论文标题
大量的渐近渐进性非本地改性的kortweg-de Vries方程，带有阶梯状边界条件
Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions
论文作者
论文摘要
We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, $u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$是一个任意的正实数。我们首先开发了直接散射理论，以建立与阶梯式初始数据相关的基本Riemann-Hilbert（RH）问题。多亏了symmetries $ x \ rightarrow-x $，$ t \ rightarrow-t $ of nonlocal mkdv方程，我们分别研究了$ t \ rightarrow- \ infty $和$ t \ rightarrow+rightarrow+\ \ \ \ \ \ \ \ infty $。我们的主要技术是使用最陡峭的下降分析将原始矩阵值的RH问题变形为相应的常规RH问题，可以明确解决。最后，我们获得了Cauchy问题解决方案解决方案的不同大型渐近行为，用于在不同的时空部门$ \ Mathcal {r} _ {i} $，$ \ MATHCAL {r} _ $ \ Mathcal {r} _ {iv} $上的整个$（x，t）$ - 平面。
We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, $u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$ is an arbitrary positive real number. We firstly develop the direct scattering theory to establish the basic Riemann-Hilbert (RH) problem associated with step-like initial data. Thanks to the symmetries $x\rightarrow-x$, $t\rightarrow-t$ of nonlocal MKdV equation, we investigate the asymptotics for $t\rightarrow-\infty$ and $t\rightarrow+\infty$ respectively. Our main technique is to use the steepest descent analysis to deform the original matrix-valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally we obtain the different large-time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space-time sectors $\mathcal{R}_{I}$, $\mathcal{R}_{II}$, $\mathcal{R}_{III}$ and $\mathcal{R}_{IV}$ on the whole $(x,t)$-plane.
