Long-time asymptotic behavior of the Hunter-Saxton equation

Luman Ju, Kai Xu, Engui Fan

Published 2023-07-30Version 1

We study the long-time asymptotic behavior for the solution to the weighed Sobolev initial value problem of the Hunter-Saxton equation, where $\omega>0$ is a constant. Based on the inverse scattering transform, the solution of this initial value problem is expressed in term of the solution of an associated Riemann-Hilbert(RH) problem. Further using the $\bar{\partial}$ steepest descent method, we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in different space-time region. As for $\xi=\frac{y}{t}>0$, the solution $u(x,t)$ decays as the speed of $\mathcal{O}(t^{-1/2})$; While for $\xi=\frac{y}{t}<0,$ the solution is approximated by the solution of a parabolic cylinder model RH problem with an residual error order as $\mathcal{O}(t^{-1+\frac{1}{2p}}),\ p>2$.