Finite amplitude electron plasma waves in a cylindrical waveguide

The nonlinear behaviour of the electron plasma wave propagating in a cylindrical plasma waveguide immersed in an infinite axial magnetic field is investigated using the Krylov-Bogoliubov-Mitropolsky perturbation method, by means of which is deduced the nonlinear Schrodinger equation governing the long-time slow modulation of the wave amplitude. From this equation the amplitude-dependent frequency and wavenumber shifts are calculated, and it is found that the electron waves with short wavelengths are modulationally unstable with respect to long-wavelength, low-frequency perturbations. It is further shown that an oscillatory solution of the Korteweg-de Vries equation, which is derived in the small wavenumber region, satisfies the small wavenumber limit of the nonlinear Schrodinger equation.