What happens to Langmuir waves at a density gradient in plasmas? I'm trying to understand a little bit of what happens to electrostatic Langmuir waves when there is an initial density gradient present. The plasma frequency depends on the density, so you would expect the frequency to change as they propagate into higher density plasma, but what does this look like?
I tried to derive the dispersion relation by linearising the electron momentum and continuity equations and Gauss' law, but very quickly ran into something that could not be converted into a dispersion relation. Clearly, this is not a linear problem and linearising wasn't likely to be successful.
If you had a very steep density gradient, where on one side the waves have a very short $\omega_p^{-1}$ and very long on the other side, do the waves propagate in some assymmetric way?
 A: 
The plasma frequency depends on the density, so you would expect the frequency to change as they propagate into higher density plasma, but what does this look like?

Langmuir waves tend to reflect off of density gradients if the gradient is sufficiently steep.  This is why some radio waves bounce off the ionosphere.
There's a nice paper by Krafft et al. [2014] that shows some simulated examples of a time series waveform.  If the density gradient is small, you tend to get amplitude modulation and sometimes even mode conversion through a nonlinear wave-wave interaction.  This nonlinear wave-wave interaction comes in a few forms but the two top-level descriptors are Langmuir wave decay [e.g., Cairns, 1989; Malaspina et al., 2013] and mode conversion to what's called a z-mode wave [e.g., Bale et al., 1998] -- the name derives from the shape of its dispersion relation on an $\omega$ vs $k$ plot.  In the large $k$ limit, the z-mode is a longitudinal wave and nearly identical to a Langmuir wave when $\tfrac{ \Omega_{ce} }{ \omega_{pe} } \ll 1$ (i.e., close magnetized bodies like Earth), where $\Omega_{ce}$ is the cyclotron frequency and $\omega_{pe}$ is the plasma frequency.
Nonlinear wave decay usually involves a Langmuir wave decaying into two daughter waves [e.g., Cairns, 1989].  Some examples include a Langmuir wave going to: another Langmuir wave at slightly lower frequency and an ion acoustic wave or an electromagnetic free mode emission plus an ion acoustic wave.

I tried to derive the dispersion relation by linearising the electron momentum and continuity equations and Gauss' law, but very quickly ran into something that could not be converted into a dispersion relation. Clearly, this is not a linear problem and linearising wasn't likely to be successful.

See the references below and references therein for some example derivations of these processes.  If you are specifically curious about a Langmuir wave running into a density gradient, look at the Krafft et al. [2014] paper for a proper dispersion relation.

If you had a very steep density gradient, where on one side the waves have a very short $\omega_{pe}^{-1}$ and very long on the other side, do the waves propagate in some assymmetric way?

Yes, the incident wave will most likely reflect off of the boundary.
References

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*Bale, S.D., et al., "Transverse z‐mode waves in the terrestrial electron foreshock," Geophys. Res. Lett. 25(1), pp. 9-12, doi:10.1029/97gl03493, 1998.

*Cairns, I.H., "Electrostatic wave generation above and below the plasma frequency by electron beams," Phys. Fluids B: Plasma Phys. 1(1), pp. 204-213, doi:10.1063/1.859088, 1989.

*Krafft, C., et al., "Waveforms of Langmuir turbulence in inhomogeneous solar wind plasmas," J. Geophys. Res. 119, pp. 9369-9382, doi:10.1002/2014JA020329, 2014.

*Malaspina, D.M., et al., "Langmuir wave harmonics due to driven nonlinear currents," J. Geophys. Res. 118, pp. 6880-6888, doi:10.1002/2013JA019309, 2013.

