Associated Project 8 • Stability of periodic waves in dissipative-dispersive systems
Periodic waves are omnipresent in dynamical processes in many scientific disciplines, such as nonlinear optics, hydrodynamics, ecology or biology. The well-known Turing mechanism, describing pattern-forming processes in spatially extended systems, asserts that periodic waves are typically the first patterns to arise after a homogeneous background state becomes unstable, and serve as the building blocks for more complicated structures.
Despite the apparent simplicity of periodic waves, establishing their nonlinear stability has proven to be challenging on spatially extended domains. The main obstruction is that the linearization about a periodic wave possesses continuous spectrum touching the imaginary axis, which prevents exponential decay on the linear level. That is, linear decay is not strong enough to close a nonlinear argument and nonlinear stability might depend on the choice of the nonlinearity and the space of perturbations under consideration.
Over the past decades several tools have been developed to study global existence and decay in such delicate situations. On the one hand, for dissipative partial differential equations, such as reaction-diffusion systems, an efficient theory is available to establish nonlinear stability of periodic traveling waves provided that the spectrum of the linearization is diffusively spectrally stable, i.e., it touches the imaginary axis only at the origin in a single quadratic tangency. This theory employs spatio-temporal phase modulation and relies on an associated decomposition of the linear solution operator in Bloch frequency space. On the other hand, for dispersive systems, such as the nonlinear Schrödinger equation, the space-time resonances method allows one to identify the nonlinear terms that might prevent global existence in time. The space-time resonances method combines the celebrated normal form method of Shatah and the vector field method of Kleinermann.
In the project AP8 we study systems, which exhibit both dissipative and dispersive effects, and try to understand how these interact in situations where the linearization possesses spectrum up to the imaginary axis. In the consecutive works [dRS20] and [dRS21] we studied reaction-diffusion-advection systems, involving respectively two or more species. There, some of the nonlinear coupling terms that would usually lead to blow-up in finite time for solutions with small initial data are handled using the dispersive effect of advection. As soon as two different species move with two distinct velocities, they are essentially noninteracting. Likewise, in the work [GdR22] we reveal that, for a model known as the viscous half Klein–Gordon equation, the presence of dispersive terms allow us to perform a normal-form transform, and thus control quadratic nonlinear terms, which would otherwise lead to finite-time blow up of solutions with small initial data.
Periodic wave traveling with speed $c$.
Typical spectrum of the linearization
about a periodic wave, in case of
diffusive spectral stability. Spectrum of the linearization about
the ground state in the viscous
Klein–Gordon equation. Two curves
of spectrum touch the imaginary axis.
