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Group classification, conservation laws and Painlevé analysis for Klein–Gordon–Zakharov equations in (3\(+\)1)-dimension

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Abstract

In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with determination of symmetry algebra, the one- and two-dimensional optimal systems are constructed, and optimality is also established using various invariant functions of full adjoint action. Apart from classification and construction of several conservation laws, the Painlevé analysis is also performed in a symbolic manner which describes the non-integrability of equations.

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Acknowledgements

Rajesh Kumar Gupta thanks the University Grants Commission for sponsoring this research under the Research Award Scheme (F. 30-105 / 2016 (SA-II)).

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Authors and Affiliations

  1. Yadavindra College of Engineering, Punjabi University, Guru Kashi Campus, Talwandi Sabo, 151 302, India

    Manjit Singh

  2. Centre for Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, 151 001, India

    R K Gupta

Authors
  1. Manjit Singh
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  2. R K Gupta
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Corresponding author

Correspondence to Manjit Singh.

Appendices

Appendix A. Maple code for full adjoint action

figure a

Appendix B

$$\begin{aligned} {\tilde{a}}_{1}= & {} a_{1},\\ {\tilde{a}}_{2}= & {} -\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{5}}\\&+\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{3}}\\&+\cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) a_{{3}}+\sin ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) a_{{5}}\\&+\cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{2}},\\ {\tilde{a}}_{3}= & {} \cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{5}}\\&-\sin ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{3}}\\&+\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) a_{{3}}+\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) a_{{5}}\\&-\sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{2}},\\ {\tilde{a}}_{4}= & {} a_{{4}}\cos ( \epsilon _{{2}} ) {\mathrm{e}^{\epsilon _{{1}}}} \cos ( \epsilon _{{3}} ) +a_{{3}}\cos ( \epsilon _{{2}} ) \epsilon _{{4}}\sin ( \epsilon _{{3}} ) \\&+\,a_{{5}}\cos ( \epsilon _{{2}} ) \epsilon _{{6}}\sin ( \epsilon _{{3} } )-a_{{3}}\cos ( \epsilon _{{2}} ) \epsilon _{{7}} \cos ( \epsilon _{{3}} )\\&+\,a_{{6}}\sin ( \epsilon _{{2}} ) {\mathrm{e}^{\epsilon _{{1}}}}\cos ( \epsilon _{{3}} ) -a_{{3}}\sin ( \epsilon _{{2}} ) \epsilon _{{6}}\sin ( \epsilon _{{3}} )\\&+\,a_{{5}}\sin ( \epsilon _{{2}} ) \epsilon _{{4}}\sin ( \epsilon _{{3}} ) -a_{{5}}\sin ( \epsilon _{{2}} ) \epsilon _{{7}}\cos ( \epsilon _{{3}} ) \\&+\,a_{{7}}{\mathrm{e}^{\epsilon _{{1}}}}\sin ( \epsilon _{{3}} ) -a_{{1}}\epsilon _{{7}}\sin ( \epsilon _{{3}} )\\&-\,a_{ {1}}\epsilon _{{4}}\cos ( \epsilon _{{3}} ) -a_{{2}}\epsilon _ {{6}}\cos ( \epsilon _{{3}} ),\\ {\tilde{a}}_{5}= & {} a_{{5}}\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{3}} ) -a_{{3}}\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{3}} )\\&+a_{{2}}\sin ( \epsilon _{{3}} ),\\ {\tilde{a}}_{6}= & {} -\cos ( \epsilon _{{2}} ) {\mathrm{e}^{\epsilon _{{1}}}}\sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{ 4}}\\&+\cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{3}}\epsilon _{{7}}\\&+\cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{3}}\epsilon _{{4}}\\&+\cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{6}}\\&-\sin ( \epsilon _{{2} } ) {\mathrm{e}^{\epsilon _{{1}}}}\sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{6}}\\&+\sin ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{7}}\\&-\sin ( \epsilon _{{2} } ) \sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3} } ) a_{{3}}\epsilon _{{6}}\\&+\sin ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{4}}\\&+\cos ( \epsilon _{{2}} ) {\mathrm{e}^{ \epsilon _{{1}}}}\cos ( \epsilon _{{5}} ) a_{{6}}-\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) a_{{ 5}}\epsilon _{{7}}\\&-\sin ( \epsilon _{{2}} ) {\mathrm{e}^{ \epsilon _{{1}}}}\cos ( \epsilon _{{5}} ) a_{{4}}+\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) a_{{ 3}}\epsilon _{{7}}\\&+\,{\mathrm{e}^{\epsilon _{{1}}}}\sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{7}}\\&+\sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{1}} \epsilon _{{4}}+\sin ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{2}}\epsilon _{{6}}\\&-\sin ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{1}}\epsilon _{{7}}\\&-\cos ( \epsilon _{{5}} ) a_{{1}}\epsilon _{{6}}+\cos ( \epsilon _{{5}} ) a_{{2}}\epsilon _{{4}},\\ {\tilde{a}}_{7}= & {} -\cos ( \epsilon _{{2}} ) {\mathrm{e}^{\epsilon _{{1}}}}\cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{ 4}}\\&+\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{3}}\epsilon _{{7}}\\&+\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{3}}\epsilon _{{4}}\\&+\cos ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{6}}\\&-\sin ( \epsilon _{{2} } ) {\mathrm{e}^{\epsilon _{{1}}}}\cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{6}}\\&+\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{7}}\\&-\sin ( \epsilon _{{2} } ) \cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3} } ) a_{{3}}\epsilon _{{6}}\\&+\sin ( \epsilon _{{2}} ) \cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{5}}\epsilon _{{4}}\\&-\cos ( \epsilon _{{2}} ) {\mathrm{e}^{ \epsilon _{{1}}}}\sin ( \epsilon _{{5}} ) a_{{6}}+\cos ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) a_{{ 5}}\epsilon _{{7}}\\&+\sin ( \epsilon _{{2}} ) {\mathrm{e}^{ \epsilon _{{1}}}}\sin ( \epsilon _{{5}} ) a_{{4}}\\&-\sin ( \epsilon _{{2}} ) \sin ( \epsilon _{{5}} ) a_{{ 3}}\epsilon _{{7}}+{\mathrm{e}^{\epsilon _{{1}}}}\cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{7}}\\&+\cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{1}} \epsilon _{{4}}\\&+\cos ( \epsilon _{{5}} ) \sin ( \epsilon _{{3}} ) a_{{2}}\epsilon _{{6}}-\cos ( \epsilon _{{5}} ) \cos ( \epsilon _{{3}} ) a_{{1}}\epsilon _{{7}}\\&+\sin ( \epsilon _{{5}} ) a_{{1}}\epsilon _{{6}}-\sin ( \epsilon _{{5}} ) a_{{2}}\epsilon _{{4}},\\ {\tilde{a}}_{8}= & {} -{\mathrm{e}^{\epsilon _{{1}}}}a_{{1}}\epsilon _{{8}}+{\mathrm{e}^{\epsilon _{{ 1}}}}a_{{8}},\\ {\tilde{a}}_{9}= & {} a_{9}. \end{aligned}$$

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Singh, M., Gupta, R.K. Group classification, conservation laws and Painlevé analysis for Klein–Gordon–Zakharov equations in (3\(+\)1)-dimension. Pramana - J Phys 92, 1 (2019). https://doi.org/10.1007/s12043-018-1665-3

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