一类可积方程的渐近分析

In this project, we aim to extend the nonlinear steepest descent method based on Riemann-Hilbert problem to the integrable equations associated with the higher order spectrum problems. On the one hand, we study the long time asymptotic behaviors of the solutions with decaying initial data for the integrable equations associated with the higher order spectrum problems. Firstly, formulating a Riemann-Hilbert problem that is equivalent to the initial value problem, then we reduce the original Riemann-Hilbert problem to a model one which plays a leading role in the neighborhood of the stationary phase points through the equivalent transformation, asymptotic deformation and scaling. Then we express the leading term of solution for the initial value problem in terms of the special functions such as the standard parabolic cylinder function or Painlevé equation in different areas. The research of this project involves two stationary phase points, three stationary phase points on a straight line and three stationary phase points with triangular distribution, follows the principle from the easier to the more advanced. On the other hand, with the help of the Fokas unified transform method and the nonlinear steepest descent method, we study the long time asymptotic behaviors of the initial boundary value problems for the integrable equations associated with multi-component AKNS spectrum problems.

本项目拟将基于Riemann-Hilbert问题的非线性最速下降法推广到与高阶谱问题相联系的可积方程。一方面，研究与高阶谱问题相联系的可积方程在快速衰减初值条件下解的长时间渐近行为。首先构造与方程初值问题等价的Riemann-Hilbert问题，然后经过一系列等价变换、渐近形变以及尺度变换将其化为一个在驻定相位点邻域起主导作用的典型Riemann-Hilbert问题，随之在不同区域利用标准抛物柱面函数或者Painlevé方程等特殊函数来表示方程初值问题解的渐近主部。本项目研究分别涉及到两个驻定相位点、在一条直线上的三个驻定相位点以及呈三角形分布的三个驻定相位点，符合认知规律的由易到难。另一方面，将Fokas统一变换理论和非线性最速下降方法结合起来，研究与多分量AKNS谱问题相联系的可积方程初边值问题解的长时间渐近行为。

本项目涉及两个方面：一方面研究与3×3矩阵谱问题相联系的耦合可积方程的Riemann-Hilbert问题的构造以及解的长时间渐近分析。基于矩阵分块的思想，将高阶矩阵谱问题低阶化，利用低阶分块矩阵Riemann-Hilbert问题提出耦合可积方程反散射变换的统一格式。利用一些技巧和严格的理论分析，并巧妙避开矩阵Riemann-Hilbert问题的不可解性这一难题，改进非线性最速下降法，对耦合可积方程初值问题解的长时间渐近行为进行本质性的研究。另一方面研究与高阶谱问题相联系的可积方程的有限亏格解、怪波解，这是为了后续研究非零边界条件下可积方程解的渐近分析作准备。