Björn Sandstede

This work was done in collaboration with Arnd Scheel (U Minnesota).

Instabilities of nonlinear waves on unbounded domains can manifest themselves in different ways. Absolute instabilities occur if perturbations grow in time at every fixed point in the unbounded domain. Convective instabilities are characterized by the fact that, even though the overall norm of the perturbation grows in time, perturbations decay locally at every fixed point in the unbounded domain; in other words, the growing perturbation is transported, or convected, towards infinity. This distinction between absolute and convective instabilities depends crucially on the choice of the spatial coordinate system: changing to a moving frame can turn a convective instability into an absolute instability, and vice versa.

On bounded domains, one has, in addition, to account for the possibility that linear waves may be reflected at the boundary. If the domain is large, the so-called absolute spectrum provides a criterion that distinguishes between absolute and convective instabilities and, in addition, measures whether reflections from the boundaries lead to instabilities. The absolute spectrum itself is not spectrum. All but possibly a fixed number of discrete eigenvalues will, however, converge, in distance, to the absolute spectrum as the domain size tends to infinity. In other words, except possibly for finitely many eigenvalues that are created by the boundary conditions or through the spatial dependence of the underlying pattern, all eigenvalues for the PDE on the bounded domain are close to the absolute spectrum. Thus, if the absolute spectrum reaches into the right half-plane, the underlying nonlinear wave is unstable. Otherwise, it is stable.

Patterns on the unbounded domain are unstable when their essential spectrum reaches into the right half-plane. Patterns that have unstable essential and absolute spectrum are absolutely unstable, while patterns with unstable essential but stable absolute spectrum are only convectively unstable. More details on absolute and essential spectra are available PDF.

The following schematic picture illustrates the various spectra. The essential spectrum is plotted in green, while the absolute spectrum is the red line. The blue bullets symbolize the actual point spectrum of the PDE on a given large bounded domain.

The fact that the essential spectrum of a nonlinear wave may not be recovered upon truncating an unbounded domain to a large but bounded domain has also consequences for the numerical computation of spectra. In actual numerical computations on bounded domains, it is the absolute, and not the essential, spectrum that is computed.

Consider the equation
u_{t}=u_{xx}+2u_{x}+0.8u
on [0,L] with Dirichlet boundary conditions
u(0)=u(L)=0. The leftmost picture shows
absolute and essential spectra for a convectively unstable pattern.
Click on the middle or right image to view a movie (<1.3MB) with the
time evolution of a solution associated with the plotted initial
condition (the PDE is the heat equation with transport and Dirichlet
boundary conditions): even though the pulse grows exponentially while
travelling towards the left boundary, it eventually decays to zero. A
comparison of the two solutions, computed on intervals of length L=20
and L=40, shows that the maximal amplitude increases drastically, in
fact, exponentially, in the interval length. This indicates that the
pattern will likely be more and more sensitive to perturbations as the
domains size increases.

Consider the equation
u_{t}=u_{xx}+2u_{x}+1.1u
on [0,L] with Dirichlet boundary conditions
u(0)=u(L)=0. The left picture shows absolute
and essential spectrum for an absolutely unstable pattern. Click on the
right image to view a movie (<1.3MB) with the time evolution of the
solution associated with the plotted initial condition (the PDE is the
heat equation with transport and Dirichlet boundary conditions): The
pulse travels to the left while growing exponentially. After the pulse
hits the boundary, the perturbation is eventually growing everywhere
without bound in the entire interval.

An interesting application of absolute spectra arises when fronts and backs with unstable essential spectrum are glued together to produce pulses.

Thus, suppose that a nonlinear PDE on the real line supports a front and a back that travel with the same speed. It is then possible to glue the front and back together to obtain a pulse that travels with a slightly different wave speed. The length of the plateau where front and back are glued together can be prescribed arbitrarily (as long as this length is sufficiently large).

If the asymptotic "top" state is unstable while the "bottom" state is stable, then both front and back are unstable. Their spectrum is plotted in the left picture below. One might then expect that the resulting pulse is also unstable as it is the concatenation of two unstable waves, the front and the back. In fact, the spectrum of the resulting pulses is the union of the stable essential spectrum of the asymptotic "bottom" state and the absolute spectrum of the "top" state. Depending whether the "top" state is convectively or absolutely unstable, the pulses can be stable or will be unstable.

As we increase the length of the "top" plateau of the pulse, more and more discrete eigenvalues move out of the essential spectrum of the "bottom" state and begin to creep along the absolute spectrum of the "top" state towards its rightmost end point.