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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 stochastic quantization method recently developed by Haba and Kleinert is extended to non-autonomous mechanical systems, in the case of the time-dependent harmonic oscillator. In comparison with the autonomous case, the quantization procedure involves the solution of a nonlinear, auxiliary equation. Using a rescaling transformation, the Schrodinger equation for the time-dependent harmonic oscillator is obtained after averaging of a classical stochastic differential equation.

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.

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

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.

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 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 didactic and systematic derivation of Noether point symmetries and conserved currents is put forward in special relativistic field theories, without a priori assumptions about the transformation laws. Given the Lagrangian density, the invariance condition develops as a set of partial differential equations determining the symmetry transformation. The solution is provided in the case of real scalar, complex scalar, free electromagnetic, and charged electromagnetic fields. Besides the usual conservation laws, a less popular symmetry is analyzed: the symmetry associated with the linear superposition of solutions, whenever applicable. The role of gauge invariance is emphasized. The case of the charged scalar particle under external electromagnetic fields is considered, and the accompanying Noether point symmetries determined. Noether point symmetries for a dynamical system in extended gravity cosmology are also deduced.

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

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

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 generalized Hamiltonian structures of several three-dimensional dynamical systems of interest in physical applications are considered. In general, Hamiltonians exist only for systems that possess at least one time-independent constant of motion. Systems with only time-dependent constants of motion may sometimes be rescaled and their constant of motion made time-independent. When this is possible, the transformed system may be cast in a generalized Hamiltonian formalism with non-canonical structure functions.

The results from the last chapter are extended to the three-dimensional situation. First, the three-dimensional quantum Zakharov system is derived from a two-time scales analysis. A Lagrangian formalism and the associated conservation laws are written down. Restricting to the adiabatic and semiclassical case, the system reduces to a quantum vector nonlinear Schrodinger equation (QVNLS) for the envelope electric field. A Lagrangian formalism for this QVNLS equation is used to investigate the behavior of Gaussian shaped solutions (Langmuir wave packets), by means of a time-dependent variational method. Quantum corrections are shown to prevent the collapse of Langmuir wave packets, in both two and three spatial dimensions. The conservation laws of the QVNLS equation are discussed. Finally, we discuss the oscillations of the width of the Langmuir wave packets, as a result from the interplay between classical refraction and quantum diffraction.

A set of quantum hydrodynamic equations are derived from the moments of the electrostatic mean-field Wigner kinetic equation. No assumptions are made on the particular local equilibrium or on the statistical ensemble wave functions. Quantum diffraction effects appear explicitly only in the transport equation for the heat flux triad, which is the third-order moment of the Wigner pseudo-distribution. The general linear dispersion relation is derived, from which a quantum modified Bohm-Gross relation is recovered in the long wave-length limit. Nonlinear, traveling wave solutions are numerically found in the one-dimensional case. The results shed light on the relation between quantum kinetic theory, the Bohm-de Broglie-Madelung eikonal approach, and quantum fluid transport around given equilibrium distribution functions., 5 pages, three figures, uses elsarticle.cls

Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner-Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein-Greene-Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner-Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed., 7 figures

The Lie point symmetry analysis of Ermakov systems is extended to the case where the frequency function in the equations of motion depends also on the dynamical variables and their derivatives. From this new standpoint, a quite general class of two dimensional oscillators with nonlinear coupling can be viewed as a Hamiltonian Ermakov system possessing a Lie point symmetry and is shown to be completely integrable.

The quantum Zakharov system is described in terms of a Lagrangian formalism. A time-dependent Gaussian trial function approach for the envelope electric field and the low-frequency part of the density fluctuation leads to a coupled, nonlinear system of ordinary differential equations. In the semiclassic case, linear stability analysis of this dynamical system shows a destabilizing r\^ole played by quantum effects. Arbitrary value of the quantum effects are also considered, yielding the ultimate destruction of the localized, Gaussian trial solution. Numerical simulations are shown both for the semiclassic and the full quantum cases., Comment: 6 figures

The Poisson structures for 3D systems possessing one constant of motion can always be constructed from the solution of a linear PDE. When two constants of the motion are available the problem reduces to a quadrature and the structure functions include an arbitrary function of them.

Ermakov systems possessing Noether point symmetry are identified among the Ermakov systems that derive from a Lagrangian formalism and, the Ermakov invariant is shown to result from an associated symmetry of dynamical character. The Ermakov invariant and the associated Noether invariant, are sufficient to reduce these systems to quadratures.

A Gaussian ansatz, for the wave function of two-dimensional harmonically trapped anisotropic Bose-Einstein condensates is shown to lead, via a variational procedure, to a coupled system of two second-order, nonlinear ordinary differential equations. This dynamical system is shown to be in the general class of Ermakov systems. Complete integrability of the resulting Ermakov system is proven. Using the exact solution, collapse of the condensate is analyzed in detail. Time dependence of the trapping potential is allowed.

We show that the basic results on the paper referred in the title [J. Phys. A: Math. Gen. v. 35 (2002) 5333-5345], concerning the derivation of the Ermakov invariant from Noether symmetry methods, are not new.

It is shown that, for a large class of statistical mixtures, the Wigner-Poisson (or Hartree) system can be reduced to an effective Schroedinger-Poisson system, in which the Schroedinger equation contains a new nonlinearity. For the case of a zero-temperature one-dimensional electron gas, this additional nonlinearity is of the form vertical bar {Psi} vertical bar{sup 4}. In the long-wavelength limit, the results obtained from the effective Schroedinger-Poisson system are in agreement with those of the Wigner-Poisson system. The reduced model is further used to describe the stationary states of a quantum electron gas and the two-stream instability.

The occurrence of rogue waves (freak waves) associated with electrostatic wavepacket propagation in a quantum electron–positron–ion plasma is investigated from first principles. Electrons and positrons follow a Fermi–Dirac distribution, while the ions are subject to a quantum (Fermi) pressure. A fluid model is proposed and analyzed via a multiscale technique. The evolution of the wave envelope is shown to be described by a nonlinear Schrodinger equation (NLSE). Criteria for modulational instability are obtained in terms of the intrinsic plasma parameters. Analytical solutions of the NLSE in the form of envelope solitons (of the bright or dark type) and localized breathers are reviewed. The characteristics of exact solutions in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather are proposed as candidate functions for rogue waves (freak waves) within the model. The characteristics of the latter and their dependence on relevant parameters (positron concentration and temperature) are investigated.

By means of the Nyquist method, we investigate the linear stability of electrostatic waves in homogeneous equilibria of quantum plasmas described by the Wigner-Poisson system. We show that, unlike the classical Vlasov-Poisson system, the Wigner-Poisson case does not necessarily possess a Penrose functional determining its linear stability properties. The Nyquist method is then applied to a two-stream distribution, for which we obtain an exact, necessary and sufficient condition for linear stability, as well as to a bump-in-tail equilibrium., Comment: 6 figures

We show that Theorems 8, 9, and 11 in the work cited in the title are incorrect in general. The existence of globally well-defined first integrals or flow invariant functions for dynamical systems in R n cannot be taken for granted.

The temporal dynamics of the quantum Zakharov equations (QZEs) in one spatial dimension, which describes the nonlinear interaction of quantum Langmuir waves (QLWs) and quantum ion-acoustic waves (QIAWs) is revisited by considering their solution as a superposition of three interacting wave modes in Fourier space. Previous results in the literature are modified and rectified. Periodic, chaotic as well as hyperchaotic behaviors of the Fourier-mode amplitudes are identified by the analysis of Lyapunov exponent spectra and the power spectrum. The periodic route to chaos is explained through an one-parameter bifurcation analysis. The system is shown to be destabilized via a supercritical Hopf-bifurcation. The adiabatic limits of the fully spatio-temporal and reduced systems are compared from the viewpoint of integrability properties., 7 pages, 8 figures. Physics of Plasmas, in press (2010)

By means of a Madelung decomposition, exact periodic traveling solutions are constructed for a modified nonlinear Schrodinger equation derived by Stenflo and Gradov, describing electrostatic surface waves in semi-infinite plasma. The condition for the existence of bi-stable equilibria is discussed. A conservation law as well as the modulational instability admitted by the model are analyzed., Comment: 8 figures