标题： 非零边界条件下的聚焦mKdV方程中的Painlevé超越函数 显示英文标题

标题： Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions

摘要： 我们考虑非聚焦修正Korteweg-de Vries (mKdV) 方程在非零边界条件\begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align}下的柯西问题，该问题可以通过逆散射变换通过黎曼-希尔伯特问题来表征。 使用$\bar\partial$的 Deift-Zhou 非线性最陡下降方法的推广，结合双尺度极限技术，我们在过渡区域$|x/t+6|t^{2/3}< C$中得到了非聚焦 mKdV 方程柯西问题解的长时间渐近行为，其中包含$C>0$。 渐近行为可以用第二 Painlevé 超越函数的解来表示。 显示英文摘要

摘要： We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlev\'{e} transcendent.