Stability of nonlinear waves: Pointwise estimates

Figure 1.  Typical spectra of linear operators that are spectrally stable in a strong sense: $\sup \mathrm{Re} \sigma(\mathcal{L}) < 0$. On the left we see a half line of essential spectrum and an isolated eigenvalue (the cross), and on the right we see a parabolic region of essential spectrum and an isolated eigenvalue.

Figure 2.  Typical spectra of linear operators that are spectrally stable in a weaker sense: $\sup \mathrm{Re} \sigma(\mathcal{L}) = 0$. On the left we see a half line of essential spectrum and an isolated eigenvalue (the cross) on the imaginary axis, and on the right we see a parabolic region of essential spectrum touching the imaginary axis and an embedded eigenvalue (denoted now in red for visual clarity) at the origin.

Figure 3.  Floquet spectrum of a spectrally stable viscous shock near the origin. Note the spectrum is non-unique, as it can be shifted by any integer multiple of $2\pi{\rm{i}}$, and hence the parabolas repeat infinitely many times up and down the imaginary axis. There are two embedded eigenvalues at the origin, due to translations in space and time.

Figure 4.  Left panel: original vertical contour with real part $\mu$ used in 3.4. Right panel: deformed contour used to obtain bounds on the Green's function. The parameters $\epsilon$ and $r$ can be chosen to be small and to optimize the resultant bounds.

Figure 5.  On the left is a diagram of the profile of a source as a function of $x$ for a fixed value of $t$, with the motion of perturbations, relative to the speed of the defect core, indicated by the red arrows and the group velocities of the asymptotic wave trains. The right panel shows the behavior of small phase $\phi$ or wave number $\phi_x$ perturbation of a wave train: to leading order, they are transported with speed given by the group velocity $c_g$ without changing their shape [7].

Figure 6.  On the left is a sketch of the space-time diagram of a perturbed source. The defect core will adjust in response to an imposed perturbation (although this is not depicted), and the emitted wave trains, whose maxima are indicated by the lines that emerge from the defect core, will therefore exhibit phase fronts that travel with the group velocities of the asymptotic wave trains away from the core towards $\pm \infty$. The right panel illustrates the profile of the anticipated phase function $\phi(x,t)$.