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A canonical Hamiltonian is found for a reduced version of the Jackiw-Pi model for bilayer graphene. From the corresponding Lagrangian, the Noether point symmetries and conserved quantities are determined. The Noether symmetry group is the same as the fifth-dimensional group for the time-dependent harmonic oscillator. The realization of the algebra is achieved in terms of just one particular solution $g_1$ of the time-dependent harmonic oscillator equation underlying the reduced Jackiw-Pi model. Some numerical solutions are worked out.

The pseudopotential method is used to derive electron hole structures in a suprathermal plasma with a regularized κ probability distribution function background. The regularized character allows the exploration of small κ values beyond the standard suprathermal case for which κ>3/2 is a necessary condition. We found the nonlinear dispersion relation yielding the amplitude of the electrostatic potential in terms of the remaining parameters, in particular the drift velocity, the wavenumber and the spectral index. Periodic, solitary wave, drifting and non-drifting solutions have been identified. In the linear limit, the dispersion relation yields generalized Langmuir and electron acoustic plasma modes. Standard electron hole structures are regained in the κ≫1 limit.

We extend de concept of Compton scattering to the case of plasmons. This concept was originally applied to electrons in vacuum. Here, we consider electrons in a plasma, and study the scattering properties of photon-plasmon interactions. We show that a number n of plasmons with frequency ω ≃ ω p is scattered by an electron, for an incident photon with frequency ω ′ ≥ ω p , where ω p is the plasma frequency. We describe the general case of arbitrary n and assume that Compton scattering of plasmons is intrinsically a nonlinear process. Our theoretical model is based on Volkov solutions of the Klein–Gordon equation describing the state of relativistic electrons, when the spin is ignored. We derive the corresponding scattering probability, as well as the recoil formula associated with arbitrary final electron states. This process can be relevant to intense laser-plasma interactions.

The purpose of this review is to revisit recent results in the literature where quantum plasmas with arbitrary degeneracy degree are considered. This is different from a frequent approach, where completely degeneracy is assumed in dense plasmas. The general reasoning in the reviewed works is to take a numerical coefficient in from of the Bohm potential term in quantum fluids, in order to fit the linear waves from quantum kinetic theory in the long wavelength limit. Moreover, the equation of state for the ideal Fermi gas is assumed, for arbitrary degeneracy degree. The quantum fluid equations allow the expedite derivation of weakly nonlinear equations from reductive perturbation theory. In this way, quantum Korteweg - de Vries and quantum Zakharov - Kuznetsov equations are derived, together with the conditions for bright and dark soliton propagation. Quantum ion-acoustic waves in unmagnetized and magnetized plasmas, together with magnetosonic waves, have been obtained for arbitrary degeneracy degree. The conditions for the application of the models, and the physical situations where the mixed dense - dilute systems exist, have been identified.

We briefly review some recent advances in the field of nonlinear dynamics of atomic clouds in magneto-optical traps. A hydrodynamical model in a three-dimensional geometry is applied and analyzed using a variational approach. A Lagrangian density is proposed in the case where thermal and multiple scattering effects are both relevant, where the confinement damping and harmonic potential are both included. For generality, a general polytropic equation of state is assumed. After adopting a Gaussian profile for the fluid density and appropriate spatial dependencies of the scalar potential and potential fluid velocity field, a set of ordinary differential equations is derived. These equations are applied to compare cylindrical and spherical geometry approximations. The results are restricted to potential flows.

We derive a generalized Hamiltonian formalism for a modified susceptible–infectious–recovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu system, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations., Journal of Physics A: Mathematical and Theoretical, 55 (22), ISSN:1751-8113, ISSN:1361-6447

An antiproton plasma confined in a quasi-1D device is described in terms of a self-consistent fluid formulation using a variational approach. Unlike previous treatments, the use of the time-dependent variational method allows to retain the thermal and Coulomb effects. A certain Ansatz is proposed for the number density and fluid velocity fields, which reduces the problem essentially to ordinary nonlinear differential equations. In adiabatic cooling, the frequency of the trap potential is slowly decreased. An adiabatic equation of state is assumed for closure. The numerical simulation of the nonlinear dynamics is performed, for realistic parameters.

The pseudo-potential method is applied to derive diverse propagating electron hole structures, in a nonthermal or $\kappa$ particle distribution function background. The associated distribution function Ansatz reproduces the Schamel distribution of \cite{Schamel2015} in the Maxwellian ($\kappa \rightarrow \infty$) limit, providing a significant generalization of it for plasmas where superthermal electrons are ubiquitous, such as space plasmas. The pseudo-potential and the nonlinear dispersion relation are evaluated. The role of the spectral index $\kappa$ on the nonlinear dispersion relation is investigated, in what concerns the wave amplitude for instance. The energy-like first integral from Poisson's equation is applied to analyze the properties of diverse classes of solutions: with the absence of trapped electrons, with a non-analytic distribution of trapped electrons, or with a surplus of trapped electrons. Special attention is therefore paid to the non-orthodox case where the electrons distribution function exhibits strong singularities, being discontinuous or non-analytic.

The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables, similarly as in the solution of the Vlasov-Poisson system by means of the Bernstein-Greene-Kruskal method. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed and shown to be immediately integrable up to a recursive chain of quadratures in position space only. As it stands, the treatment of the self-consistent, Wigner-Poisson system is beyond the scope of the method, which assumes a given smooth time-independent external potential. Accuracy tests for the series expansion are also provided. Examples of anharmonic potentials are worked out up to a high order on the quantum diffraction parameter.

We set up a proposal to extend significantly recent works on neutrino-plasma interaction, allowing the possibility of deep degenerate and relativistic electrons, which are often present in compact stars such as high-density white dwarfs. The methodology involves the covariant hydrodynamic formulation of ultradense plasmas. We propose the generalization of previous studies, on the interaction between ion-acoustic waves and resonant neutrino flavor oscillations in a mixed neutrino beam, admitting degenerate and relativistic electron populations. Destabilization of the ion acoustic wave has higher growth rates, thanks to the very high densities present in plasmas under these extreme conditions. We take into account applications to white dwarf stars in the process of collapsing, producing type II supernovas.

Using a two-species quantum hydrodynamic model, we derive the quantum counterpart of magnetosonic waves, in a plasma with arbitrary degree of degeneracy and taking into account quantum diffraction effects due to the matter-wave character of the charge carriers. The weakly nonlinear aspects of the associated quantum magnetosonic wave are accessed by means of perturbation theory, with the derivation of a nonlinear evolution equation admitting solitons, namely, the Korteweg-de Vries equation. The degeneracy and quantum diffraction effects on soliton propagation are determined. A qualitative change on weakly nonlinear magnetosonic waves appears when quantum diffraction matches certain conditions, producing shock solutions instead of solitons, within the approximation level. We also include explicit numeric estimates and a discussion on the coupling (nonideality) parameter for quantum plasmas with intermediate degeneracy degree.

We analyze the role of collisional effects on the coupling between ion-acoustic waves and neutrino flavor oscillations, discussing its relevance for plasma instabilities in extreme plasma environments like in type II supernovae, where intense neutrino bursts exist. Electrons (leptons) are coupled to the electron-neutrino fluid through the weak Fermi force, but the electron-neutrinos are allowed to convert to other neutrino flavors and vice versa. Due to the typically slow frequency of neutrino flavor oscillations, many orders smaller than, e.g., the plasma frequency, an effective energy transfer between plasma waves and neutrino flavor oscillations takes place at the low-frequency electrostatic branch, viz., the ion-acoustic mode. We show the destabilization of ion-acoustic waves in dense astrophysical scenarios, with a focus on the collisional effects mediated by electron-ion scattering. The maximal instability growth-rate is evaluated and compared to the characteristic inverse times of type II supernova explosions. The results can be used for independent experimental verification of the non-zero neutrino mass, in a plasma physics context. Published by AIP Publishing.

In a previous work [Haas et al., Phys. Plasmas 23, 012104 (2016)], a new model was introduced, taking into account the role of the Fermi weak force due to neutrinos coupled to magnetohydrodynamic plasmas. The resulting neutrino-magnetohydrodynamics was investigated in a particular geometry associated with the magnetosonic wave, where the ambient magnetic field and the wavevector are perpendicular. The corresponding fast, short wavelength neutrino beam instability was then obtained in the context of supernova parameters. The present communication generalizes these results, allowing for arbitrary direction of wave propagation, including fast and slow magnetohydrodynamic waves and the intermediate cases of oblique angles. The numerical estimates of the neutrino-plasma instabilities are derived in extreme astrophysical environments where dense neutrino beams exist. Published by AIP Publishing.

Understanding the transport properties of charged particle beams is important not only from a fundamental point of view but also due to its relevance in a variety of applications. A theoretical model is established in this article, to model the interaction of a tenuous positively charged ion beam with an ultradense quantum electron-ion plasma, by employing a rigorous relativistic quantum-hydrodynamic (fluid plasma) electrostatic model proposed in McKerr et al. [M. McKerr, F. Haas, and I. Kourakis, Phys. Rev. E 90, 033112 (2014)]. A nonlinear analysis is carried out to elucidate the propagation characteristics and the existence conditions of large amplitude electrostatic solitary waves propagating in the plasma in the presence of the beam. Anticipating stationary profile excitations, a pseudomechanical energy balance formalism is adopted to reduce the fluid evolution equation to an ordinary differential equation. Exact solutions are thus obtained numerically, predicting localized excitations (pulses) for all of the plasma state variables, in response to an electrostatic potential disturbance. An ambipolar electric field form is also obtained. Thorough analysis of the reality conditions for all variables is undertaken in order to determine the range of allowed values for the solitonic pulse speed and how it varies as a function of the beam characteristics (beam velocity and density).

A relativistic quantum-hydrodynamic plasma model is proposed, to model the propagation ofelectrostatic waves in an ultradense quantum electron-ion plasma in the presence of an ion beam. Adispersion relation is derived for harmonic waves, and the stability of electrostatic wavepackets isinvestigated. Three types of waves are shown to exist, representing a modified electron plasma(Langmuir-type) mode, a low-frequency ion acoustic mode, and an ion-beam driven mode, respectively.Stability analysis reveals the occurrence of an imaginary frequency part in three regions.The dependence of the instability growth rate on the ion beam parameters (concentration andspeed) has been investigated.

The work investigates the coupling between ion-acoustic waves and neutrino flavor oscillations in a non-relativistic electron-ion plasma under the influence of a mixed neutrino beam. Neutrino oscillations are mediated by the flavor polarization vector dynamics in a material medium. The linear dispersion relation around homogeneous static equilibria is developed. When resonant with the ion-acoustic mode, the neutrino flavor oscillations can transfer energy to the plasma exciting a new fast unstable mode in extreme astrophysical scenarios. The growth rate and the unstable wavelengths are determined in typical type II supernovae parameters. The predictions can be useful for a new indirect probe on neutrino oscillations in nature.

Using a variation of the celebrated Volkov solution, the Klein-Gordon equation for a charged particle is reduced to a set of ordinary differential equations, exactly solvable in specific cases. The new quantum relativistic structures can reveal a localization in the radial direction perpendicular to the wave packet propagation, thanks to a non-vanishing scalar potential. The external electromagnetic field, the particle current density, and the charge density are determined. The stability analysis of the solutions is performed by means of numerical simulations. The results are useful for the description of a charged quantum test particle in the relativistic regime, provided spin effects are not decisive.

The destabilizing role of neutrino beams on the Trivelpiece-Gould modes is considered, assuming electrostatic perturbations in a magnetized plasma composed by electrons in a neutralizing ionic background, coupled to a neutrino species by means of an effective neutrino force arising from the electroweak interaction. The magnetic field is found to significantly improve the linear instability growth rate, as calculated for supernova type II environments. On the formal level, for wave vectors parallel or perpendicular to the magnetic field, the instability growth rate is found from the unmagnetized case replacing the plasma frequency by the appropriated Trivelpiece-Gould frequency. The growth rate associated with oblique propagation is also obtained.

Nonlinear ion-acoustic waves are analyzed in a nonrelativistic magnetized quantum plasma with arbitrary degeneracy of electrons. Quantum statistics is taken into account by means of the equation of state for ideal fermions at arbitrary temperature. Quantum diffraction is described by a modified Bohm potential consistent with finite-temperature quantum kinetic theory in the long-wavelength limit. The dispersion relation of the obliquely propagating electrostatic waves in magnetized quantum plasma with arbitrary degeneracy of electrons is obtained. Using the reductive perturbation method, the corresponding Zakharov-Kuznetsov equation is derived, describing obliquely propagating two-dimensional ion-acoustic solitons in a magnetized quantum plasma with degenerate electrons having an arbitrary electron temperature. It is found that in the dilute plasma case only electrostatic potential hump structures are possible, while in dense quantum plasma, in principle, both hump and dip soliton structures are obtainable, depending on the electron plasma density and its temperature. The results are validated by comparison with the quantum hydrodynamic model including electron inertia and magnetization effects. Suitable physical parameters for observations are identified.

Shielding effects in non-degenerate and degenerate plasmas are compared. A detailed derivation of the Wigner-Poisson system is provided for electrostatic quantum plasmas in which relativistic, spin, and collisional effects are not essential. A detailed derivation of a quantum hydrodynamic model starting from the Wigner-Poisson system is presented. The route for this derivation considers the eikonal decomposition of the one-body wavefunctions of the quantum statistical mixture. The merits and limitations of the resulting quantum hydrodynamic model are discussed.

An effective self-consistent model is derived and used to study the dynamics of an electron gas confined in a nonparabolic and anisotropic quantum well. This approach is based on the equations of quantum hydrodynamics, which incorporate quantum and nonlinear effects in an approximate fashion. The effective model consists of a set of six coupled differential equations (dynamical system) for the electric dipole and the size of the electron gas. Using this model we show that: (i) high harmonic generation is related to the appearance of chaos in the phase space, as attested to by related Poincar´e sections; (ii) higher order harmonics can be excited efficiently and with relatively weak driving fields by making use of chirped electromagnetic waves.

A new transverse mode in a two-stream magnetized quantum plasma is studied by means of a quantum hydrodynamic model, under non-relativistic and ideal Fermi gas assumptions. It is found that Fermi pressure effects induce a minimum cutoff wavelength for instability, unlike the classical case which is unstable for larger wavenumbers. The external magnetic field is also shown to produce a stabilizing effect. Conditions for the applicability of the model and specific parameters for experimental observations are thoroughly discussed.

A collisional trapped non-neutral plasma is described by a hydrodynamical model in onedimensional geometry. For suitable initial conditions and velocity fields, the Lagrangian variables method reduces the pressure dominated problem to a damped autonomous Pinney equation, representing a dissipative nonlinear oscillator with an inverse cubic force. An accurate approximate analytic solution derived from Kuzmak-Luke perturbation theory is applied, allowing the assessment of the fully nonlinear dynamics. On the other hand, in the cold plasma case, the Lagrangian variables approach allows the derivation of exact damped nonlinear oscillations. The conditions for the applicability of the hot, pressure dominated or cold gas assumptions are derived. Published by AIP Publishing.

Linear and nonlinear ion-acoustic waves are studied in a fluid model for nonrelativistic, unmagnetized quantum plasma with electrons with an arbitrary degeneracy degree. The equation of state for electrons follows from a local Fermi-Dirac distribution function and applies equally well both to fully degenerate and classical, nondegenerate limits. Ions are assumed to be cold. Quantum diffraction effects through the Bohm potential are also taken into account. A general coupling parameter valid for dilute and dense plasmas is proposed. The linear dispersion relation of the ion-acoustic waves is obtained and the ion-acoustic speed is discussed for the limiting cases of extremely dense or dilute systems. In the long-wavelength limit, the results agree with quantum kinetic theory. Using the reductive perturbation method, the appropriate Korteweg-de Vries equation for weakly nonlinear solutions is obtained and the corresponding soliton propagation is analyzed. It is found that soliton hump and dip structures are formed depending on the value of the quantum parameter for the degenerate electrons, which affect the phase velocities in the dispersive medium.

We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.

The dispersion properties of electrostatic waves propagating in ultrahigh density plasma are investigated, from first principles, in a one-dimensional geometry. A self-consistent multispecies plasma fluid model is taken as starting point, incorporating electron degeneracy and relativistic effects. The inertia of all plasma components is retained, for rigor. Exact expressions are obtained for the oscillation frequency, and the phase and group velocity of electrostatic waves is computed. Two branches are obtained, viz. an acoustic low-frequency dispersion branch and an upper (optic-like) branch: these may be interpreted as ion-acoustic and electron plasma(Langmuir) waves, respectively, as in classical plasmas, yet bearing an explicit correction in account of relativistic and electron degeneracy effects. The electron plasma frequency is shown to reduce significantly at high values of the density, due to the relativistic effect. The result is compared with approximate models, wherein either electrons are considered in ertialess (low frequency ionic scale) or ions are considered to be stationary (Langmuir-wave limit).

We demonstrate the fact that linearity is a meaningful symmetry in the sense of Lie and Noether. The role played by that `linearity symmetry' in the quadrature of linear ordinary second-order differential equations is reviewed, by the use of canonical coordinates and the identification of a Wronskian-like conserved quantity as Lie invariant. The Jacobi last multiplier associated with two independent linearity symmetries is applied to derive the Caldirola-Kanai Lagrangian from symmetry principles. Then the symmetry is recognized to be also a Noether one. Finally, the study is extended to higher-order linear ordinary differential equations, derivable or not from an action principle., Comment: 16 pages

The collective dynamics of nonlinear electron waves in an one-dimensional degenerate electron gas is treated using the Lagrangian fluid approach. A new class of solutions with a nontrivial space and time dependence is derived. Both analytical and numerical results demonstrate the formation of stable, breather-like modes, provided certain conditions are met. For large amplitudes of the initial density perturbation, a catastrophic collapse of the plasma density is predicted, even in the presence of the quantum statistical pressure and quantum diffraction dispersive effects. The results are useful for the understanding of the properties of general nonlinear structures in dense plasmas.

The nonlinear electron dynamics in metallic nanoparticles is studied using a hydrodynamic model that incorporates most quantum many-body features, including spill-out and nonlocal effects as well as electron exchange and correlations. We show that, by irradiating the nanoparticle with a chirped laser pulse of modest intensity (autoresonance), it is possible to drive the electron dynamics far into the nonlinear regime, leading to enhanced energy absorption and complete ionization of the nanoparticle on a time scale of the order of 100 fs. The accompanying radiated power spectrum is rich in high-order harmonics.

We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assurately restricted to a two-dimensional foliation. In some cases, Frobenius theorem allows the explicit construction of an associated invariant. It is proven the inverse result that, if an invariant is known, then it always can be furnished by Frobenius theorem.

We present a Lie point symmetry analysis for non-relativistic two-dimensional charged-particle motion. The resulting symmetries comprise a quasi-invariance transformation, a time-dependent rotation, a time-dependent spatial translation and a dilation. We also find that the associated electromagnetic fields must satisfy a system of first-order linear partial differential equations. This system is solved exactly, yielding four classes of compatible electromagnetic fields.

We find the Noether point symmetries for non-relativistic two-dimensional charged particle motion. These symmetries are composed of a quasi-invariance transformation, a time-dependent rotation and a time-dependent spatial translation. The associated electromagnetic field satisfy a system of first-order linear partial differential equations. This system is solved exactly, yielding three classes of electromagnetic fields compatible with Noether point symmetries. The corresponding Noether invariants are derived and interpreted.

A linearization procedure is proposed for Ermakov systems with frequency depending on dynamic variables. The procedure applies to a wide class of generalized Ermakov systems which are linearizable in a manner similar to that applicable to usual Ermakov systems. The Kepler--Ermakov systems belong into this category but others, more generic, systems are also included.

The symmetry analysis of Ermakov systems is extended to the generalized case where the frequency depends on the dynamical variables besides time. In this extended framework, a whole class of nonlinearly coupled oscillators are viewed as Hamiltonian Ermakov system and exactly solved in closed form.

The influence of localized water currents on the nonlinear dynamics and stability of large amplitude, statistically distributed gravity waves is investigated theoretically and numerically by means of an evolution equation for a Wigner function governing the spectrum of waves. It is shown that water waves propagating in the opposite direction of a localized current channel can be trapped in the channel, which can lead to the amplification of the wave intensity. Under certain conditions the wave intensity can be further localized due to a self-focusing (Benjamin-Feir) instability. The localized amplification of the wave intensity may increase the probability of extreme events in the form of freak waves, which have been observed in connection with ocean currents.

A self-consistent relativistic two-fluid model is proposed for electron-ion plasma dynamics. A one-dimensional geometry is adopted. Electrons are treated as a relativistically degenerate fluid, governed by an appropriate equation of state. The ion fluid is also allowed to be relativistic, but is cold, nondegenerate, and subject only to an electrostatic potential. Exact stationary-profile solutions are sought, at the ionic scale, via the Sagdeev pseudopotential method. The analysis provides the pulse existence region, in terms of characteristic relativistic parameters, associated with the (ultrahigh) particle density.

It is shown that the two-flavor neutrino oscillation equations admit an exact analytic solution for arbitrarily chosen normalized electron neutrino population, provided the electron plasma density is adjusted in a certain way. The associated formula for the electron plasma density is applied to the cases of exponentially decaying or oscillating electron neutrino populations., 4 figures

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.

Recently, there has been great interest around quantum relativistic models for plasmas. In particular, striking advances have been obtained by means of the Klein–Gordon–Maxwell system, which provides a first-order approach to the relativistic regimes of quantum plasmas. The Klein–Gordon–Maxwell system provides a reliable model as long as the plasma spin dynamics is not a fundamental aspect, to be addressed using more refined (and heavier) models involving the Pauli–Schrödinger or Dirac equations. In this work, a further simplification is considered, tracing back to the early days of relativistic quantum theory. Namely, we revisit the square-root Klein–Gordon–Poisson system, where the positive branch of the relativistic energy–momentum relation is mapped to a quantum wave equation. The associated linear wave propagation is analyzed and compared with the results in the literature. We determine physical parameters where the simultaneous quantum and relativistic effects can be noticeable in weakly coupled electrostatic plasmas.

A self-consistent relativistic two-fluid model is proposed for one-dimensional electron-ion plasma dynamics. A multiple scales perturbation technique is employed, leading to an evolution equation for the wave envelope, in the form of a nonlinear Schr\"odinger type equation (NLSE). The inclusion of relativistic effects is shown to introduce density-dependent factors, not present in the non-relativistic case - in the conditions for modulational instability. The role of relativistic effects on the linear dispersion laws and on envelope soliton solutions of the NLSE is discussed., Comment: Submitted to Physics of Plasmas

Electron-positron clusters are studied using a quantum hydro- dynamic model that includes Coulomb and exchange interactions. A variational Lagrangian method is used to determine their stationary and dynamical properties. The cluster static features are validated against existing Hartree-Fock calculations. In the linear response regime, we investigate both dipole and monopole (breathing) modes. The dipole mode is reminiscent of the surface plasmon mode usually observed in metal clusters. The nonlinear regime is explored by means of numerical simulations. We show that, by exciting the cluster with a chirped laser pulse with slowly varying frequency (autoresonance), it is possible to efficiently separate the electron and positron populations on a timescale of a few tens of femtoseconds.

Using a variational approach based on a Lagrangian formulation and Gaussian trial functions, we derive a simple dynamical system that captures the main features of the time-dependent Schr\"odinger-Newton equations. With little analytical or numerical effort, the model furnishes information on the ground state density and energy eigenvalue, the linear frequencies, as well as the nonlinear long-time behaviour. Our results are in good agreement with those obtained through analytical estimates or numerical simulations of the full Schr\"odinger-Newton equations., Comment: 14 pages, 7 figures

The quantum relativistic Buneman instability is investigated theoretically using a collective Klein-Gordon model for the electrons and a cold fluid model for the ions. The growth rate and unstable wave spectrum is investigated in different parameter regimes corresponding to various degrees of relativistic and quantum effects. The results may be important for streaming instabilities involving ion dynamics in very dense plasmas.

A multistream model for spinless electrons in a relativistic quantum plasma is introduced by means of a suitable fluid-like version of the Klein-Gordon-Maxwell system. The one and two-stream cases are treated in detail. A new linear instability condition for two-stream quantum plasmas is obtained, generalizing the previously known non-relativistic results. In both the one and two-stream cases, steady-state solutions reduce the model to a set of coupled nonlinear ordinary differential equations, which can be numerically solved, yielding a manifold of nonlinear periodic and soliton structures. The validity conditions for the applicability of the model are addressed.

The three-dimensional nonlinear dynamics of an electron gas in a semiconductor quantum well is analyzed in terms of a self-consistent fluid formulation and a variational approach. Assuming a time-dependent localized profile for the fluid density and appropriated spatial dependences of the scalar potential and fluid velocity, a set of ordinary differential equations is derived. In the radially symmetric case, the prominent features of the associated breathing mode are characterized.