M. R. Feix

Expansion of a quantum electron gas

The expansion of a quantum electron gas (non-relativistic, no spin) is investigated via the one-particle Schrodinger–Poisson model. Classically, the nonlinear term enhances the formation of a very regular asymptotic state. By means of rescaling methods, we conjecture that the quantum asymptotic solution is identical to the classical one. Subsequent numerical simulations confirm the above conjecture and define precisely the way in which the classical limit is approached.

Families of invariants of the motion for the Lotka–Volterra equations: The linear polynomials family

The modified Carleman embedding method already introduced by the authors to find first integrals (invariants of the motion) of polynomial form to the Lotka–Volterra system is described in detail, and its efficiency to treat the N‐dimensional system proved. Using this method, an extensive investigation is performed for polynomials of the first degree, which allow a classification of the integrals in three families. For some systems possessing one invariant it is possible to find a second invariant using rescaling methods. They represent very restrictive solutions, implying that there exists a great number of conditions among the equation’s coefficients to satisfy. A proof is given that the Volterra invariants can be deduced as a limit. Finally, the interesting properties of the solutions of these systems are studied in detail.

Analysis and solution of a nonlinear second‐order differential equation through rescaling and through a dynamical point of view

The solutions of the equation y+yy+βy3=0, where β is a free parameter, are investigated. For β= (1)/(9) the equation is linearizable through an eight‐parameter symmetry group and is completely integrable. For β≠ (1)/(9) only two symmetries subsist, but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of β and according to the initial conditions.

Rescaling methods and plasma expansions into vacuum

The problem of a two‐component, collisionless plasma expansion into vacuum is investigated from the viewpoint of the Vlasov–Poisson model. The set of equations is treated both analytically (through the rescaling transformations) and numerically, using a one‐dimensional Eulerian code. In planar geometry, the rescaling allows to conjecture the existence of a self‐similar expansion over long times. Numerical results subsequently confirm the conjecture and show that the plasma becomes neutral over a smaller and smaller scale. A few thermodynamical properties are studied: the temperature is shown to decrease as t−2; the polytropic relation (d/dt)(pn−γ)=0 (with γ=3) is verified asymptotically via a semianalytical argument. Finally, the same problem is studied in a spherical one‐dimensional geometry. The time‐asymptotic solution is again self‐similar. Numerical simulations show that a non‐neutral, multiple‐layer structure appears, which is proved to be stable over long times.

Exponential nonlocal symmetries and nonnormal reduction of order

The conventional approach to double reduction of the order of an ordinary differential equation using Lie symmetries is via the normal subgroups of point symmetries. We show that, provided that one is prepared to use nonlocal symmetries, initial reduction by the nonnormal subgroup does not prevent the double reduction. We further illustrate our results with the general third-order equations invariant under the nonsolvable algebras, sl(2, R) (of which the Chazy equation is a noted example) and so(3).

Properties of second‐order ordinary differential equations invariant under time translation and self‐similar transformation

The general form of a second‐order ordinary differential equation invariant under time translation and self‐similarity is obtained together with its solution in parametric form. Next two representatives of two different families of such equations are studied. In terms of rescaled phase space variables, the first example has a simple physical interpretation and its limit cycle properties and period are easily derived. Similar results are found for the second example, but the physical interpretation is less obvious. The invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self‐similar solution (SSS) exists.

Nonlinear time‐dependent anharmonic oscillator: Asymptotic behavior and connected invariants

The motion of a particle in a potential decreasing with time as ‖X‖n is considered. Different time and space rescaling are considered in order to obtain the asymptotic solutions. The validity of adiabatic invariants is discussed. The classical critical case corresponds to the obtainment of self‐similar solutions for the quantum problem.

Time-independent invariants of motion for the quadratic system

A Hamiltonian method is developed to obtain first integrals of the form P Qmu and P Qmu Rnu for a general system of two-dimensional autonomous ordinary differential equations with quadratic terms, where P, Q and R are linear or quadratic polynomials, and mu , nu are two real numbers. It is found that there is an intimate relationship between the polynomials and the equilibrium points of the system, which is useful for determining the existence of periodic orbits and asymptotic behaviour.

Hamiltonian method and invariant search for 2D quadratic systems

The Hamiltonian formalism for a two-dimensional system of ODE possessing an invariant of the motion not containing the time explicitly has suggested a method for the search of first integrals. Applied to the quadratic system it leads to the finding of two phase space configurations.

Study of the diffusion across a magnetic field in a beam-plasma interaction using a drift-kinetic Vlasov code

A drift‐kinetic Eulerian Vlasov code, with fluid equations for the ions, is developed to study the problem of the injection of an electron beam into a two‐dimensional magnetized plasma, often referred to as direct current (dc) helicity injection. The diffusion of electrons across a magnetic field in the presence of a beam–plasma instability is studied. The case of a magnetic field tilted with respect to the beam direction is considered. The competition between the velocity shear Kelvin–Helmholtz (KH) and the beam–plasma (BP) instabilities is investigated in order to analyze the plasma heating and current drive mechanism induced by the beam injection. The KH instability generates low‐frequency plasma convection motion associated with cE×B/B2 drift. In particular, the diffusion coefficients Dy and Dv ∥ describing, respectively, the anomalous diffusion process induced in space across the magnetic field by the KH instability, and the velocity diffusion process due to the kinetic effects induced in velocity sp...