求基本孤立波的共轭梯度法
Conjugate gradient method for finding fundamental solitary waves
摘要：
共轭梯度法（CGM）是求解对称符号定矩阵线性方程组的最快通用迭代法。本文对这种方法进行了改进，使之能求解非线性哈密顿方程的基本孤立波。这种修正的CGM克服的主要障碍是关于孤立波线性化的方程的算子不是符号定的。相反，它有一个有限数量的本征值在零的相对侧比其频谱的其余部分。我们提出了修改后的CGM的版本，可以找到孤立波的传播常数或功率的规定值。我们也将这些方法推广到多分量非线性波方程。给出了所提出的方法的收敛条件，并讨论了它们的实际意义。我们证明，我们修改后的CGMs收敛速度比，比如说，Petviashvili或类似的方法，特别是当后者收敛缓慢。
英文摘要：
The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it could find fundamental solitary waves of nonlinear Hamiltonian equations. The main obstacle that such a modified CGM overcomes is that the operator of the equation linearized about a solitary wave is not sign definite. Instead, it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum. We present versions of the modified CGM that can find solitary waves with prescribed values of either the propagation constant or power. We also extend these methods to handle multi-component nonlinear wave equations. Convergence conditions of the proposed methods are given, and their practical implications are discussed. We demonstrate that our modified CGMs converge much faster than, say, Petviashvili's or similar methods, especially when the latter converge slowly.
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