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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17586/2220-8054-2018-9-5-581-585</article-id><article-id custom-type="elpub" pub-id-type="custom">najo-664</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>MATHEMATICS</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>МАТЕМАТИКА</subject></subj-group></article-categories><title-group><article-title>New extended Jacobi elliptic function expansion scheme for wave-wave interaction in ionic media</article-title><trans-title-group xml:lang="ru"><trans-title></trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Pankaj</surname><given-names>R. D.</given-names></name></name-alternatives><bio xml:lang="en"><p>Jodhpur</p></bio><email xlink:type="simple">drrdpankaj@yahoo.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Singh</surname><given-names>B.</given-names></name></name-alternatives><bio xml:lang="en"><p>Jodhpur</p></bio><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Kumar</surname><given-names>A.</given-names></name></name-alternatives><bio xml:lang="en"><p>Kota (Raj.)</p></bio><xref ref-type="aff" rid="aff-2"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>Department of Mathematics, J. N. V. University</institution><country>India</country></aff><aff xml:lang="en" id="aff-2"><institution>2Department of Mathematics, Government College Kota</institution><country>India</country></aff><pub-date pub-type="collection"><year>2018</year></pub-date><pub-date pub-type="epub"><day>12</day><month>08</month><year>2025</year></pub-date><volume>9</volume><issue>5</issue><fpage>581</fpage><lpage>585</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Pankaj R.D., Singh B., Kumar A., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Pankaj R.D., Singh B., Kumar A.</copyright-holder><copyright-holder xml:lang="en">Pankaj R.D., Singh B., Kumar A.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://nanojournal.ifmo.ru/jour/article/view/664">https://nanojournal.ifmo.ru/jour/article/view/664</self-uri><abstract><p>New Jacobi Elliptic functions expansion scheme, more general than the hyperbolic tangent function method, is derived to construct the exact wave solutions in terms of Jacobi Elliptic functions. The coupled 1D nonlinear Schrodinger–Zakharov (CNLSZ) system is taken as the model ¨ equation for wave-wave interaction in ionic media. It is shown that more new solutions can be obtained at their limit condition. </p></abstract><kwd-group xml:lang="en"><kwd>the coupled 1D nonlinear Schrodinger–Zakharov (CNLSZ) system</kwd><kwd>Jacobi elliptic function expansion scheme</kwd><kwd>hyperbolic tangent function expansion</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Langmuir I. Oscillations in Ionized Gases. Proc. Natl. Acad. Sci. U.S.A., 1928, 14 (8), P. 627–637.</mixed-citation><mixed-citation xml:lang="en">Langmuir I. Oscillations in Ionized Gases. Proc. Natl. Acad. Sci. U.S.A., 1928, 14 (8), P. 627–637.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Tonks L., Langmuir I. Oscillations in Ionized Gases. Physical Review, 1929, 33, P. 195–210.</mixed-citation><mixed-citation xml:lang="en">Tonks L., Langmuir I. Oscillations in Ionized Gases. Physical Review, 1929, 33, P. 195–210.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Chen F.F. Introduction to Plasma Physics, Plenum, New York, 1974.</mixed-citation><mixed-citation xml:lang="en">Chen F.F. Introduction to Plasma Physics, Plenum, New York, 1974.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Scott C., Chu F.Y.F., Mcglaughlin D.W. The soliton: a new concept in applied science. Proc. IEEE, 1973, 61, P. 1443–1483.</mixed-citation><mixed-citation xml:lang="en">Scott C., Chu F.Y.F., Mcglaughlin D.W. The soliton: a new concept in applied science. Proc. IEEE, 1973, 61, P. 1443–1483.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Zakharov V.E. Collapse of Langmuir Waves. JEPT, 1972, 35 (5), P. 908–914.</mixed-citation><mixed-citation xml:lang="en">Zakharov V.E. Collapse of Langmuir Waves. JEPT, 1972, 35 (5), P. 908–914.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Pereira N.R., Sudan R.N., Denavit J. ”Numerical study of two-dimensional generation and collapse of Langmuir solitons. The Physics of Fluids, 1977, 20 (6), P. 936–945.</mixed-citation><mixed-citation xml:lang="en">Pereira N.R., Sudan R.N., Denavit J. ”Numerical study of two-dimensional generation and collapse of Langmuir solitons. The Physics of Fluids, 1977, 20 (6), P. 936–945.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Nicholson D.R., et al. Nonlinear Langmuir waves during type III solar radio bursts. The Astrophysical Journal, 1978, 223, P. 605–619.</mixed-citation><mixed-citation xml:lang="en">Nicholson D.R., et al. Nonlinear Langmuir waves during type III solar radio bursts. The Astrophysical Journal, 1978, 223, P. 605–619.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Goldman M.V., Nicholson D.R. Virial theory of direct Langmuir collapse. Physical Review Letters, 1978, 41 (6), P. 406.</mixed-citation><mixed-citation xml:lang="en">Goldman M.V., Nicholson D.R. Virial theory of direct Langmuir collapse. Physical Review Letters, 1978, 41 (6), P. 406.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Nicholson R., Goldman M.V. Cascade and collapse of Langmuir waves in two dimensions. Phys. Fluids, 1978, 21 (10), P. 1766–1776.</mixed-citation><mixed-citation xml:lang="en">Nicholson R., Goldman M.V. Cascade and collapse of Langmuir waves in two dimensions. Phys. Fluids, 1978, 21 (10), P. 1766–1776.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Hasegawa A. Stimulated Modulational Instabilities of Plasma Waves. Phys. Rev. A, 1970, 1 (6), P. 1746.</mixed-citation><mixed-citation xml:lang="en">Hasegawa A. Stimulated Modulational Instabilities of Plasma Waves. Phys. Rev. A, 1970, 1 (6), P. 1746.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Thyagaraja A. Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrodinger equation. ¨ Phys. of Fluids, 1981, 24 (11), P. 1973–1975.</mixed-citation><mixed-citation xml:lang="en">Thyagaraja A. Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrodinger equation. ¨ Phys. of Fluids, 1981, 24 (11), P. 1973–1975.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Russell D.A., Ott E. Chaotic (strange) and periodic behavior in instability saturation by the oscillating twostream instability. The Physics of Fluids, 1981, 24 (11), P. 1976–1988.</mixed-citation><mixed-citation xml:lang="en">Russell D.A., Ott E. Chaotic (strange) and periodic behavior in instability saturation by the oscillating twostream instability. The Physics of Fluids, 1981, 24 (11), P. 1976–1988.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Weatherall J.C., et al. Solitons and ionospheric heating. Journal of Geophysical Research: Space Physics, 1982, 87 (A2), P. 823–832.</mixed-citation><mixed-citation xml:lang="en">Weatherall J.C., et al. Solitons and ionospheric heating. Journal of Geophysical Research: Space Physics, 1982, 87 (A2), P. 823–832.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Vakhnenko V.O., Parkes E.J., Morrison A.J. A Backlund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons &amp; Fractals, 2003, 17 (4), P. 683–692.</mixed-citation><mixed-citation xml:lang="en">Vakhnenko V.O., Parkes E.J., Morrison A.J. A Backlund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos, Solitons &amp; Fractals, 2003, 17 (4), P. 683–692.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Ye-peng Sun, Jin-bo Bi, Deng-yuan Chen. N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation. Chaos, Solitons &amp; Fractals, 2005, 26 (3), P. 905–912.</mixed-citation><mixed-citation xml:lang="en">Ye-peng Sun, Jin-bo Bi, Deng-yuan Chen. N-soliton solutions and double Wronskian solution of the non-isospectral AKNS equation. Chaos, Solitons &amp; Fractals, 2005, 26 (3), P. 905–912.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Sakka A. Backland transformations for Painleve I and II equations to Painleve-type equations of second order and higher degree. Phys. Lett. A, 2002, 300 (2–3), P. 228–232.</mixed-citation><mixed-citation xml:lang="en">Sakka A. Backland transformations for Painleve I and II equations to Painleve-type equations of second order and higher degree. Phys. Lett. A, 2002, 300 (2–3), P. 228–232.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Yao R.X., Li Z.B. New exact solutions for three nonlinear evolution equations. Phys. Lett. A, 2002, 297, P. 196–204.</mixed-citation><mixed-citation xml:lang="en">Yao R.X., Li Z.B. New exact solutions for three nonlinear evolution equations. Phys. Lett. A, 2002, 297, P. 196–204.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A. An analytical solution for a coupled partial differential equation. Applied Mathematics and Computation, 2009, 212, P. 245–250.</mixed-citation><mixed-citation xml:lang="en">Kumar A. An analytical solution for a coupled partial differential equation. Applied Mathematics and Computation, 2009, 212, P. 245–250.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Liu Z., Chen C. Compactons in a general compressible hyper elastic rod. Chaos, Solitons &amp; Fractals, 2004, 22 (3), P. 627–640.</mixed-citation><mixed-citation xml:lang="en">Liu Z., Chen C. Compactons in a general compressible hyper elastic rod. Chaos, Solitons &amp; Fractals, 2004, 22 (3), P. 627–640.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Kaya D., El-Sayed S.M. An application of the decomposition method for the generalized KdV and RLW equations. Chaos, Solitons &amp; Fractals, 2003, 17 (5), P. 869–877.</mixed-citation><mixed-citation xml:lang="en">Kaya D., El-Sayed S.M. An application of the decomposition method for the generalized KdV and RLW equations. Chaos, Solitons &amp; Fractals, 2003, 17 (5), P. 869–877.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Hagtae Kim, Dong Pyo Hong, Kil to Chong. A numerical solution of point kinetics equations using the Adomian Decomposition Method. Systems and Informatics (ICSAI), 2012 International Conference on IEEE 19–20 May 2012, 12835470.</mixed-citation><mixed-citation xml:lang="en">Hagtae Kim, Dong Pyo Hong, Kil to Chong. A numerical solution of point kinetics equations using the Adomian Decomposition Method. Systems and Informatics (ICSAI), 2012 International Conference on IEEE 19–20 May 2012, 12835470.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Ram D.P. Solitary Wave Solutions of Schrodinger Equation by Laplace–Adomian Decomposition Method. ¨ Physical Review &amp; Research International, 2013, 3 (4), P. 702–712.</mixed-citation><mixed-citation xml:lang="en">Kumar A., Ram D.P. Solitary Wave Solutions of Schrodinger Equation by Laplace–Adomian Decomposition Method. ¨ Physical Review &amp; Research International, 2013, 3 (4), P. 702–712.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Ghoreishi M., Ismail A.I.B., Rashid A. Numerical Solution of Klein–Gordon–Zakharov Equations using Chebyshev Cardinal Functions. Journal of Computational Analysis &amp; Applications, 2012, 14 (1), P. 574–582.</mixed-citation><mixed-citation xml:lang="en">Ghoreishi M., Ismail A.I.B., Rashid A. Numerical Solution of Klein–Gordon–Zakharov Equations using Chebyshev Cardinal Functions. Journal of Computational Analysis &amp; Applications, 2012, 14 (1), P. 574–582.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Zhao Xiaofei, Ziyi Li. Numerical Methods and Simulations for the Dynamics of One-Dimensional Zakharov-Rubenchik Equations. Journal of Scientific Computing, 2013, 59 (2), P. 412–438.</mixed-citation><mixed-citation xml:lang="en">Zhao Xiaofei, Ziyi Li. Numerical Methods and Simulations for the Dynamics of One-Dimensional Zakharov-Rubenchik Equations. Journal of Scientific Computing, 2013, 59 (2), P. 412–438.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Ram D.P., Manish G. Finite difference scheme of a model for nonlinear wave-wave interaction in ionic media. Computational Mathematics and Modeling, 2011, 22 (3), P. 255–265.</mixed-citation><mixed-citation xml:lang="en">Kumar A., Ram D.P., Manish G. Finite difference scheme of a model for nonlinear wave-wave interaction in ionic media. Computational Mathematics and Modeling, 2011, 22 (3), P. 255–265.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Ram D.P. Finite Difference Scheme for the Zakharov Equation as a Model for Nonlinear Wave-Wave Interaction in Ionic Media. International Journal of Scientific &amp; Engineering Research, 2014, 5 (2), P. 759–762.</mixed-citation><mixed-citation xml:lang="en">Kumar A., Ram D.P. Finite Difference Scheme for the Zakharov Equation as a Model for Nonlinear Wave-Wave Interaction in Ionic Media. International Journal of Scientific &amp; Engineering Research, 2014, 5 (2), P. 759–762.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Kumar A., Ram D.P. Solitary Wave Solutions of Schrodinger Equation by Laplace–Adomian Decomposition Method. ¨ Physical Review &amp; Research International, 2013, 3 (4), P. 702–712.</mixed-citation><mixed-citation xml:lang="en">Kumar A., Ram D.P. Solitary Wave Solutions of Schrodinger Equation by Laplace–Adomian Decomposition Method. ¨ Physical Review &amp; Research International, 2013, 3 (4), P. 702–712.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
