Description of the nonlinear and collective behavior of charged-particle beams by means of a nonlinear Schrodinger equation

Charged-particle beams are employed for a number of scientific and technological applications. The conventional description of their collective behavior is usually given in terms of the Vlasov equation. In the last 15 years some alternative descriptions have been developed in terms of a nonlinear Schrodinger equation governing the collective dynamics of the beam while interacting with the surrounding medium. This approach gives new insights, providing an alternative "key of reading" of the charged-particle beam dynamics, and have been applied to a number of physical problems concerning conventional particle accelerating machines as well as plasma-based accelerator schemes. Remarkably, it is based on a mathematical formalism fully similar to those used for the propagation of e.m. radiation beams in nonlinear media a well as the nonlinear dynamics of the Bose-Einsten condensates. In this paper, a presentation of some significant nonlinear collective effects of a charged-particle beam in particle accelerators, that have been recently investigated in the framework of the above Schrodinger-like descriptions, is given.