving Vlasov-Poisson-Fokker-Planck Equations using NRxx method." Communications in Computational Physics 21, no. 3 (February 7, 2017): 782–807. http://dx.doi.org/10.4208/cicp.220415.080816a.

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AbstractWe present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the NRxx method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the NRxx method when applied to the VPFP system.
22

Wei, Dongming. "1D Vlasov-Poisson equations with electron sheet initial data." Kinetic & Related Models 3, no. 4 (2010): 729–54. http://dx.doi.org/10.3934/krm.2010.3.729.

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23

Lee, Shyi‐Shiun, Shih‐Tuen Lee, and Jaw‐Yen Yang. "Numerical solution of the system of Vlasov‐Poisson equations." Journal of the Chinese Institute of Engineers 22, no. 3 (April 1999): 341–50. http://dx.doi.org/10.1080/02533839.1999.9670471.

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24

Crouseilles, Nicolas, Mohammed Lemou, and Florian Méhats. "Asymptotic Preserving schemes for highly oscillatory Vlasov–Poisson equations." Journal of Computational Physics 248 (September 2013): 287–308. http://dx.doi.org/10.1016/j.jcp.2013.04.022.

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25

Casas, Fernando, Nicolas Crouseilles, Erwan Faou, and Michel Mehrenberger. "High-order Hamiltonian splitting for the Vlasov–Poisson equations." Numerische Mathematik 135, no. 3 (June 22, 2016): 769–801. http://dx.doi.org/10.1007/s00211-016-0816-z.

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26

Barré, Julien, David Chiron, Thierry Goudon, and Nader Masmoudi. "From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge." Journal de l’École polytechnique — Mathématiques 2 (2015): 247–96. http://dx.doi.org/10.5802/jep.24.

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27

Fimin, Nikolay Nikolaevich, and Aisen Gavril'evich Nikoforov. "Energetic and hydrodynamic substitutions for Vlasov-Poisson and its astrophysical consequences." Keldysh Institute Preprints, no. 29 (2022): 1–22. http://dx.doi.org/10.20948/prepr-2022-29.

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This preprint describes the system of Vlasov–Poisson equations in the self-consistent gravitational potential of cosmological genesis, and shows a case that leads to the formation of coherent pseudochaotically distributed ”walls” of the cosmological structure.
28

LUNDBERG, JONAS, and TOR FLÅ. "A perturbation method for the Vlasov–Poisson system." Journal of Plasma Physics 60, no. 1 (August 1998): 181–92. http://dx.doi.org/10.1017/s0022377898006503.

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A perturbation method for the Vlasov–Poisson system is presented. It is self-consistent and entirely based on Lie transformations, which are considered as active transformations, generating the dynamics of the particle distribution function in the space of distribution functions. The main result is a set of three equations that forms a good starting point for a wide variety of problems concerning nonlinear wave propagation. Besides being efficient, the new perturbation method is systematic and therefore also suited for the use of computer algebra.
29

BONILLA, LUIS L., and JUAN S. SOLER. "HIGH-FIELD LIMIT OF THE VLASOV–POISSON–FOKKER–PLANCK SYSTEM: A COMPARISON OF DIFFERENT PERTURBATION METHODS." Mathematical Models and Methods in Applied Sciences 11, no. 08 (November 2001): 1457–68. http://dx.doi.org/10.1142/s0218202501001410.

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A reduced drift-diffusion (Smoluchowski–Poisson) equation is found for the electric charge in the high-field limit of the Vlasov–Poisson–Fokker–Planck system, both in one and three dimensions. The corresponding electric field satisfies a Burgers equation. Three methods are compared in the one-dimensional case: Hilbert expansion, Chapman–Enskog procedure and closure of the hierarchy of equations for the moments of the probability density. Of these methods, only the Chapman–Enskog method is able to systematically yield reduced equations containing terms of different order.
30

Badsi, Mehdi, Martin Campos-Pinto, Bruno Després, and Ludovic Godard-Cadillac. "A variational sheath model for stationary gyrokinetic Vlasov–Poisson equations." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 6 (November 2021): 2609–42. http://dx.doi.org/10.1051/m2an/2021067.

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We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.
31

Brizard, Alain J., and Natalia Tronko. "Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations." Physics of Plasmas 18, no. 8 (August 2011): 082307. http://dx.doi.org/10.1063/1.3625554.

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32

Nguyen van yen, Romain, Éric Sonnendrücker, Kai Schneider, and Marie Farge. "Particle-in-wavelets scheme for the 1D Vlasov-Poisson equations." ESAIM: Proceedings 32 (October 2011): 134–48. http://dx.doi.org/10.1051/proc/2011017.

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33

CARPIO, A., E. CEBRIAN, and F. J. MUSTIELES. "LONG TIME ASYMPTOTICS FOR THE SEMICONDUCTOR VLASOV–POISSON–BOLTZMANN EQUATIONS." Mathematical Models and Methods in Applied Sciences 11, no. 09 (December 2001): 1631–55. http://dx.doi.org/10.1142/s0218202501001513.

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In this paper we analyze the long time behavior of solutions to the one-dimensional Vlasov–Poisson–Boltzmann (VPB) equations for semiconductors in unbounded domains when only one type of carriers (electrons) are considered. We prove that the distribution of electrons tends for large times to a steady state of the VPB equations with vanishing collision term and the same total charge as the initial data. In the proof of the main result, the conservation law of charge, the balance of energy and entropy inequalities are rigorously derived. An important argument in the proof is to use a Lyapunov-type functional related to these physical quantities.
34

Carmack, Lori, and George Majda. "Concentrations in the one-dimensional Vlasov-Poisson equations: additional regularizations." Physica D: Nonlinear Phenomena 121, no. 1-2 (October 1998): 127–62. http://dx.doi.org/10.1016/s0167-2789(98)00029-3.

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35

Cottet, G. H., and P. A. Raviart. "On particle-in-cell methods for the Vlasov-Poisson equations." Transport Theory and Statistical Physics 15, no. 1-2 (February 1986): 1–31. http://dx.doi.org/10.1080/00411458608210442.

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36

HAAS, FERNANDO. "On quantum plasma kinetic equations with a Bohmian force." Journal of Plasma Physics 76, no. 3-4 (January 8, 2010): 389–93. http://dx.doi.org/10.1017/s0022377809990572.

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AbstractThe dispersion relation arising from a Vlasov–Poisson system with a Bohmian force term is examined and compared to the more fundamental Bohm and Pines dispersion relation for quantum plasmas. Discrepancies are found already when considering the leading order thermal effects. The time-averaged energy densities for longitudinal modes are also shown to be noticeably different.
37

Gasser, I., P. E. Jabin, and B. Perthame. "Regularity and propagation of moments in some nonlinear Vlasov systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 6 (December 2000): 1259–73. http://dx.doi.org/10.1017/s0308210500000676.

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We introduce a new variant to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame. We combine it with moment lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov–Poisson system, and we solve a long-standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov–Stokes system, where we prove the same result for fairly singular kernels.
38

Croci, Riccardo. "Asymptotic solution of the Vlasov and Poisson equations for an inhomogeneous plasma." Journal of Plasma Physics 45, no. 1 (February 1991): 59–70. http://dx.doi.org/10.1017/s002237780001549x.

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The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter ε goes to zero (the kernel becoming proportional to a Dirac δ function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on εx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter.
39

Kozhevnikov, Vasily, Andrey Kozyrev, Aleksandr Kokovin, and Natalia Semeniuk. "Kinetic simulation of vacuum plasma expansion beyond the "plasma approximation"." Vojnotehnicki glasnik 70, no. 3 (2022): 650–63. http://dx.doi.org/10.5937/vojtehg70-37337.

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Introduction/purpose: One of the key approaches to solving an entire class of modern plasma physics problems is the so-called "plasma approximation". The most general definition of the "plasma approximation" is a theoretical approach to the electric field calculation of a system of charges under the electric quasi-neutrality condition. The purpose of this paper is to compare the results of the numerical simulation of the kinetic processes of the quasi-neutral plasma bunch expansion to the analytical solution of a similar kinetic model but in the "plasma approximation". Methods: The given results are obtained by the methods of deterministic modeling based on the numerical solution of the system of Vlasov-Poisson equations. Results: The provided comparison of the analytical expressions for the solution of kinetic equations in the "plasma approximation" and the numerical solutions of the Vlasov-Poisson equations system convincingly show the limitations of the "plasma approximation" in some important cases of the considered problem of plasma formation decay. Conclusion: The theoretical results of this work are of great importance for understanding the shortcomings of the "plasma approximation", which can manifest themselves in practical applications of computational plasma physics.
40

ESEN, OĞUL, and HASAN GÜMRAL. "LIFTS, JETS AND REDUCED DYNAMICS." International Journal of Geometric Methods in Modern Physics 08, no. 02 (March 2011): 331–44. http://dx.doi.org/10.1142/s0219887811005166.

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We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.
41

Hosseini Jenab, S. M., and G. Brodin. "Head-on collision of nonlinear solitary solutions to Vlasov-Poisson equations." Physics of Plasmas 26, no. 2 (February 2019): 022303. http://dx.doi.org/10.1063/1.5078865.

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42

Brenier, Y. "convergence of the vlasov-poisson system to the incompressible euler equations." Communications in Partial Differential Equations 25, no. 3-4 (January 2000): 737–54. http://dx.doi.org/10.1080/03605300008821529.

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43

Pulvirenti, M., and J. Wick. "On the statistical solutions of Vlasov-Poisson equations in two dimensions." ZAMP Zeitschrift f�r angewandte Mathematik und Physik 36, no. 4 (July 1985): 508–19. http://dx.doi.org/10.1007/bf00945293.

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44

Perez, Jérôme. "Vlasov and Poisson Equations in the Context of Self‐Gravitating Systems." Transport Theory and Statistical Physics 34, no. 3-5 (August 2005): 391–406. http://dx.doi.org/10.1080/00411450500274691.

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45

Karimov, A. R. "Nonlinear solutions of a Maxwellian type for the Vlasov–Poisson equations." Physics of Plasmas 8, no. 5 (May 2001): 1533–37. http://dx.doi.org/10.1063/1.1356439.

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46

Ambrosio, Luigi, Maria Colombo, and Alessio Figalli. "On the Lagrangian structure of transport equations: The Vlasov–Poisson system." Duke Mathematical Journal 166, no. 18 (December 2017): 3505–68. http://dx.doi.org/10.1215/00127094-2017-0032.

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47

Channell, Paul J. "Systematic solution of the Vlasov–Poisson equations for charged particle beams." Physics of Plasmas 6, no. 3 (March 1999): 982–93. http://dx.doi.org/10.1063/1.873339.

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48

Skubachevskii, A. L. "Nonlocal problems for the Vlasov—Poisson equations in an infinite cylinder." Functional Analysis and Its Applications 49, no. 3 (July 2015): 234–38. http://dx.doi.org/10.1007/s10688-015-0112-1.

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49

Degond, P., and H. Neunzert. "Local existence of solutions of the vlasov-maxwell equations and convergence to the vlasov-poisson equations for infinite light velocity." Mathematical Methods in the Applied Sciences 8, no. 1 (1986): 533–58. http://dx.doi.org/10.1002/mma.1670080135.

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50

Fimin, Nikolay Nikolaevich, and Valery Mihailovich Chechetkin. "Determinism of genesis of large-scale structures in astrophysics." Keldysh Institute Preprints, no. 67 (2023): 1–24. http://dx.doi.org/10.20948/prepr-2023-67.

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Abstract:
The criteria for the formation of non-stationary pseudo-periodic structures in a system of gravitating particles, described by the Vlasov--Poisson system of equations. Conditions of branching of solutions of a nonlinear integral equation for a generalized gravitational potential, leading to the emergence of coherent complex states of relative equilibrium in non-stationary systems of massive particles, is studied.
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