Joao Goedert

On the generalized Hamiltonian structure of 3D dynamical systems

Abstract 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 motion are available the problem reduces to a quadrature and the structure functions include an arbitrary function of them.

Generalized Hamiltonian structures for systems in three dimensions with a rescalable constant of motion

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.

Invariants for models of interacting populations

Abstract The generalised Lotka-Volterra system is studied. We use a modification of the Carleman embedding method. The position of the equilibrium point, the possibility of obtaining invariants, the asymptotic cyclic motions and the connection to the Volterra model are discussed.

Numerical simulation of a negative ion plasma expansion into vacuum

The expansion into vacuum of a one-dimensional, collisionless, negative ion plasma is investigated in the framework of the Vlasov–Poisson model. The basic equations are written in a “new time space” by use of a rescaling transformation and, subsequently, solved numerically through a fully Eulerian code. As in the case of a two species plasma, the time-asymptotic regime is found to be self-similar with the temperature decreasing as t−2. The numerical results exhibit clearly the physically expected effects produced by the variation of parameters such as initial temperatures, mass ratios and charge of the negative ions.

On the Hamiltonian structure of Ermakov systems

A canonical Hamiltonian formalism is derived for a class of Ermakov systems specified by several different frequency functions. This class of systems comprises all known cases of Hamiltonian Ermakov systems and can always be reduced to quadratures. The Hamiltonian structure is explored to find exact solutions for the Calogero system and for a non-central potential with dynamic symmetry. Some generalizations of these systems possessing exact solutions are also identified and solved.

On the Lie symmetries of a class of generalized Ermakov systems

Abstract 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 a Hamiltonian Ermakov system and exactly solved in closed form.

Second constant of motion for generalized Ermakov systems

Abstract We derive a second invariant for generalized Ermakov-Ray-Reid systems. This invariant and the one previously known form a non-linear system of equations that can be solved for the velocities. The results are general solutions to one of the equations. This exhibits openly the nonlinear superposition law.

Quantum Zakharov Equations

In this paper we investigate the existence of soliton soluti ons for a modified form of the Zakharov equations describing modulational instabilities i n quantum plasmas. In particular, we show that quantum effects suppress the easily identifiable s oliton solutions, obtained in the adiabatic limit. In this limit, the quantum Zakharov equations become a coupled fourth order system, not amenable to straightforward integration as it was the case for the integrable nonlinear Schrodinger equation. By considering the simultaneous adiabatic and semiclassical limits, we obtain more detailed results through a variational solutio n. Specifically these results show that quantum effects enhance the dispersion and smear out the classical one soliton solution.