Particle-beam approach to collective instabilities -- application to space-charge dominated beams

Nonlinear dynamics deals with parametric resonances and di#11;usion. The phenomena are usually beam-intensity independent and rely on a particle Hamiltonian. Collective instabilities deal with beam coherent motion, where the Vlasov equation is frequently used in conjunction with a beam-intensity dependent Hamiltonian. We ad- dress the questions: Are the two descriptions the same? Are collective instabilities the results of encountering parametric resonances whose driving force is intensity depen- dent? We study here the example of a space-charge dominated beam governed by the Kapchinskij-Vladimirskij (K-V) envelope equation [1]. The stability and instability regions as functions of tune depression and envelope mismatch are compared in the two approaches. The study has been restricted to the simple example of a uniformly focusing channel