8 research outputs found
Phase Space Derivation of a Variational Principle for One Dimensional Hamiltonian Systems
We consider the bifurcation problem u'' + \lambda u = N(u) with two point
boundary conditions where N(u) is a general nonlinear term which may also
depend on the eigenvalue \lambda. A new derivation of a variational principle
for the lowest eigenvalue \lambda is given. This derivation makes use only of
simple algebraic inequalities and leads directly to a more explicit expression
for the eigenvalue than what had been given previously.Comment: 2 pages, Revtex, no figure
On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation
We consider the problem of the speed selection mechanism for the one
dimensional nonlinear diffusion equation . It has been
rigorously shown by Aronson and Weinberger that for a wide class of functions
, sufficiently localized initial conditions evolve in time into a monotonic
front which propagates with speed such that . The lower value is that predicted
by the linear marginal stability speed selection mechanism. We derive a new
lower bound on the the speed of the selected front, this bound depends on
and thus enables us to assess the extent to which the linear marginal selection
mechanism is valid.Comment: 9 pages, REVTE
Propagation and Structure of Planar Streamer Fronts
Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.
In the present long paper, you find the physics of the model and the
interfacial approach further explained. As a first ingredient of this approach,
we here analyze planar fronts, their profile and velocity. We encounter a
selection problem, recall some knowledge about such problems and apply it to
planar streamer fronts. We make analytical predictions on the selected front
profile and velocity and confirm them numerically.
(abbreviated abstract)Comment: 23 pages, revtex, 14 ps file