The theory and applications of the unstable nonlinear Schrödinger equation

Abstract

Recent works on the unstable nonlinear Schrödinger (UNS) equation, iq x + q n +2| q| 2 q=0, are reviewed. It is shown that initial value problems can be solved by the inverse scattering method. Exact N soliton solutions gives an information on the soliton interaction in unstable media. Namely, the position shifts due to the collisions have opposite signs compared with the conventional nonlinear Schrödinger equation, iq t + q xx + 2| q| 2 q = 0. As applications of the UNS equation, two physical systems, the Rayleigh-Taylor instability problem and electron beam plasma, are discussed. In both systems, the dispersion relation has a critical wave number below which frequency becomes a complex number. It is shown that near the critical wave number the wave amplitude obeys the UNS equation. It is concluded that the UNS equation is a canonical equation which describes nonlinear modulations of wave amplitude in unstable media.