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Ion-Acoustic Solitons in Magnetized Plasma Under Weak Relativistic Effects on the Electrons

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Abstract

Investigating ion-acoustic disturbances in a magnetized plasma, consisting of relativistic electrons and non-thermal ions, entails a comprehensive study into the nonlinear wave structure. By condensing the fundamental set of fluid equations for the flow variables, a singular equation known as the Sagdeev potential equation is derived using the pseudopotential approach. In this investigation of the magnetized relativistic plasma, we have observed only dip (rarefactive) \(\left( {N < 1} \right)\) soliton under both subsonic \(\left( {M < 1} \right)\) and supersonic \(\left( {M > 1} \right)\) conditions. The occurrence of the soliton depends on the wave velocities in different propagation directions. The magnitude of amplitudes of the relativistic solitons is higher for higher Mach number \(\left( {M > 1} \right)\) irrespective of the wave’s propagation direction. Furthermore, the magnitude of amplitudes of the solitary wave is seen to increase near the direction of the magnetic field.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India

    B. Madhukalya

  2. Department of Mathematics, Arya Vidyapeeth College, Guwahati, 781016, Assam, India

    R. Das

  3. Department of Mathematics, Near East University TRNC, Mersin 10, Turkey

    K. Hosseini

  4. Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, 06530, Turkey

    D. Baleanu

  5. Institute of Space Sciences, Magurele, Bucharest, Romania

    D. Baleanu

  6. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    D. Baleanu & S. Salahshour

  7. Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

    S. Salahshour

  8. Faculty of Science and Letters, Piri Reis University, Tuzla, Istanbul, Turkey

    S. Salahshour

Authors
  1. B. Madhukalya
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  2. R. Das
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  3. K. Hosseini
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  4. D. Baleanu
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  5. S. Salahshour
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Contributions

BM—Writing—original draft, Methodology, Formal analysis; RD—Conceptualization, Writing—original draft, Formal analysis; KH—Writing – original draft, Methodology, Writing—review & editing; DB—Writing—review & editing; SS—Writing—review & editing

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Correspondence to K. Hosseini.

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Appendix

Appendix

Multiplying both sides of Eq. (17) by the expression in the parenthesis of left hand side, we obtain

$$ \frac{1}{2}\frac{d}{d\xi }\left\{ {f\left( n \right)\frac{1}{n}\frac{dn}{{d\xi }}} \right\}^{2} = \frac{1}{{k_{x} }}\left( {1 - \frac{1}{n}} \right)\left\{ {1 + Q - \frac{{k_{z}^{2} }}{{M^{2} }}.\,n + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}.\,\left( {1 - \frac{1}{n}} \right)^{2} } \right\}f\left( n \right)\frac{dn}{{d\xi }} $$

By integrating, we get

$$ \begin{gathered} \frac{1}{2}\left\{ {f\left( n \right)\frac{1}{n}\frac{dn}{{d\xi }}} \right\}^{2} = \frac{1}{{k_{x} }}\int {\left( {1 - \frac{1}{n}} \right)\left\{ {1 + Q - \frac{{k_{z}^{2} }}{{M^{2} }}.\,n + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}.\,\left( {1 - \frac{1}{n}} \right)^{2} } \right\}f\left( n \right)dn} \hfill \\ \quad \quad \quad \quad \quad \quad \quad = \,\frac{1}{{k_{x} }}\int {\left( {1 - \frac{1}{n}} \right)\left\{ {1 + Q - \frac{{k_{z}^{2} }}{{M^{2} }}.\,n + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}.\,\left( {1 - \frac{1}{n}} \right)^{2} } \right\}\left( {T_{1} + \frac{{T_{2} }}{{n^{2} }} + \frac{{T_{3} }}{{n^{3} }} + \frac{{T_{4} }}{{n^{4} }}} \right)dn} \hfill \\ \end{gathered} $$
$$\begin{gathered} \frac{1}{2}\left\{ {f\left( n \right)} \right\}^{2} \frac{1}{{n^{2} }}\left( {\frac{dn}{{d\xi }}} \right)^{2} = \frac{1}{{k_{x} }}\left[ {\left\{ {\left( {1 + Q + \frac{{k_{z}^{2} }}{{M^{2} }} + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{1} } \right\}\left( {n - 1} \right) - \left( {\frac{{T_{1} k_{z}^{2} }}{{2M^{2} }}} \right)} \right.\left( {n^{2} - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left\{ {\frac{{T_{2} k_{z}^{2} }}{{M^{2} }} + \left( {1 + Q + \frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{1} } \right\}\log n \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left\{ {\left( {1 + Q + \frac{{k_{z}^{2} }}{{M^{2} }} + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{2} + \left( {\frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{1} - \frac{{T_{3} k_{z}^{2} }}{{M^{2} }}} \right\}\left( {\frac{1}{n} - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{1}{2}\left\{ {\left( {1 + Q + \frac{{k_{z}^{2} }}{{M^{2} }} + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{3} - \left( {1 + Q + \frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{2} - \left( {\frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{1} - \frac{{T_{4} k_{z}^{2} }}{{M^{2} }}} \right\}\left( {\frac{1}{{n^{2} }} - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{1}{3}\left\{ {\left( {1 + Q + \frac{{k_{z}^{2} }}{{M^{2} }} + \frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{4} - \left( {1 + Q + \frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{3} + \left( {\frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{2} } \right\}\left( {\frac{1}{{n^{3} }} - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{1}{4}\left\{ {\left( {1 + Q + \frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{4} - \left( {\frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{3} + \left( {\frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{2} } \right\}\left( {\frac{1}{{n^{4} }} - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{1}{5}\left\{ {\left( {\frac{{3QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{4} - \left( {\frac{{QM^{2} }}{{2c^{2} k_{z}^{2} }}} \right)T_{3} } \right\}\left( {\frac{1}{{n^{5} }} - 1} \right) + \left. {\left( {\frac{{T_{4} QM^{2} }}{{12c^{2} k_{z}^{2} }}} \right)\left( {\frac{1}{{n^{6} }} - 1} \right)} \right] \, \hfill \\ \end{gathered}$$

After simplification, we find the exact from of Eq. (18).

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Madhukalya, B., Das, R., Hosseini, K. et al. Ion-Acoustic Solitons in Magnetized Plasma Under Weak Relativistic Effects on the Electrons. Int. J. Appl. Comput. Math 9, 102 (2023). https://doi.org/10.1007/s40819-023-01579-3

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