Department of Mathematics - SEMINAR ON PDE - Asymptotic stability of the sine-Gordon kink outside symmetry

We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction
potentials. Prime examples include the φ4 model and the sine-Gordon model. These models feature
simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a
rich class of problems owing to the combination of weak dispersion in one space dimension, low power
nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances
or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon
kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof
combines a space-time resonances approach based on the distorted Fourier transform to capture
modified scattering effects with modulation techniques to take into account the invariance under Lorentz
transformations and under spatial translations. A major difficulty is the slow local decay of the radiation
term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the
modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic
nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.
The entire framework of our proof, including the systematic development of the distorted Fourier theory,
is general and not specific to the sine-Gordon model. This is a joint work with Jonas Lührmann (Texas
A&M).