Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.
Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.
The two-dimensional dynamics of nonlinear ion-acoustic waves in plasmas are considered. A new nonlinear equation is found which is valid for unmagnetized as well as magnetized plasmas. For exactly vanishing magnetic fields, the Kadomtsev–Petviashvili equation is recovered; for weak magnetic fields, however, the dynamics are qualitatively different from that equation, depending on amplitude. With increasing magnetic field, the new equation becomes similar (but not identical) to the Zakharov–Kuznetsov equation which was derived for very strong magnetic fields. The stability behavior of plane solitons propagating along the magnetic field is also discussed. The transition from stable to unstable behavior and some quantitative new results for the instability growth rates are presented.
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