Landau-Type Damping in Nonlinear Wavepacket Propagation

Abstract

A modulational instability analysis is carried out for a new kind of integro-differential nonlinear Schrödinger equation which describes nonlinear wave–envelope propagation for several different situations, such as the transport of very intense charged–particle beams in accelerating machines, large amplitude electromagnetic wavepacket propagation in nonlinear media (optical fibres, plasmas, etc.), Langmuirndash;wavendash;envelope propagation in warm plasmas, and collective dynamics in mesoscopic physics. The modulational instability analysis is extended from configuration space into phase space, by using the Wigner transform. It is shown that the propagation of a wavepacket in a nonlinear medium, governed by the above nonlinear Schrödinger equation, can be described, in phase space, in terms of a kinetic-like theory similar to the one based on the Vlasov equation which is used for describing both collective plasma dynamics and collective longitudinal dynamics of chargedndash;particle beams in accelerating machines. Remarkably, the phenomenon of Landau damping is recovered for the longitudinal charged-particle beam dynamics (extending in this way a previous analysis carried out within the Thermal Wave Model (R Fedele and G Miele 1991 Il Nuovo Cim. D 13 1527)) but is also predicted for other physical situations concerning electromagnetic nonlinear wave envelope propagation and mesoscopic physics. Furthermore, the concept of a coupling impedance associated with the wavepacket propagation is also introduced in analogy to the one of charged-particle bunches. This approach provides stability charts fully similar to the ones describing charged-particle beams in accelerating machines. These new results generalize the conventional theory of the modulational instability associated with the nonlinear Schrödinger equation and show clearly the stabilizing role of Landau damping during the development of the modulational instability.