Quasi-periodic traveling electron layers

Emeric Roulley; Luca Franzoi

Quasi-periodic traveling electron layers

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Small-amplitude traveling quasi-periodic electron layers exist in the Vlasov-Poisson equations for most strip areas.

desk verdict The paper gives the first rigorous construction of small-amplitude quasi-periodic traveling electron layers in 1D Vlasov-Poisson near flat strips, for most strip areas, via Nash-Moser plus KAM reducibility, with a byproduct for cubic Euler-Poisson. read the letter →

arxiv 2605.25885 v1 pith:ZE7HHMER submitted 2026-05-25 math.AP physics.plasm-ph

Luca Franzoi , Emeric Roulley This is my paper · ORCID

classification math.APphysics.plasm-ph

keywords Vlasov-Poissonequationsquasi-periodicsolutionstravelingwaveselectronlayersNash-MosertheoremKAMtheoryEuler-Poissonsystem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs small amplitude traveling quasi-periodic solutions that appear as strip-shaped patches of electrons in phase space for the one-dimensional periodic Vlasov-Poisson equations. These solutions are found close to symmetric flat velocity strips and exist for most values of the strip area. The result also provides the first rigorous quasi-periodic solutions for kinetic models and extends directly to the related electronic Euler-Poisson system with cubic pressure law. A sympathetic reader would care because it demonstrates existence of complex time-dependent structures in plasma models that were previously inaccessible by rigorous methods.

What carries the argument

Nash-Moser construction with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions, applied after a linear transformation connecting the electron patch problem to the Euler-Poisson system.

What would settle it

A calculation or simulation that exhibits breakdown of the KAM reduction or absence of such solutions on a positive-measure set of strip areas would falsify the existence claim for most areas.

Watch

Extended reading notes

Core claim

We construct, close to symmetric flat velocity strips, small amplitude traveling quasi-periodic electron-layers, namely strip-shaped patches of electrons in the phase space, for the one dimensional space-periodic Vlasov-Poisson equations. These solutions are found for most values of the strip area. The proof uses a Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. A suitable linear transformation reveals a connection to the electronic Euler-Poisson system with cubic pressure law, yielding small-amplitude quasi-periodic traveling waves for this model as well.

Load-bearing premise

The Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions can be carried through without breakdown for the chosen strip areas and small amplitudes.

Editorial extensions

If this is right

Quasi-periodic traveling waves exist for the electronic Euler-Poisson system with cubic pressure law at small amplitudes.
The construction applies for most values of the strip area in the Vlasov-Poisson model.
This provides the first rigorous example of quasi-periodic solutions for kinetic models.
Small amplitude solutions can be constructed near symmetric flat velocity strips.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The linear transformation technique may allow transfer of other Vlasov-Poisson results to the Euler-Poisson setting.
Quasi-periodic electron layers could appear in more general initial data if the small-amplitude condition can be relaxed.
Numerical continuation from these solutions might test whether the layers persist at moderate amplitudes.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core result is a construction of traveling quasi-periodic electron patches in phase space for the space-periodic Vlasov-Poisson system, sitting close to symmetric flat velocity strips and existing for most values of the strip area. A linear change of variables turns the problem into a cubic-pressure Euler-Poisson system, so the same argument yields quasi-periodic traveling waves there as well.

What is actually new is the kinetic-model case; earlier rigorous quasi-periodic work stayed in fluid or periodic settings. The reduction to the fluid system is a clean observation that buys both results at once. The technical route—Nash-Moser iteration with pseudo-differential homogeneous expansions and KAM-style reducibility of the linearized operator—matches the standard toolkit for small-divisor problems in these equations, and the abstract states the setup without obvious circularity or free parameters beyond the strip area itself.

The soft spot is the usual one for this style of argument: the measure of admissible parameters and the precise control of derivative loss in the iteration need to be checked in the full proof. Nothing in the outline suggests the homological equations break down, but the details matter. The claim is stated for “most” areas, which is the expected form.

This is aimed at people working on quasi-periodic solutions in kinetic or fluid plasma models. A reader who follows KAM or Nash-Moser constructions in PDEs will get value from the reduction and the extension to the kinetic setting. The work shows clear engagement with the literature and the technical repertoire, so it deserves a serious referee even if the estimates require heavy checking.

Referee Report

0 major / 3 minor

Summary. The paper constructs small-amplitude traveling quasi-periodic electron layers (strip-shaped patches in phase space) for the one-dimensional space-periodic Vlasov-Poisson equations, close to symmetric flat velocity strips and for most values of the strip area. The proof employs a Nash-Moser iteration together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. A linear change of variables links the problem to the cubic-pressure electronic Euler-Poisson system, yielding quasi-periodic traveling waves for that model as a byproduct. The abstract states this is the first rigorous construction of quasi-periodic solutions for kinetic models.

Significance. If the Nash-Moser/KAM construction is complete and the measure of admissible strip areas is positive, the result is significant: it supplies the first rigorous quasi-periodic solutions in a kinetic model and simultaneously produces new traveling waves for the physically relevant Euler-Poisson system. The self-contained nature of the construction (no fitted parameters beyond the strip area) and the explicit connection between kinetic and fluid regimes add value.

minor comments (3)

The abstract and introduction should explicitly state the precise function space (e.g., Gevrey or analytic class) in which the quasi-periodic solutions are constructed, as this determines the scope of the small-divisor estimates.
Notation for the linear transformation mapping Vlasov-Poisson to Euler-Poisson (mentioned in the abstract) should be introduced with an equation number in §2 or §3 so that the subsequent reducibility analysis can be traced directly to the transformed system.
The statement 'for most values of the strip area' should be accompanied by a quantitative lower bound on the measure of the admissible set (e.g., in terms of the amplitude parameter) already in the introduction, rather than deferred to the final theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. We are pleased that the novelty of the first rigorous quasi-periodic solutions in a kinetic model, as well as the connection to the Euler-Poisson system, is recognized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper constructs quasi-periodic solutions via Nash-Moser iteration combined with pseudo-differential expansions and KAM reducibility after a linear change of variables that maps the Vlasov-Poisson electron-patch problem onto the cubic-pressure Euler-Poisson system. No load-bearing step reduces by definition or by construction to its own fitted inputs; the admissible strip areas are stated to be obtained for most values without any parameter-fitting loop described. No self-citations are invoked to justify uniqueness theorems or ansatzes, and the claim of providing the first rigorous kinetic quasi-periodic example is presented as a consequence of the new construction rather than a renaming of prior results. The method is standard for such problems and does not collapse to tautology.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

Review based on abstract only; the Vlasov-Poisson system and the applicability of Nash-Moser/KAM are taken as given.

free parameters (1)

strip area
Solutions exist for most values of the strip area; the measure of admissible areas is not quantified in abstract.

assumptions (2)

domain assumption One dimensional space-periodic Vlasov-Poisson equations
The model under consideration.
domain assumption Existence of symmetric flat velocity strips as base solutions
The construction is performed close to these strips.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quasi-periodic traveling electron layers." pith.science (2026). https://pith.science/paper/ZE7HHMER

@misc{pith2026260525885,
  author       = {Pith},
  title        = {Pith review of: Quasi-periodic traveling electron layers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZE7HHMER}},
  note         = {Machine review of arXiv:2605.25885}
}

read the original abstract

We consider the one dimensional space-periodic Vlasov-Poisson equations and construct, close to symmetric flat velocity strips, small amplitude traveling quasi-periodic electron-layers, namely strip-shaped patches of electrons in the phase space. These solutions are found for most values of the strip area. The proof uses a Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. Thanks to a suitable linear transformation of the unknowns, we reveal a connection between the electron patch problem and the classical, physically relevant, electronic Euler-Poisson system with cubic pressure law. As a direct, non trivial consequence, we obtain small-amplitude quasi-periodic traveling waves for this model as well. To the best of our knowledge, this work provides the first rigorous example of construction of quasi-periodic solutions for kinetic models.

Figures

Figures reproduced from arXiv: 2605.25885 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Time-periodic vortices near translating symmetric dipole patches
math.AP 2026-07 accept novelty 7.0 of 10

Non-rigid time-periodic vortex patch solutions bifurcating from translating symmetric dipoles are constructed for the 2D Euler equations using Lyapunov-Schmidt reduction and Nash-Moser methods.

Reference graph

Works this paper leans on

94 extracted references · 9 canonical work pages · cited by 1 Pith paper

[1]

Ambrosio, M

L. Ambrosio, M. Colombo, A. Figalli,On the Lagrangian structure of transport equations: The Vlasov–Poisson system,Duke Mathematical Journal 166 (2017), no. 18, 3505–3568

2017
[2]

Armstrong, D

T. Armstrong, D. Montgomery,Asymptotic state of the two-stream instability,Journal of Plasma Physics 1 (1967), 425–433

1967
[3]

A. A. Arsen’ev,Existence in the large of a weak solution of Vlasov’s system of equations,Zhurnal Vychis- litel’noi Matematiki i Matematicheskoi Fiziki 15 (1975), no. 276, 136–147

1975
[4]

J. Bae, B. Kwon,Small amplitude limit of solitary waves for the Euler-Poisson system,Journal of Differ- ential Equations 266 (2019), no. 6, 3450–3478

2019
[5]

J. Bae, B. Kwon,Linear stability of solitary waves for the isothermal Euler-Poisson system,Archive for Rational Mechanics and Analysis 243 (2022), no. 1, 257–327

2022
[6]

Baldi, M

P. Baldi, M. Berti, E. Haus, R. Montalto,Time quasi-periodic gravity water waves in finite depth,Inven- tiones Mathematicae 214 (2018), no. 2, 739–911

2018
[7]

Baldi, R

P. Baldi, R. Montalto,Quasi-periodic incompressible Euler flows in 3D,Advances in Mathematics 384 (2021), 107730

2021
[8]

Bardos, P

C. Bardos, P. Degond,Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data,Annales de l’Institut Henri Poincar´ e Analyse Non Lin´ eaire 2 (1985), 101–118

1985

Show all 94 references

[9]

Batt,Global symmetric solutions of the initial value problem of stellar dynamics,Journal Differential Equations 25 (1977), 342–364

J. Batt,Global symmetric solutions of the initial value problem of stellar dynamics,Journal Differential Equations 25 (1977), 342–364

1977
[10]

Bedrossian, N

J. Bedrossian, N. Masmoudi,Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations, Publications Math´ ematiques de l’IHES 122 (2015), 195–300

2015
[11]

I. B. Bernstein, J. M. Greene, M. D. Kruskal,Exact nonlinear plasma oscillations,Physical Review 108 (1957), 546–550

1957
[12]

Hamiltonian PDEs and Applications

M. Berti, P. Bolle,A Nash-Moser approach to KAM theory, Fields Institute Communications, special volume “Hamiltonian PDEs and Applications”, (2015), 255–284

2015
[13]

Berti, L

M. Berti, L. Franzoi, A. Maspero,Traveling quasi-periodic water waves with constant vorticity, Archive for Rational Mechanics and Analysis, 240 (2021), 99–202

2021
[14]

Berti, L

M. Berti, L. Franzoi, A. Maspero,Pure gravity traveling quasi-periodic water waves with constant vorticity, Communications on Pure and Applied Mathematics 77 (2024), no. 2, 990–1064

2024
[15]

Berti, Z

M. Berti, Z. Hassainia, N. Masmoudi,Time quasi-periodic vortex patches of Euler equation in the plane, Inventiones Mathematicae 233 (2023), 1279–1391

2023
[16]

Berti, T

M. Berti, T. Kappeler, R. Montalto,Large KAM tori for quasi-linear perturbations of KdV,Archive for Rational Mechanics 239 (2021), 1395–1500

2021
[17]

Berti, R

M. Berti, R. Montalto,Quasi-periodic standing wave solutions of gravity-capillary water waves, MEMO, Volume 263, 1273, Memoirs of the American Mathematical Society, ISSN 0065-9266, (2020)

2020
[18]

A. L. Bertozzi, A. J. Majda,Vorticity and Incompressible Flow,Cambridge texts in applied Mathematics, Cambridge University Press, Cambridge, (2002)

2002
[19]

D. Bian, E. Grenier, W. Huang, Benoˆ ıt Pausader,Stability and instability of small BGK waves,arXiv preprint, arXiv:2601.10030
[20]

Bianchini, L

R. Bianchini, L. Franzoi, R. Montalto, S. Terracina,Large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids,Communications in Mathematical Physics 406 (2025), no. 66, 1–67

2025
[21]

D. Bohm, E. P. Gross,Theory of plasma oscillations. A. Origin of medium-like behavior,Physical Review 75 (1949), 1851–1864

1949
[22]

Buchanan, J

M. Buchanan, J. Dorning,Nonlinear electrostatic waves in collisionless plasmas,Physical Review E 52 (1995), 3015–3033. 78

1995
[23]

Bostan, F

M. Bostan, F. Poupaud,Periodic solutions of the Vlasov-Poisson system with boundary conditions,Math- ematical Models and Methods in Applied Sciences 10 (2000), no. 5, 651–672

2000
[24]

A. C.-L. Chian, P. C. Clemmow,Nonlinear, periodic waves in a cold plasma: a quantitative analysis, Journal of Plasma Physics 14 (1975), no. 3, 505–527

1975
[25]

Cordier, P

S. Cordier, P. Degond, P. Markowich, C. Schmeiser,Travelling wave analysis and jump relations for Eu- ler–Poisson model in the quasineutral limit,Asymptotic Analysis 11 (1995), no. 3, 209–240

1995
[26]

Cordier, P

S. Cordier, P. Degond, P. Markowich, C. Schmeiser,Travelling wave analysis of an isothermal Euler- Poisson model,Annales de la Facult´ e des sciences de Toulouse: Math´ ematiques, S´ erie 6, Tome 5 (1996), no. 4, 599–643

1996
[27]

G. H. Cottet, P. A. Raviart,Particle methods for the 1-D Vlasov-Poisson equations,SIAM Journal on Numerical Analysis, 21 (1984), no. 1, 52–76

1984
[28]

Crouseilles, E

N. Crouseilles, E. Faou,Quasi-periodic solutions of the 2D Euler equations,Asymptotic analysis 81 (2013), no. 1, 31–34

2013
[29]

Y. Deng, N. Masmoudi,Long time instability of the Couette flow in low Gevrey spaces, Communincations on Pure and Applied Mathematics 76 (2023), no. 10, 2804–2887

2023
[30]

R. J. DiPerna, P.-L. Lions,Solutions globales d’´ equations du type Vlasov-Poisson,Comptes rendus de l’Acad´ emie des Sciences de Paris S´ erie I Math´ ematiques 307 (1988), 655–658

1988
[31]

R. J. DiPerna, P.-L. Lions,Global weak solutions of Vlasov-Maxwell systems,Communications on Pure and Applied Mathematics 42 (1989), 729–757

1989
[32]

R. S. Dziurzynski,Patches of electrons and electron sheets for the 1-D Vlasov-Poisson equation,PhD thesis University of California, Berkeley, (1987)

1987
[33]

Enciso, D

A. Enciso, D. Peralta-Salas, F. Torres de Lizaur,Quasi-periodic solutions to the incompressible Euler equations in dimensions two and higher,Journal of Differential Equations 354 (2023), 170–182

2023
[34]

Feola, L

R. Feola, L. Franzoi, R. Montalto,Long time dynamics close to large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids,arXiv preprint, arXiv:2604.09302

arXiv
[35]

Feola, F

R. Feola, F. Giuliani,Quasi-periodic traveling waves on an infinitely deep perfect fluid under gravity, Memoirs of the American Mathematical Society, (2021)

2021
[36]

Feola, F

R. Feola, F. Giuliani, R. Montalto, M. Procesi,Reducibility of first order linear operators on tori via Moser’s theorem, Journal of Functional Analysis 276 (2019), no. 3, 932–970

2019
[37]

Feola, R

R. Feola, R. Montalto, S. Terracina,Time quasi-periodic three-dimensional traveling gravity water waves, arXiv preprint, arXiv:2509.10318
[38]

Franzoi, R

L. Franzoi, R. Montalto,A KAM approach to the inviscid limit for the 2D Navier-Stokes equations,Annales Henri Poincar´ e 25 (2024), no. 12, 5231–5275

2024
[39]

Franzoi, R

L. Franzoi, R. Montalto,Time almost-periodic solutions of the incompressible Euler equations,Mathematics in Engineering 6 (2024), no. 1, 394–406

2024
[40]

Franzoi, N

L. Franzoi, N. Masmoudi, R. Montalto,Space quasi-periodic steady Euler flows close to the inviscid Couette flow,Archive for Rational Mechanics and Analysis 248 (2024), no. 81, 1–79

2024
[41]

Y. Fu, C. Qu, X. Wu,Reducibility for the linearized two-component intermediate long-wave equation, Journal of Functional Analysis (2025), no. 288, 110832

2025
[42]

Gagnebin, M

A. Gagnebin, M. Iacobelli,Landau damping on the torus for the Vlasov-Poisson system with massless electronsJournal of Differential Equations 376 (2023), 154–203

2023
[43]

Gasser, P.-E

I. Gasser, P.-E. Jabin, B. Perthame,Regularity and propagation of moments in some nonlinear Vlasov systems.Proceedings of the Royal Society of Edinburgh Section A 130 (2000), 1259–1273

2000
[44]

Ghizzo, B

A. Ghizzo, B. Izrar, P. Bertrand, E. Fijalkow, M.R. Feix, M. Shoucri,Stability of Bernstein–Greene–Kruskal plasma equilibria. Numerical experiments over a long time,Physics of Fluids 31 (1988), 72–82

1988
[45]

Goldman,Theory of stability of large periodic plasma waves,Physics of Fluids 13 (1970), 1281–1289

M. Goldman,Theory of stability of large periodic plasma waves,Physics of Fluids 13 (1970), 1281–1289. 79

1970
[46]

Guelmame, T

B. Guelmame, T. Hmidi, H. Houamed, F. Rousset,Singular traveling waves for the Euler-Poisson system, arXiv preprint, arXiv:2604.14997

arXiv
[47]

Guo,Smooth irrotational flows in the large to the Euler–Poisson system inR 3+1, Communications in Mathematical Physics 195 (1998), 249–265

Y. Guo,Smooth irrotational flows in the large to the Euler–Poisson system inR 3+1, Communications in Mathematical Physics 195 (1998), 249–265

1998
[48]

Y. Guo, Z. Lin,The existence of stable BGK waves,Communications in Mathematical Physics 352 (2017), 1121–1152

2017
[49]

Y. Guo, L. Han, J. Zhang,Absence of shocks for one dimensional Euler–Poisson system, Archive of Rational Mechanics and Analysis 223 (2017), 1057–1121

2017
[50]

Y. Guo, W. Strauss,Instability of periodic BGK equilibria,Communications on Pure and Applied Mathe- matics 48 (1995), 861–894

1995
[51]

Y. Guo, W. Strauss,Unstable BGK solitary waves and collisionless shocks,Communications in Mathemat- ical Physics 195 (1998), 267–293

1998
[52]

Hadˇ zi´ c, M

M. Hadˇ zi´ c, M. Moreno,On absence of embedded eigenvalues and stability of BGK waves, arXiv preprint, arXiv:2412:07025
[53]

Hadˇ zi´ c, G

M. Hadˇ zi´ c, G. Rein, M. Schrecker, C. Straub,Damping versus oscillations for a gravitational Vlasov–Poisson system, Archive for Rational Mechanics and Analysis 249 (2025), no. 45. DOI: 10.1007/s00205-025-02114-y

2025 doi
[54]

Hannibal, E

L. Hannibal, E. Rebhan, C. Kielhorn,Bifurcation of BGK waves in a plasma of cold ions and electrons, Journal of Plasma Physics 52 (1994), 1–22

1994
[55]

Hˇ arˇ agu¸ s, D

M. Hˇ arˇ agu¸ s, D. P. Nicholls, D. H. Sattinger,Solitary wave interactions of the Euler-Poisson equations, Journal of Mathematical Fluid Mechanics 5 (2003), no. 1, 92–118

2003
[56]

Hˇ arˇ agu¸ s, A

M. Hˇ arˇ agu¸ s, A. Scheel,Linear stability and instability of ion-acoustic plasma solitary waves,Physica D 170 (2002), no. 1, 13–30

2002
[57]

Hassainia, T

Z. Hassainia, T. Hmidi, N. Masmoudi,KAM theory for active scalar equations,Memoirs of the American Mathematical Society 314 (2025), no. 1596

2025
[58]

Hassainia, T

Z. Hassainia, T. Hmidi, E. Roulley,Invariant KAM tori around annular vortex patches for 2D Euler equations,Communications in Mathematical Physics 405 (2024), no. 270, 1–127

2024
[59]

Hassainia, E

Z. Hassainia, E. Roulley,Boundary effects on the existence of quasi-periodic solutions for Euler equations, Nonlinearity 38 015016 (2025), no. 1, 81pp

2025
[60]

Hmidi, E

T. Hmidi, E. Roulley,Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751. To appear in M´ emoires de la Soci´ et´ e Math´ ematique de France. DOI: 10.24033/msmf.498

doi
[61]

Horst,On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation

E. Horst,On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory.Mathematical Methods in the Applied Sciences 3 (1981), 229–248

1981
[62]

Horst,On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation

E. Horst,On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II: Special cases.Mathematical Methods in the Applied Sciences 4 (1982), 19–32

1982
[63]

Horst, R

E. Horst, R. Hunze,Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equa- tion,Mathematical Methods in the Applied Sciences 6 (1984), 262–279

1984
[64]

Illner, H

R. Illner, H. Neunzert,An existence theorem for the unmodified Vlasov equation,Mathematical Methods in the Applied Sciences 1 (1979), 530–544

1979
[65]

A. D. Ionescu, H. Jia,Inviscid damping near the Couette flow in a channel, Communications in Mathe- matical Physics 374 (2020), 2015–2096

2020
[66]

A. D. Ionescu, B. Pausader,The Euler–Poisson system in 2D: global stability of the constant equilibrium solution, International Mathematics Research Notices 2013 (2013), no. 4, 761–826

2013
[67]

A. D. Ionescu, B. Pausader, X. Wang, K. Widmayer,Nonlinear Landau damping for the Vlasov–Poisson system inR 3: the Poisson equilibrium, Annals of PDEs 10 (2024), no. 2. DOI: 10.1007/s40818-023-00161-w

2024 doi
[68]

S. V. Iordanskii,The Cauchy problem for the kinetic equation of plasma,Trudy Matematicheskogo Instituta imeni V.A. Steklova 60 (1961), 181–194. 80

1961
[69]

J. D. Jackson,Classical Electrodynamics, John Wiley & Sons Inc, (1962)

1962
[70]

N. G. V. Kampen,On the theory of stationary waves in plasmas,Physica 21 (1955), 949–963

1955
[71]

Landau,On the vibration of the electronic plasma,Journal of Physics USSR 10, 25 (1946)

L. Landau,On the vibration of the electronic plasma,Journal of Physics USSR 10, 25 (1946)

1946
[72]

D. Li, Y. Wu,The Cauchy problem for the two-dimensional Euler–Poisson system, Journal of the European Mathematical Society 16 (2014), 2211–2266

2014
[73]

Lin,Instability of periodic BGK waves,Mathematical Research Letters 8 (2001) 521–534

Z. Lin,Instability of periodic BGK waves,Mathematical Research Letters 8 (2001) 521–534

2001
[74]

Lin,Nonlinear instability of periodic BGK waves for Vlasov–Poisson system,Communications on Pure and Applied Mathematics 58 (2005), 505–528

Z. Lin,Nonlinear instability of periodic BGK waves for Vlasov–Poisson system,Communications on Pure and Applied Mathematics 58 (2005), 505–528

2005
[75]

Z. Lin, C. Zeng,Small BGK waves and nonlinear Landau damping,Communications in Mathematical Physics 306 (2011), 291–331

2011
[76]

Lions, B

P.-L. Lions, B. Perthame,Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system,Inventiones Mathematicae 105 (1991), 415–430

1991
[77]

Loeper,Uniqueness of the solution to the Vlasov-Poisson system with bounded density,Journal de Math´ ematiques Pures et Appliqu´ ees 86 (2006), 68–79

G. Loeper,Uniqueness of the solution to the Vlasov-Poisson system with bounded density,Journal de Math´ ematiques Pures et Appliqu´ ees 86 (2006), 68–79

2006
[78]

Manfredi, P

G. Manfredi, P. Bertrand,Stability of Bernstein–Greene–Kruskal modes,Physics of Fluids 7 (2000), 2425– 2431

2000
[79]

Mouhot, C

C. Mouhot, C. Villani,On Landau damping,Acta Mathematica 207 (2011), no. 1, 29–201

2011
[80]

Y. Mu, D. Wang,Global existence for the relativistic Vlasov-Poisson system in a two-dimensional bounded domain, arXiv preprint, arXiv:2511.06595
[81]

Noble, L

P. Noble, L. M. Rodrigues, C. Sun,Spectral instability of small-amplitude periodic waves of the electronic Euler–Poisson system,Nonlinearity 36 4615 (2023), no. 9

2023
[82]

Pallard,Space moments of the Vlasov-Poisson system: propagation and regularity,SIAM Journal on Mathematical Analysis 46 (2014), 1754–1770

C. Pallard,Space moments of the Vlasov-Poisson system: propagation and regularity,SIAM Journal on Mathematical Analysis 46 (2014), 1754–1770

2014
[83]

Pankavich, R

S. Pankavich, R. Allen,Instability conditions for some periodic BGK waves in the Vlasov–Poisson system, European Physical Journal D 68 (2014), 1–7

2014
[84]

Pfaffelmoser,Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,Journal of Differential Equations 95 (1992), no

K. Pfaffelmoser,Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,Journal of Differential Equations 95 (1992), no. 2, 281–303

1992
[85]

Rein,Collisionless kinetic equations from astrophysics: the Vlasov-Poisson system,Handbook of differ- ential equations: evolutionary equations

G. Rein,Collisionless kinetic equations from astrophysics: the Vlasov-Poisson system,Handbook of differ- ential equations: evolutionary equations. Vol. III, 383–476, Elsevier/North-Holland, Amsterdam, 2007

2007
[86]

Roulley,Periodic and quasi-periodic Euler-αflows close to Rankine vortices,Dynamics of Partial Dif- ferential Equations 20 (2023), no

E. Roulley,Periodic and quasi-periodic Euler-αflows close to Rankine vortices,Dynamics of Partial Dif- ferential Equations 20 (2023), no. 4, 311–366

2023
[87]

Roulley,Local and global bifurcation of electron states,Discrete and Continuous Dynamical Systems 45 (2025), no

E. Roulley,Local and global bifurcation of electron states,Discrete and Continuous Dynamical Systems 45 (2025), no. 8, 2381–2419

2025
[88]

Roulley,Ions-electrons-states for the two-component Vlasov-Poisson equation,arXiv preprint, arXiv:2602.09293

E. Roulley,Ions-electrons-states for the two-component Vlasov-Poisson equation,arXiv preprint, arXiv:2602.09293
[89]

R¨ ussmann,Invariant tori in non-degenerate nearly integrable Hamiltonian systems,Regular and Chaotic Dynamics 6 (2001), no

H. R¨ ussmann,Invariant tori in non-degenerate nearly integrable Hamiltonian systems,Regular and Chaotic Dynamics 6 (2001), no. 2, 119–204

2001
[90]

Schaeffer,Global existence for the Poisson-Vlasov system with nearly symmetric data,Journal of Differ- ential Equations 69 (1987), 111–148

J. Schaeffer,Global existence for the Poisson-Vlasov system with nearly symmetric data,Journal of Differ- ential Equations 69 (1987), 111–148

1987
[91]

J. L. Schwarzmeier, H. R. Lewis, B. Abraham-Shrauner, K. R. Symon,Stability of Bern- stein–Greene–Kruskal equilibria,Physics of Fluids 22 (1979), 1747–1760

1979
[92]

Suzuki, M

M. Suzuki, M. Takayama, K. Z. Zhang,Traveling waves of the Vlasov–Poisson system,Journal of Differ- ential Equations 428 (2025), 230–290

2025
[93]

S. Ukai, T. Okabe,On classical solutions in the large in time of two-dimensional Vlasov’s equation,Osaka Journal of Mathematics 15 (1978), 245–261

1978
[94]

Wollman,Global-in-time solutions of the two-dimensional Vlasov-Poisson system,Communications on Pure and Applied Mathematics 33 (1980), 173–197

S. Wollman,Global-in-time solutions of the two-dimensional Vlasov-Poisson system,Communications on Pure and Applied Mathematics 33 (1980), 173–197. 81

1980

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.

[1] [1]

Ambrosio, M

L. Ambrosio, M. Colombo, A. Figalli,On the Lagrangian structure of transport equations: The Vlasov–Poisson system,Duke Mathematical Journal 166 (2017), no. 18, 3505–3568

2017

[2] [2]

Armstrong, D

T. Armstrong, D. Montgomery,Asymptotic state of the two-stream instability,Journal of Plasma Physics 1 (1967), 425–433

1967

[3] [3]

A. A. Arsen’ev,Existence in the large of a weak solution of Vlasov’s system of equations,Zhurnal Vychis- litel’noi Matematiki i Matematicheskoi Fiziki 15 (1975), no. 276, 136–147

1975

[4] [4]

J. Bae, B. Kwon,Small amplitude limit of solitary waves for the Euler-Poisson system,Journal of Differ- ential Equations 266 (2019), no. 6, 3450–3478

2019

[5] [5]

J. Bae, B. Kwon,Linear stability of solitary waves for the isothermal Euler-Poisson system,Archive for Rational Mechanics and Analysis 243 (2022), no. 1, 257–327

2022

[6] [6]

Baldi, M

P. Baldi, M. Berti, E. Haus, R. Montalto,Time quasi-periodic gravity water waves in finite depth,Inven- tiones Mathematicae 214 (2018), no. 2, 739–911

2018

[7] [7]

Baldi, R

P. Baldi, R. Montalto,Quasi-periodic incompressible Euler flows in 3D,Advances in Mathematics 384 (2021), 107730

2021

[8] [8]

Bardos, P

C. Bardos, P. Degond,Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data,Annales de l’Institut Henri Poincar´ e Analyse Non Lin´ eaire 2 (1985), 101–118

1985