Modulational instability of ion-acoustic soliton in electron-positron-ion plasma with dust particulates

Abstract

Theoretical investigation has been made to study the envelope excitations of ion-acoustic solitons (IAS) in plasma composed of electrons, positrons, ions and dust particles. A nonlinear Schrödinger equation which describes the modulational instability of the IAS is derived by using the multiple scale method. The dispersive and nonlinear coefficients are obtained which depend upon the temperature of the ions, concentration of the positrons, electrons and dust particles. The modulationally stable and unstable regions are studied numerically for a wide range of parameters. It is found that these parameters play significant role in the formation of bright and dark envelope solitons in this plasma system.

Appendix

For the second order (n=2) reduced equation with l=1, the following equations are obtained

$$ -\iota\omega n_{1}^{(2)}+\iota k u_{1}^{(2)}=v_{g}\frac{\partial n_{1}^{(1)}}{\partial\xi}- \frac {\partial u_{1}^{(1)}}{\partial\xi} $$

(19)

$$ -\iota\omega u_{1}^{(2)}+ \iota k\phi_{1}^{(2)}+ \iota k 3\sigma n_{1}^{2} = v_{g}\frac{\partial u_{1}^{(1)}}{\partial \xi}-\frac{\partial \phi_{1}^{(1)}}{\partial\xi}- 3 \sigma\frac{\partial n_{1}^{(1)}}{\partial\xi} $$

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$$ (k^2+\lambda_{1})\phi_{1}^{(2)}-n_{1}^{(2)}=2\iota k\frac{\partial \phi_{1}^{(1)}}{\partial\xi} $$

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For the m=2 and l=0 components of the third-order part of the reduced equations determine the following second order quantities in the zeroth harmonic:

$$ \begin{array}{@{}l} n_{0}^{(2)}= A_{1}\bigl|\phi_{1}^{(1)}\bigr|^2,\qquad u_{0}^{(2)}= A_{2}\bigl|\phi_{1}^{(1)}\bigr|^2,\\[2mm] \phi_{0}^{(2)}= A_{3}\bigl|\phi_{1}^{(1)}\bigr|^2 \end{array} $$

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The components of l=2 for second order (m=2) equations determine the second harmonic quantities as

$$ \begin{array}{@{}l} n_{2}^{(2)}= B_{1}\bigl(\phi_{1}^{(1)} \bigr)^2,\qquad u_{2}^{(2)}= B_{2}\bigl( \phi_{1}^{(1)}\bigr)^2,\\[2mm] \phi_{2}^{(2)}= B_{3}\bigl(\phi_{1}^{(1)}\bigr)^2 \end{array} $$

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where the coefficients A ₁, A ₂, A ₃, B ₁, B ₂ and B ₃ appearing in the above equations are given as:

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