标题： 具有阶梯初始数据的衰减mkdV方程的长时间渐近行为 显示英文标题

标题： Long-time asymptotics of the defocusing mKdV equation with step initial data

摘要： 本工作研究了一类阶梯型初始数据的非聚焦修正Korteweg-de Vries方程解的长时间渐近性。 通过应用Deift-Zhou非线性最速下降法，对该Riemann-Hilbert问题进行了严格的渐近分析。 在此过程中，推广了$g$-函数的构造，并通过保角变换和对称性提出了到Riemann-theta函数的亏格约简方法。 结果显示，对于足够大的时间，解表现出三部分时空结构：在左平面波区域，解衰减为具有振荡校正的调制平面波；在中央色散冲击波区域，解由调制椭圆波支配；在右平面波区域，解以指数方式收敛到一个常数。 从长时间渐近分析得出的结果与直接数值模拟的结果非常吻合。 显示英文摘要

摘要： This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by applying Deift-Zhou nonlinear steepest descent method. In this process, the construction of the $g$-function is generalized and the method of genus reduction to Riemann-theta function is proposed via conformal transformation and symmetries. It is revealed that for sufficiently large time, the solution manifests a tripartite spatiotemporal structure, i.e., in the left plane-wave region, the solution decays to a modulated plane wave with oscillatory correction; in the central dispersive shock wave region, the solution is governed by a modulated elliptic wave; in the right plane-wave region, the solution converges exponentially to a constant. The results from the long-time asymptotic analysis have been shown to match remarkably well with direct numerical simulations.