Haïssinski方程数值解法的改进
Improved Numerical Method for Solution of the Haïssinski Equation
摘要：
电子束在其平衡态的纵向电荷密度由海辛斯基方程的解给出，海辛斯基方程提供了弗拉索夫-福克-普朗克方程的稳态解。物理输入是纵向尾流势。我们将Haïssinski方程表示为一个非线性积分方程，其中归一化积分表示为解的泛函。这个方程可以通过牛顿迭代的矩阵形式以一种简单的方式求解，从高斯作为第一个猜测开始。我们说明了几个准现实的唤醒潜力。收敛是非常强大的，即使在电流远高于标称的存储环考虑。该方法克服了以往方法的局限性，并提供了解决方案的自动归一化的方便。
英文摘要：
The longitudinal charge density of an electron beam in its equilibrium state is given by the solution of the Haïssinski equation, which provides a stationary solution of the Vlasov-Fokker-Planck equation. The physical input is the longitudinal wake potential. We formulate the Haïssinski equation as a nonlinear integral equation with the normalization integral stated as a functional of the solution. This equation can be solved in a simple way by the matrix version of Newtons's iteration, beginning with the Gaussian as a first guess. We illustrate for several quasi-realistic wake potentials. Convergence is extremely robust, even at currents much higher than nominal for the storage rings considered. The method overcomes limitations of earlier procedures, and provides the convenience of automatic normalization of the solution.
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