Integral form of the beam envelope equation for electron beam propagation in vacuum and its relation to the K-V equation

Integral form of the beam envelope equation for electron beam propagation in vacuum and its relation to the K-V equation里从图1到图6的数据，X轴的单位为km，代表beam沿着轴向的传输距离；Y轴的单位为m，代表beam的包络半径。