Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations
Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations
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发表时间：
1984
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通讯作者：
T. Taha;M. Ablowitz
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作者：
T. Taha;M. Ablowitz
得到了以非线性Schriidinger方程、Korteweg-deVries方程和修正的Korteweg-deVries方程为极限形式的非线性偏差分方程。这些差分方程有许多特殊的性质。它们是通过与逆散射变换相关的方法构造的。它们可以作为相关非线性发展方程数值格式的基础。实验表明，他们比较非常有利的其他已知的数值方法（论文二，三）。在第二篇论文中，将非线性薛定谔方程的Ablowitz-Ladik格式与其他已知的数值格式进行了比较，一般证明其速度比所有使用的有限差分格式快，但比有限傅立叶（伪谱）方法慢。在第三篇论文中，Korteweg-deVries方程的一个建议方案被证明比已经考虑的有限差分和有限傅立叶方法都快。
Nonlinear partial difference equations are obtained which have as limiting forms the nonlinear Schriidinger, Korteweg-deVries and modified Korteweg-deVries equations. These difference equations have a number of special properties. They are constructed by methods related to the inverse scattering transform. They can be used as a basis for numerical schemes to the associated nonlinear evolution equations. Experiments have shown that they compare very favorably with other known numerical methods (papers II, III). In paper II, the Ablowitz-Ladik scheme for the nonlinear Schrodinger equation is compared to other known numerical schemes, and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite Fourier (pseudospectral) methods. In paper III, a proposed scheme for the Korteweg-deVries equation proved to be faster than both the finite difference and finite Fourier methods already considered.