We consider complex-valued parabolic evolution equations in one space dimension that are equivariant under spatial translation, and, in addition, admit a symmetry under complex phase-shift.
A major example is the complex quintic Ginzburg-Landau equation.
In this talk we are interested in traveling oscillating front solutions. The solutions admit a fixed profile, which travels in space and oscillates in the time evolution, by multiplication with a time-dependent rotational term. Besides, the profile converges to the zero steady state on the side and to a complex-valued but nonzero steady state on the other. The stability behaviour of the solution is quite delicate since the essential spectrum of the linearization touches the imaginary axis at the origin.
I give an introduction into the topic and present a nonlinear stability result in exponentially weighted spaces.
I conclude by applying the so-called freezing method to those solutions and show numerical experiments.
Within the CRC this talk is associated to the project(s): B3