Course info

Traveling waves are solutions to nonlinear partial differential equations that propagate over time with a fixed speed without changing their profile. These special solutions arise in many applied problems where they model, for instance, water waves, nerve impulses in axons or pulses in optical fibers. Therefore, the naturally associated question of their dynamic stability is of interest: do solutions whose initial conditions are small perturbations of the wave stay close to the original solution of not? Indeed, only those waves that are stable against (localized) perturbations are observed in practice. The first step in the stability analysis is to linearize the underlying partial differential equation about the wave and compute the associated spectrum, which is in general a nontrivial task. To approximate the spectra of various waves, we introduce the following tools:
- Sturm-Liouville theory
- exponential dichotomies
- Fredholm theory
- the Evans function
- parity arguments
- Lyapunov-Schmidt reduction
- exploiting (reversible) symmetries
The next step is to derive stability of the wave from its spectral properties. We find that stability depends on the type of perturbations leading to the concepts of transient vs. remnant instabilities and absolute vs. convective instabilities. As an additional complicating factor, any non-constant traveling waves has spectrum up to the imaginary axis prohibiting a `standard stability argument' as used for steady states. All in all, we consider various nonlinear iteration arguments employing:
- (pointwise) weights
- energy estimates
- Hamiltonian structure
- temporal/spatial Green's function estimates

Kurs aus TUMOnline ( S2020)