Landau-type damping in nonlinear wavepacket propagation

A modulational instability analysis is carried out for a new kind of integr o-differential nonlinear Schrodinger equation which describes nonlinear wav e-envelope propagation for several different situations, such as the transp ort of very intense charged-particle beams in accelerating machines, large amplitude electromagnetic wavepacket propagation in nonlinear media (optica l fibres, plasmas, etc.), Langmuir-wave-envelope propagation in warm plasma s, and collective dynamics in mesoscopic physics. The modulational instabil ity analysis is extended from configuration space into phase space, by usin g the Wigner transform. It is shown that the propagation of a wavepacket in a nonlinear medium, governed by the above nonlinear Schrodinger 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 charged- particle beams in accelerating machines. Remarkably, the phenomenon of Land au 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, Il Nuovo Cim. D13, 1527 (1991)]) but is also predicted for other physical situations concerning electromagnetic no nlinear 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 cha rged-particle beams in accelerating machines. These new results generalize the conventional theory of the modulational instability associated with the nonlinear Schrodinger equation and show clearly the stabilizing role of La ndau damping during the development of the modulational instability.