Nonlinear dynamics of intense ion beam envelopes

The nonlinear envelope equations for an intense Kapchinskij-Vladimirskij (KV) beam equilibrium are studied self-consistently for the case of a periodic quadrupole focusing lattice. First, the linearized solutions to the two nonlinearly coupled envelope equations for a matched KV beam are obtained in the smooth-beam approximation. A comparison between the solutions of the linearized equations and the (numerically solved) nonlinear equations is presented. Second, the nonlinear evolution of the envelopes for a mismatched beam is studied numerically. It is found that the oscillation of the beam envelope exhibits chaotic behavior in certain regions of the parameter space (KS/ɛ,${\mathrm{\sigma }}_{\mathit{v}}$). Here, K is the self-field perveance, ɛ is the unnormalized beam emittance, S is the axial periodicity length, and ${\mathit{J}}_{\mathit{v}}$ is the vacuum phase advance. Detailed numerical results are presented and the stable regime in the parameter space (KS/ɛ,${\mathrm{\sigma }}_{\mathit{v}}$) is determined numerically. It is found that the threshold condition for the onset of unstable oscillations of the envelope functions is independent of the filling factor η of the quadrupole focusing lattice. © 1996 The American Physical Society.