I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:

$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}(f(x,t))$, where $\mathbf{N}$ is a nonlinear second order operator.

Now, I have been able to establish that $f_0(x)$ is exponentially **linearly** stable for all initial perturbations that have frequency greater than some constant $c$. This has been shown by converting the problem into Fourier domain, in which **the **linearized** PDE (around $f_0(x)$) **decouples into (infinite) system of ODEs, one for each frequency**.**

**However, the decoupling doesn't hold for the nonlinear PDE**. So an initial perturbation which is linearly stable may excite modes of frequency lower than $c$ due to coupling in the nonlinear equation. Hence, we cannot naively claim that
"linear stability => nonlinear stability" in this case.

So I am looking for examples where such a situation has been studied. Probably in fluid mechanics or other physical phenomenon ? I am hoping to either prove or disprove the notion of nonlinear stability for the system I am dealing with.

"On Landau damping", Acta Mathematica, 2011, V. 207, Issue 1, pp 29–201link.springer.com/article/10.1007/s11511-011-0068-9, or the same article in arXiv. $\endgroup$1more comment