Long-time asymptotics of the Hunter-Saxton equation on the line
JOURNAL OF DIFFERENTIAL EQUATIONS（2024）
摘要
With partial derivative-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 utxx-2 omega ux+2uxuxx+uuxxx=0,x is an element of R,t>0 u(x,0)=u0(x), where u0 is an element of H3,4(R) and omega >0 is a constant. Via 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 kinds of space-time regions under a new scaleu (x,t). The solution of the HS equation decays as a 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 the solution of a parabolic cylinder model with an residual error order O(t-1+1/2p) with 2更多
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关键词
Hunter-Saxton equation,Riemann-Hilbert problem,8-Steepest descent method,Long-time asymptotics
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