OK so maybe I didn’t explain myself very clearly… So here’s another way of looking at it:

If your oscillations in density & temperature grow in amplitude, at some point the radial gradient from a region of high to low density will grow beyond the point where the simple linear equations cease to fit, and non-linear interactions become important.

Generally there are only two classes of instability involving gradients: Rayleigh-Taylor; where the driving force is parallel to the gradient, and Kelvin-Helmholtz; where the driving force is perpendicular to the gradient (anything else is just a mix of the two). So although it is not exactly the same scenario as the classical Rayleigh-Taylor description, characteristics such as equation of the growth rate is the same.

If you want the peak density/temperature to be in the range for fusion then the gradient away from that peak must be very steep – too steep to be stable.