Coupled mode theory

In this thesis we have studied the global dynamics which spontaneously emerge in ensembles of interacting disparate nonlinear oscillators. Collective phenomena exhibited in these systems include synchronization, quasiperiodicity, chaos, and death. Exact analytical solutions are presented for two and three coupled oscillators with phase and amplitude variations. A phenomenon analogous to a phase-transition is found as a function of interaction-range in a one-dimensional lattice: for coupling exponents larger than some critical value, alpha c, synchronization is shown to be impossible. Massively parallel computer simulations in conjunction with finite-size scaling were used to extrapolate to the infinite-size limit.

The coupled equations of motion describing electrostatic plasma oscillations
are derived using a collective coordinate approach. The
boundary between the physical and nonphysical regions of phase space
is discussed. The two mode case is studied in detail for both small
and large initial disturbances. For small initial disturbances the
motion was found to be periodic. For large initial disturbances, non-periodic
growing solutions were found that approached one of two
asymptotic limits. For these solutions, the variables became infinitely
large in a finite time.

A coherent nonlinear wave-wave coupling effect which is consistent with
a more complete description of damping effects of a plane wave disturbance
of a finite plasma initially spatially uniform and Maxwellian in velocity
space is considered. A smeared out ion background is assumed and the
coupling between the ions and electrons is neglected. A self consistent
field and collective coordinate approach is used to obtain a dispersion
relation for mode coupling in a plasma. The equations for the amplitudes
and the frequencies are solved numerically both by direct time integration
and by a perturbation method for two and three modes. The perturbation
method solutions are obtained for the n mode equations. The perturbation
equations for two modes are solved analytically. For the two mode case
the resulting coupling shows that the energy oscillates between the modes
and that the periodicity of the amplitudes and the frequencies is associated
with the initial parameters. Energy feeding between the modes is also
observed for three or more modes. However, phase mixing occurs for more
than two modes.

The collective coordinate treatment of mode coupling
in a beam plasma is investigated by computer simulation. Two methods are derived generalized and programmed. The
programming is done for a two mode, one species, stationary
plasma. The first method integrates the second order collective
Lagrangian equations for the mode amplitudes. This
method neglects the plasma temperature and coupling is not
observed. The second method integrates the first order
Hamiltonian equations for the mode amplitudes and their
canonical momentum. This takes account of the plasma temperature
and coupling is observed. A perturbation theory
approach is then suggested which would smooth out the detail
of the mode interactions and shorten the calculation time.