Modulational instability of an ion wave packet in a cylindrical plasma-filled waveguide

The method of multiple scales is used to derive a nonlinear Schrödinger equation describing the nonlinear evolution of an ion wave packet propagating along a cylindrical plasma-filled waveguide. Numerical evaluation of nonlinear and dispersive terms shows that the wave is modulationally unstable if the wavenumber exceeds a certain critical value. On comparing with the case of an unbounded plasma, it is shown that finite geometry causes a significant shift of this critical value towards smaller wavenumbers, where Landau damping is relatively small.