Georgia Institute of Technology College of Sciences

I discuss some recent results, obtained jointly with David Wallauch, on the stability of self-similar wave maps under minimal regularity assumptions on the perturbation. More precisely, we prove the asymptotic stability of an explicitly known self-similar wave map in corotational symmetry. The key tool are Strichartz estimates for the linearized equation in similarity coordinates.

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.

The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.

We will talk about our results on the elasticity and stability of the 
collision of two kinks with low speed v>0 for the nonlinear wave 
equation of dimension 1+1 known as the phi^6 model. We will show that 
the collision of the two solitons is "almost" elastic and that, after 
the collision, the size of the energy norm of the remainder and the size 
of the defect of the speed of each soliton can be, for any k>0, of the 
order of any monomial v^{k} if v is small enough.

References:
This talk is based on our current works:
On the collision problem of two kinks for the phi^6 model with low speed 
   [https://arxiv.org/abs/2211.09749]
Approximate kink-kink solutions for the phi^6 model in the low-speed 
limit [https://arxiv.org/abs/2211.09714]