Long-time asymptotic behavior of the Hunter-Saxton equation

2023/07/30 by Ju, Luman, Xu, Kai, Fan, Engui
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2307.16172

Abstract

With ∂-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 amp;amp;u_txx-2ωu_x+2u_xu_xx+uu_xxx=0, x∈ ℝ, tgt;0,
amp;amp;u(x,0)=u₀(x), where u₀∈ H^3,4(ℝ) and ω>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 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 O(t^-1+(1)/(2p)) with p>2.

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