@InProceedings{karney81a,
  author =       {Charles F. F. Karney},
  title =        {Temporal Evolution of Nonlinear Lower Hybrid Waves},
  booktitle =    {Heating in Toroidal Plasmas},
  pages =        {455-461},
  year =         1981,
  editor =       {E. Canobbio and H. P. Eubank and G. G. Leotta and
                  A. Malein and E. Sindoni},
  volume =       {I},
  publisher =    {Commission of the European Communities, Brussels,
                  Belgium  (EUR 7424 EN)},
  reprint =      {htp81},
  note =         {Proc. Second Joint Varenna--Grenoble International
                  Symposium, Como, Italy, Sept. 3--12, 1980},
  abstract =     {The nonlinear evolution of a single lower hybrid wave
                  in two dimensions and time is considered.  If the
                  potential is taken to have the form &phi;(<i>x</i>,
                  <i>z</i>, <i>t</i>&nbsp;)
                  exp(&minus;<i>i</i>&omega;<i>t</i>&nbsp;), the
                  equation describing the fields is
                  <i>i</i><i>v</i><sub>&tau;</sub> &minus; &int;
                  <i>v</i><sub>&xi;</sub> <i>d</i>&zeta; +
                  <i>v</i><sub>&zeta;&zeta;</sub> +
                  |<i>v</i>|<sup>2</sup><i>v</i> = 0 where <i>v</i> is
                  proportional to the electric field, &tau; to time, and
                  &xi; and &zeta; measure distances across and along the
                  lower hybrid ray.  The properties of this equation are
                  investigated numerically.  When the amplitude of the
                  injected lower hybrid waves is sufficiently large, the
                  following phenomena are observed: the fields do not
                  reach a steady state even though steady-state boundary
                  conditions are imposed; the density modulations are
                  sufficient to cause an appreciable fraction of the
                  incident power to be reflected; the average wavenumber
                  of the transmitted wave is larger than that of the
                  incident wave.}
}