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1. 研究目的与意义及国内外研究现状

耦合非线性Schrdinger方程在物理学的众多领域都有非常广泛的应用，例如非线性光学、凝聚态物理、等离子体物理等。因此关于该方程的求解就是一个极有意义的课题，然而关于它的精确解往往很难得到，在这种情况下，数值求解就变得极为重要。该方程满足多个重要的守恒性质，如质量守恒和能量守恒。因此，在对该方程的数值求解中，既要使得所构造的格式具有较高精度，又能尽可能多的保持原问题的一些守恒性质。基于以上考虑，本文拟对耦合非线性Schrdinger方程提出一个新的紧致差分格式，并证明新格式在离散意义下保持原问题的质量守恒和能量守恒。

国内外研究现状

已有大量文献对非线性Schrdinger方程做了研究，大部分非线性Schrdinger方程的数值方法经简单修改都可以推广到对非线性耦合Schrdinger方程的数值求解，但对算法的误差估计和稳定性分析等理论分析会因为耦合项的存在而产生很多新的困难。除此之外，也有很多文献对耦合非线性Schrdinger方程有关孤立波的碰撞进行了详细研究。关于耦合非线性Schrdinger方程的数值研究，M .S .Ismail 等人先后提出了几个具有二阶精度的线性和非线性隐式有限差分格式、四阶高精度隐式有限差分格式和Galerkin有限元格式，并运用Von-Neumann方法对算法的稳定性进行了分析。徐岩和舒其望运用局部间断有限元方法对非线性耦合Schrdinger方程进行了数值求解，并做了大量数值实验。Manakov S. V. 运用逆散射变换法对耦合非线性Schrdinger方程进行了半解析研究，Takayuki Tsuchida和Yi jinag Chen等在此基础上进一步的研究了耦合非线性Schrdinger方程的逆散射法，Christov和Sonnier等人提出了一个两层守恒差分格式，并对孤立波的碰撞进行了数值模拟。然而，以上算法大多是非线性隐式的，实际计算中不可避免地需要迭代，而且都没有给出格式的严格的收敛性分析。为此，王廷春等对[21-22]中的格式建立了严格的误差估计，并提出和分析了一个高效的迭代算法，特别给出了计算中不需迭代的具有二阶精度的线性化守恒差分格式。同年，王廷春、张鲁明和陈芳启还对耦合非线性Schrdinger方程提出了一个非耦合的线性化差分格式，并运用能量方法建立了格式的最优误差估计。王丽娟在其硕士学位论文中对耦合非线性Schrdinger方程建立了一个非线性隐式差分格式和一个线性化三层十一点隐式差分格式，并分析了格式的守恒性、稳定性、收敛性。杨磊基于二阶时间分裂结合有限差分格式对耦合非线性Schrdinger方程建立了一个高效的数值算法，并对孤立波的弹性碰撞和非弹性碰撞进行了数值模拟。谢树森等人对具有耗散的耦合非线性Schrdinger方程够构造了一个有限差分格式，证明了格式在时间和空间两个方向上都具有二阶精度，并分析了格式的无条件稳定性。然而，关于耦合非线性Schrdinger方程的紧致差分格式尚没有文献进行研究。

2. 研究的基本内容

1、首先介绍一下耦合非线性Schrdinger方程的物理背景及其守恒律，并综述一下研究现状；

2、提出一个新的有限差分格式，证明新格式能在离散意义下保持原问题的两个守恒性质；3、运用Taylor展开，详细推导出新格式的局部截断误差；4、通过算例给出测试新格式的精度，验证格式的守恒律，并对孤立波的运行进行数值模拟；5、对本文的研究结果进行简单总结。

3. 实施方案、进度安排及预期效果

1.实行方案：

我们对耦合非线性schrdinger方程构造一种紧致差分格式，分析该格式的局部截断误差，证明该格式的离散守恒律，引入追赶法对差分方程进行数值求解，并对孤波碰撞等现象进行了数值模拟。

2.进度：

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