论文标题
在经过逐步的初始数据的经过修改的Camassa-Holm方程的长期渐近学上
On the long-time asymptotics of the modified Camassa-Holm equation with step-like initial data
论文作者
论文摘要
我们研究了经过逐步的camassa-holm（MCH）方程的Cauchy问题的长时间渐近行为，并具有类似逐步的初始数据\ BEGIN \ begin {align}＆m_ {t}+\ lest（m \ left（u^{2}} -U_______________ {x}}^}^{2}^{2} \ right）\ right） m = u-u_ {xx}，\ nonumber \\＆u（x，0）= u_0（x）\ to \ left \ webt \ {\ begin {array} {ll} {ll} a_1，＆\ x \ to+\ infty，\\ [5pt] a_2，＆\ x \ to- \ infty， \ end {array} \ right。\ nonumber \ end {align}其中$ a_1 $和$ a_2 $是两个正常数。我们的主要技术工具是与相关的矩阵Riemann-Hilbert（RH）问题以及随之而来的RH问题的渐近分析的代表。基于与MCH方程和散射矩阵相关的LAX对的光谱分析，通过在新量表$（y，t）$中的RH问题解决方案来表征阶梯式初始问题的解决方案。我们采用双坐标$（ξ，c）$将半平面$ \ {（ξ，c）：ξ\ in \ mathbb {r}，\ c> 0，\ c> 0，\ξ= y/t \} $分为四个渐近区域。进一步使用Deift-Zhou最陡峭的下降方法，我们通过不同的时空区域中不同的时空区域的溶液$ u（y，t）$的长期渐近膨胀。相应的前导渐近近似是在属-0区域中的慢速/快速衰减阶梯式背景波和Genus-2区域的椭圆波中给出的。渐近学的第二项以通风函数或抛物线缸模型为特征。他们的剩余错误顺序分别为$ \ Mathcal {O}（T^{ - 1}）$或$ \ MATHCAL {O}（T^{ - 2}）$。
We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial data \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}=0, \quad m=u-u_{xx}, \nonumber \\ &u(x,0)=u_0(x)\to \left\{ \begin{array}{ll} A_1, &\ x\to+\infty,\\[5pt] A_2, &\ x\to-\infty, \end{array}\right.\nonumber \end{align} where $A_1$ and $A_2$ are two positive constants. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the step-like initial problem is characterized via the solution of a RH problem in the new scale $(y,t)$. We adopt double coordinates $(ξ, c)$ to divide the half-plane $\{ (ξ,c): ξ\in \mathbb{R}, \ c> 0, \ ξ=y/t\}$ into four asymptotic regions. Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time regions by the different choice of g-function. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterized by Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-1})$ or $\mathcal{O}(t^{-2})$ respectively.
