Nonlinear orbital stability conjecture for modulated traveling waves

Let $(\tilde{\mathbf{v}},\bar c,T)$ be a modulated traveling wave solution of the modulated traveling-wave equation, with $\tilde{\mathbf{v}}\in H^2_{per}([0,T],\mathrm{H}^1)$ and $\bar c>0$. Let $\mathbf S$ be its monodromy operator. Assume that, apart from the eigenvalue $1$, the eigenvalues of $\mathbf S$ lie in a compact subset of the open unit disk, and that $1$ has geometric and algebraic multiplicities two.

Nonlinear stability conjecture. Under these assumptions, the modulated traveling wave is exponentially and asymptotically orbitally stable.

The preceding proposition establishes the corresponding linear stability criterion through the spectrum of the monodromy operator. The conjecture asserts that this spectral condition also yields nonlinear stability; the source notes that proving this is beyond its scope.