Face to Face Collisions of Ion Acoustic Multi-Solitons and Phase Shifts in a Dense Plasma

Abstract

This work investigates the interactions among ion acoustic (IA) single- and multi-soliton and their corresponding phase shifts in an unmagnetized plasma composed of degenerate electrons, positrons, and positive ions. Two-sided Korteweg-de Vries (KdV) equations are derived by employing the extended Poincaré-Lighthill-Kuo (PLK) method for the stretched coordinates. The single- and multi-soliton solutions of the KdV equations are constructed by using the Hirota’s method. The phase shifts are determined for two-, four-, six-, and eight-IA scattering solitons. The effect of positron concentration on electrostatic IA resonances due to the interactions among solitons and their corresponding phase shifts are investigated.

Appendix

$$ \frac{\partial {n}_i^{(2)}}{\partial \tau }-{V}_p\ \left(\frac{\partial }{\partial \xi }-\frac{\partial }{\partial \eta}\right){n}_i^{(4)}+\left(1-\alpha \right)\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right){u}_i^{(4)}+\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right)\left[{n}_i^{(2)}\ {u}_i^{(2)}\right]+{V}_{p\kern0.5em }\left(\frac{\partial {P}_0}{\partial \eta }-\frac{\partial {P}_0}{\partial \xi}\right)\frac{\partial {n}_i^{(2)}}{\partial \xi }+\left(1-\alpha \right)\left(\frac{\partial {P}_0}{\partial \eta }+\frac{\partial {P}_0}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \xi }+{V}_{p\kern0.5em }\left(\frac{\partial {Q}_0}{\partial \eta }-\frac{\partial {Q}_0}{\partial \xi}\right)\frac{\partial {n}_i^{(2)}}{\partial \eta }+\left(1-\alpha \right)\left(\frac{\partial {Q}_0}{\partial \eta }+\frac{\partial {Q}_0}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \eta }=0\kern0.5em $$

(43)

$$ \frac{\partial {u}_i^{(2)}}{\partial \tau }-{V}_p\ \left(\frac{\partial }{\partial \xi }-\frac{\partial }{\partial \eta}\right){u}_i^{(4)}+\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right){\varphi}^{(4)}+\frac{1}{2}\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right){\left\{{u}_i^{(2)}\right\}}^2+{V}_{p\kern0.5em }\left(\frac{\partial {P}_0}{\partial \eta }-\frac{\partial {P}_0}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \xi }+{V}_{p\kern0.5em }\left(\frac{\partial {Q}_0}{\partial \eta }-\frac{\partial {Q}_0}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \eta }+\left(\frac{\partial {P}_0}{\partial \eta }+\frac{\partial {P}_0}{\partial \xi}\right)\frac{\partial {\varphi}^{(2)}}{\partial \xi }+\left(\frac{\partial {Q}_0}{\partial \eta }+\frac{\partial {Q}_0}{\partial \xi}\right)\frac{\partial {\varphi}^{(2)}}{\partial \eta }=0 $$

(44)

$$ \frac{\partial^2{\varphi}^{(2)}}{\partial {\xi}^2}\kern0.75em +2\frac{\partial^2{\varphi}^{(2)}}{\partial \xi \partial \eta }+\frac{\partial^2{\varphi}^{(2)}}{\partial {\eta}^2}\kern0.5em =\frac{3}{2}\left(1+\alpha {\sigma}_F\right){\varphi}^{(4)}+\frac{3}{8}\left(1-\alpha {\sigma}_F^2\right)\ {\left\{{\varphi}^{(2)}\right\}}^2-{n}_i^{(4)} $$

(45)

$$ R=\iint \frac{\partial }{\partial \xi}\left\{\left(C\frac{\partial {P}_0}{\partial \eta }-D{\varphi}_{\eta}^{(2)}\right)\frac{\partial {\varphi}_{\xi}^{(2)}}{\partial \xi}\right\} d\xi d\eta -\iint \frac{\partial }{\partial \eta}\left\{\left(C\frac{\partial {Q}_0}{\partial \xi }-D{\varphi}_{\xi}^{(2)}\right)\frac{\partial {\varphi}_{\eta}^{(2)}}{\partial \eta}\right\} d\xi d\eta +\iint \frac{\partial^2}{\partial \xi \partial \eta}\left\{\widehat{\varphi^{(4)}}+\left(\frac{3\left(1-\alpha {\sigma}_F^2\right){V}_p^2}{8\left(1-\alpha \right)}-\frac{1}{2{V}_p^2}\right){\varphi}_{\xi}^{(2)}{\varphi}_{\eta}^{(2)}\right\} d\xi d\eta $$

(46)

$$ -{V}_p\ \left(\frac{\partial }{\partial \xi }-\frac{\partial }{\partial \eta}\right){n}_i^{(5)}+\left(1-\alpha \right)\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right){u}_i^{(5)}+{V}_{p\kern0.5em }\left(\frac{\partial {P}_1}{\partial \eta }-\frac{\partial {P}_1}{\partial \xi}\right)\frac{\partial {n}_i^{(2)}}{\partial \xi }+{V}_{p\kern0.5em }\left(\frac{\partial {Q}_1}{\partial \eta }-\frac{\partial {Q}_1}{\partial \xi}\right)\frac{\partial {n}_i^{(2)}}{\partial \eta }+\left(1-\alpha \right)\left(\frac{\partial {P}_1}{\partial \eta }+\frac{\partial {P}_1}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \xi }+\left(1-\alpha \right)\left(\frac{\partial {Q}_1}{\partial \eta }+\frac{\partial {Q}_1}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \eta }=0 $$

(47)

$$ -{V}_p\ \left(\frac{\partial }{\partial \xi }-\frac{\partial }{\partial \eta}\right){u}_i^{(5)}+\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta}\right){\varphi}^{(5)}+{V}_{p\kern0.5em }\left(\frac{\partial {P}_1}{\partial \eta }-\frac{\partial {P}_1}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \xi }+{V}_{p\kern0.5em }\left(\frac{\partial {Q}_1}{\partial \eta }-\frac{\partial {Q}_1}{\partial \xi}\right)\frac{\partial {u}_i^{(2)}}{\partial \eta }+\left(\frac{\partial {P}_1}{\partial \eta }+\frac{\partial {P}_1}{\partial \xi}\right)\frac{\partial {\varphi}^{(2)}}{\partial \xi }+\left(\frac{\partial {Q}_1}{\partial \eta }+\frac{\partial {Q}_1}{\partial \xi}\right)\frac{\partial {\varphi}^{(2)}}{\partial \eta }=0 $$

(48)

$$ {n}_i^{(5)}=\frac{3}{2}\left(1+\alpha {\sigma}_F\right){\varphi}^{(5)} $$

(49)

Simplifying Eqs. (47), (48), and (49), one obtains

$$ \iint 2\left(1-\alpha \right)\frac{\partial {P}_1}{\partial \eta}\frac{\partial {\varphi}_{\xi}^{(2)}}{\partial \xi } d\xi d\eta +..=0,\kern0.5em $$

(50)

which is indicated that the remaining part R as in Eq. (46) would be generated secularities. Hence, the remaining part R must be vanished to reduce the secularities.