Long-time asymptotic behavior of the Hunter-Saxton equation

TL;DR

This paper analyzes the long-time behavior of solutions to the Hunter-Saxton equation using a novel Riemann-Hilbert approach, revealing decay rates and detailed asymptotics in different space-time regions.

Contribution

It introduces a $ar{ullpartial}$-generalized steepest descent method to derive precise long-time asymptotics for the Hunter-Saxton equation.

Findings

Solution decays as $t^{-1/2}$ in $y/t > 0$ region

Solution is described by a parabolic cylinder model in $y/t < 0$ region

Residual error order is $O(t^{-1+1/(2p)})$ for $p>2$

Abstract

With $\bar{\partial}$-generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation \begin{eqnarray} &&u_{txx}-2\omega u_x+2u_xu_{xx}+uu_{xxx}=0,\quad x\in \mathbb{R},\ t>0,\nonumber\\ &&u(x,0)=u_0(x), \nonumber \end{eqnarray} where $u_0\in H^{3,4}(\mathbb{R})$ and $\omega>0$ is a constant. Using the new scale $(y,t)$ and a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in two space-time regions: The solution of the HS equation decays as the speed of $\mathcal{O}(t^{-1/2})$ in the region $y/t >0$; While in the region $y/t<0$, the solution of the HS equation is depicted by a parabolic cylinder model with an residual error order $\mathcal{O}(t^{-1+\frac{1}{2p}})$ with $ p>2$.