双线性时空共振系统的重正化群方法

Renormalization group (RG) method is a novel singular perturbation method to obtain the approximate solutions of differential equations. RG method, which is conceptually simple and elegant, can eliminate the secular term by constructing the RG equation without additional assumption. Now RG method is a hot issue in mathematical physics field. In this project, we will construct long-time approximate solutions of a kind of important space-time resonant systems by RG method, discuss how the space-time resonance impact on bilinear dispersive systems, which can help us understand the nonlinear interaction of free waves. Firstly, we will establish RG equation of this bilinear space-time resonant system in two and three dimensional space, whose solution can eliminate the secular term generated by space-time resonance. Then, we will get the long-time approximate solutions of above systems, and compare the difference between the approximate solutions and the true solutions. Secondly, the first order long-time approximate solution of this kind of bilinear space-time resonant systems has been obtained by us. In this project, we will get the second order long-time approximate solutions by combining RG method and averaging operator.

重正化群方法是微分方程奇异摄动领域的新方法. 该方法通过构造重正化群方程消去系统共振产生的长期项, 得到方程的近似解，其优点在于概念简单, 形式优美, 且不需要额外的结构假设, 已经成为数学物理领域研究的热点. 本项目拟研究双线性时空共振系统的重正化群方法，构造一类重要双线性时空共振系统的重正化群方法近似解，探讨时空共振对双线性色散系统的影响，从而理解自由波的非线性交互作用. 首先, 建立这类系统在二维和三维情形的重正化群方程, 利用重正化群方程的解消去时空共振产生的长期项, 得到这类双线性时空共振系统在二维和三维情形的长时间近似解, 并比较近似解和真解的近似程度; 其次，对于一维情形，申请人已经得到这类系统的一阶长时间近似解. 在本项目中，我们拟结合平均化算子和重正化群方法得到系统的二阶近似解。

重正化群方法起源于理论物理学中的量子场论，是研究系统在不同的能量尺度下演化的数学方法，在固体物理学、流体力学、宇宙学以及纳米技术等领域都有重要应用。Schrodinger 方程是量子力学中的基本方程，具有广泛的物理背景。本项目主要研究二次非线性Schrödinger方程解的动力学行为，探讨如何用偏微分方程的重正化群方法构造方程的近似解。.在本项目的资助下，项目组成员取得了比较满意的进展，主要成果包括：1.借助于偏微分方程的重正化群方法，在适当的存在空间中，分别构造出二次非线性Schrödinger方程在2维情形和3维情形的长时间近似解，并比较了近似解和真解的近似程度。2.应用RPS方法分别得到了在Caputo求导意义下和在Riemann-Liouville积分意义下分数阶时空耦合系统的渐近解。.在本项目的资助下，项目组成员及其研究生展开了深入的合作，多次去国内知名大学访问学习，取得了比较满意的成果，已发表SCI论文1篇，处于投稿中的SCI论文2篇。.