Electrostatic Rogue Waves in a Degenerate Thomas-Fermi Plasma

Abstract

In this paper, we search for the modulational instability (MI) and the possibility of rogue wave generation in a degenerated Thomas-Fermi plasma, consisting of cold inertial ions and Thomas-Fermi distributed electrons and positrons. A multiscale perturbation method is used to derive a nonlinear Schrödinger equation (NLSE) for the envelope amplitude, based on which the occurrence of MI and the generation of rogue as well as super rogue waves is investigated at length. The plasma configurational parameters (namely the positron concentration and the ratio of electron Fermi temperature to positron Fermi temperature) are shown to affect the conditions for MI significantly, and in fact altering the associated threshold. In particular, the positron concentration as well as the temperature ratio leads to an increase in the critical wavenumber, suggesting that MI sets in for larger values of the wavenumber.

Appendix

Here, the coefficients for the second harmonic components are provided. ${ {D_{2}^{0}}}_{1/2/3}$ are associated with the second-order, zeroth harmonic components of the ion density, speed, and potential respectively while ${ {D_{2}^{2}}}_{1/2/3}$ stand for the second-order, second harmonic components.

$$ \begin{array}{@{}rcl@{}} D_{2_{1}}^{0}& =&\frac{d_{1}\left( d_{1}+k^{2}\right)^{2}\left( \frac{ 2v_{g}\omega} {k}+\frac{\omega^{2}}{k^{2}}\right) -\frac{2d_{2}}{d_{1}}}{ d_{1}{v_{g}^{2}}-1} \\ D_{2_{3}}^{0}& =&\frac{1}{d_{1}}D_{2_{1}}^{0}-\frac{2d_{2}}{d_{1}} \\ D_{2_{2}}^{0}& =& v_{g}D_{2_{1}}^{0}-2\left( d_{1}+k^{2}\right)^{2}\frac{ \omega} {k} \end{array} $$

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$$ \begin{array}{@{}rcl@{}} D_{2_{3}}^{2}& =&\frac{3\left( \frac{(d_{1}+k^{2})^{2}}{2}\right) -d_{2}}{ 3k^{2}} \\ D_{2_{1}}^{2}& =&\left( d_{1}+4k^{2}\right) D_{2_{3}}^{2}+d_{2} \\ D_{2_{2}}^{2}&=&\frac{\omega} {k}D_{2_{1}}^{2}-\left( d_{1}+k^{2}\right)^{2} \frac{\omega} {k} \end{array} $$

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