Localized structures in complex plasmas in the presence of a magnetic field

Abstract

In this work, the general framework in which fits our investigation is that of modeling the dynamics of dust grains therein dusty plasma (complex plasma) in the presence of electromagnetic fields. The generalized discrete complex Ginzburg-Landau equation (DCGLE) is thus obtained to model discrete dynamical structure in dusty plasma with Epstein friction. In the collisionless limit, the equation reduces to the modified discrete nonlinear Schrödinger equation (MDNLSE). The modulational instability phenomenon is studied and we present the criterion of instability in both cases and it is shown that high values of damping extend the instability region. Equations thus obtained highlight the presence of soliton-like excitation in dusty plasma. We studied the generation of soliton in a dusty plasma taking in account the effects of interaction between dust grains and theirs neighbours. Numerical simulations are carried out to show the validity of analytical approach.

Appendix

The matrix elements $m_{ij}$ ($i,j=1,2,3,4$) are given as follows:

$$\begin{aligned} &m_{1}=\bigl(-4\cos(Q+2q)+6\cos(q+Q)-4\cos(q) \\ &\phantom{m_{1}=}-2\cos(Q-q) +4\cos(2q)\bigr)R_{i}{\psi_{{0}}}^{2} \\ &\phantom{m_{1}=}+2\bigl(\cos(q)-\cos(q+Q)\bigr)P_{i}, \\ &m_{2}=\bigl(S_{r}+\bigl(-8\cos(q+Q)-2\cos(2q)+4\cos(Q) \\ &\phantom{m_{2}=}+4\cos(Q+2q)\bigr)R_{r}\bigr){\psi_{{0}}}^{2} \\ &\phantom{m_{2}=}+\bigl(2\cos(q+Q)-2\cos(q)\bigr)P_{r}, \\ &m_{4}=-m_{10}=\bigl(\bigl(4\cos(Q)-4\cos(q)-2\cos(Q-q) \\ &\phantom{m_{4}=}-2\cos(q+Q)+2\cos(2q)\bigr)R_{r}+S_{r}\bigr){\psi_{{0}}}^{2}, \\ &m_{5}=\bigl(2\cos(Q-q)+4\cos(Q+2q)-4\cos(2q) \\ &\phantom{m_{5}=}-6\cos(q+Q)+4\cos(q)\bigr)R_{r}{\psi_{{0}}}^{2} \\ &\phantom{m_{5}=}+2\bigl(\cos(q+Q)-\cos(q)\bigr)P_{r}, \\ &m_{6}=\bigl(\bigl(-8\cos(q+Q)-2\cos(2q)+4\cos(Q) \\ &\phantom{m_{6}=}+4\cos(Q+2q)\bigr)R_{i}+S_{i}\bigr){\psi_{{0}}}^{2} \\ &\phantom{m_{6}=}+\bigl(2\cos(q+Q)-2\cos(q)\bigr)P_{i}, \\ &m_{8}=-m_{14}=\bigl(\bigl(4\cos(Q)-4\cos(q)-2\cos(Q-q) \\ &\phantom{m_{8}=}-2\cos(q+Q)+2\cos(2q)\bigr)R_{i}+S_{i}\bigr){\psi_{{0}}}^{2}, \\ &m_{11}=-\bigl(-2\cos(q+Q) -4\cos(q)+6\cos(Q-q) \\ &\phantom{m_{11}=}+4\cos(2q) -4\cos(Q-2q)\bigr)R_{i}{\psi_{{0}}}^{2} \\ &\phantom{m_{11}=}-2\bigl(\cos(q)-\cos(Q-q)\bigr)P_{i}, \\ &m_{12}=-\bigl(S_{r}+\bigl(4\cos(Q-2q)+4\cos(Q)-8\cos(Q-q) \\ &\phantom{m_{12}=}-2\cos(2q)\bigr)R_{r}\bigr){\psi_{{0}}}^{2}-2\bigl(\cos(Q-q)-\cos(q)\bigr)P_{r}, \\ &m_{15}=\bigl(-2\cos(q+Q)-4\cos(q)+6\cos(Q-q) \\ &\phantom{m_{15}=}+4\cos(2q)-4\cos(Q-2q)\bigr) R_{r}{\psi_{{0}}}^{2} \\ &\phantom{m_{15}=}+2\bigl(\cos(q)-\cos(Q-q)\bigr) P_{r}, \\ &m_{16}=\bigl(\bigl(8\cos(Q-q)+2\cos(2q)-4\cos(Q-2q) \\ &\phantom{m_{16}=}-4\cos(Q)\bigr)R_{i}-S_{i}\bigr){\psi_{{0}}}^{2} \\ &\phantom{m_{16}=}+2\bigl(\cos(q)-\cos(Q-q)\bigr)P_{i}; \\ &K_{3}=-m_{15}-m_{5}-m_{2}-m_{12}, \\ &K_{2}=m_{4}^{2}+m_{2}m_{12}+m_{5}m_{15}-m_{1}m_{6}+m_{5}m_{2}+m_{2}m_{15} \\ &\phantom{K_{2}=}-m_{11}m_{16}+m_{12}m_{15}+m_{5}m_{12}, \\ &K_{1}=m_{{1}}m_{{4}}m_{{8}}-m_{{5}}m_{{2}}m_{{12}}+m_{{1}}m_{{6}}m_{{15}}+m_{{5}}m_{{11}}m_{{16}} \\ &\phantom{K_{2}=}-m_{{5}}m_{{2}}m_{{15}}+m_{{2}}m_{{11}}m_{{16}}+m_{{8}}m_{{11}}m_{{4}}-m_{{5}}{m_{{4}}}^{2} \\ &\phantom{K_{2}=}-m_{{2}}m_{{12}}m_{{15}}-{m_{{4}}}^{2}m_{{15}}+m_{{1}}m_{{6}}m_{{12}}-m_{{5}}m_{{12}}m_{{15}}, \\ &K_{0}=-m_{{1}}m_{{4}}m_{{8}}m_{{15}}+m_{{5}}m_{{2}}m_{{12}}m_{{15}}-m_{{5}}m _{{2}}m_{{11}}m_{{16}} \\ &\phantom{K_{2}=}-m_{{1}}m_{{6}}m_{{12}}m_{{15}}+m_{{1}}m_{{6}}m_{{11}}m_{{16}}-m_{{5}}m_{{8}}m_{{11}}m_{{4}} \\ &\phantom{K_{2}=}+m_{{1}}{m_{{8}}}^{2}m_{{ 11}}+m_{{5}}{m_{{4}}}^{2}m_{{15}}. \end{aligned}$$