Nonlinear Stability Theorem for High-Intensity Charged Particle Beams

Global conservation constraints based on the nonlinear Vlasov-Maxwell equations are used to derive a three-dimensional kinetic stability theorem for an intense non-neutral ion beam (or charge bunch) propagating with average axial velocity $v_b=\mathrm{const}$. It is shown that a sufficient condition for linear and nonlinear stability for perturbations with arbitrary polarization is that the equilibrium distribution be a monotonically decreasing function of the single-particle energy $H^{\prime }$ in the beam frame, i.e., $\partial f_{\mathrm{eq}}\left(H^{\prime }\right)/\partial H^{\prime }\le 0$.