Langmuir dark solitons in dense ultrarelativistic electron-positron gravito-plasma in pulsar magnetosphere

Abstract

Nonlinear propagation of electrostatic modes in ultrarelativistic dense elelectron-positron gravito-plasma at the polar cap region of pulsar magnetosphere is considered. A nonlinear Schrödinger equation is obtained from the reductive perturbation method which predicts the existence of Langmuir dark solitons. Relevance of the propagating dark solitons to the pulsar radio emission is discussed.

Appendix: The compatibility condition

It can be shown that the 1st harmonic of the 2nd-order electron and positron densities can be found to be

After finding $n_{e1}^{(2)}$–$n_{p1}^{(2)}$ and substituting it in the following 1st-harmonic of the 2nd-order part of the Poisson’s equation:

we obtain the following compatibility condition:

which is exactly the same as the expression of the group velocity, Eq. (19), obtained by differentiating ω with respect to k from the linear dispersion relation, Eq. (17).