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Year 2025, Volume: 6 Issue: 2, 218 - 234, 30.07.2025

Fatma Nur Kaya Sağlam

https://doi.org/10.54974/fcmathsci.1652460

Abstract

References

  • Ala V., Demirbilek U., Mamedov K.R., An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation, Aims Mathematics, 5(4), 3751- 3761, 2020.
  • Ahmad J., Mustafa Z., Anwar M., Kouki M., Shah N.A., Exploring solitonic wave dynamics in the context of nonlinear conformable Kairat-X equation via unified method, AIMS Mathematics, 10(5), 10898-10916, 2025.
  • Ali S., Ahmad S., Ullah A., Ahmad S., Analyzing abundant optical soliton solutions for coupled nonlinear Helmholtz system arising in optics communication, Qualitative Theory of Dynamical Systems, 24(3), 115, 2025.
  • Aljoufi L.S., Almatrafi M.B., Seadawy A.R., Dynamical analysis of discrete time equations with a generalized order, Alexandria Engineering Journal, 64, 937-945, 2023.
  • Alquran M., Al-Khaled K., The tanh and sine–cosine methods for higher order equations of Korteweg– de Vries type, Physica Scripta, 84(2), 025010, 2011.
  • Behera S., Analysis of traveling wave solutions of two space-time nonlinear fractional differential equations by the first-integral method, Modern Physics Letters B, 38(04), 2350247, 2024.
  • Bibi I., Muhammad S., Shakeel M., Ceesay B., Optical soliton structure solutions, sensitivity, and modulation stability analysis in the chiral nonlinear Schr¨odinger equation with Bohm potential, Advances in Mathematical Physics, 2025(1), 9185387, 2025.
  • Biswas, A., Temporal 1-soliton solution of the complex Ginzburg-Landau equation with power law nonlinearity, Progress in Electromagnetics Research, 96, 1-7, 2009.
  • Debnath L., Nonlinear Partial Differential Equations for Scientists and Engineers, Birkh¨auser, 2005.
  • Dokuyucu M.A., Celik E., Bulut H., Baskonus H.M., Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133, 92, 2018.
  • Durur H., Exact solutions of the (3+1)-dimensional Khokhlov-Zabolotskaya-Kuznetsov equation via the modified sub-equation method, Modern Physics Letters B, 2450502, 2024.
  • Fan E., Zhang J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305(6), 383-392, 2002.
  • Faridi W.A., Tipu G.H., Riaz M.B., Mostafa A.M., Al Qahtani S.A., Myrzakulov R., Umurzakhova Z., Analyzing optical soliton solutions in Kairat-X equation via new auxiliary equation method, Optical and Quantum Electronics, 56(8), 1317, 2024.
  • Gündoğdu H., Gözükızıl Ö.F., Solving Benjamin-Bona-Mahony equation by using the sn-ns method and the tanh-coth method, Mathematica Moravica, 21(1), 95-103, 2017.
  • Hussain A., Chahlaoui Y., Zaman F.D., Parveen T., Hassan A.M., The Jacobi elliptic function method and its application for the stochastic NNV system, Alexandria Engineering Journal, 81, 347-359, 2023.
  • Isah M.A., Külah¸cı M.A., Special curves according to bishop frame in minkowski 3-space, Applied Mathematics and Nonlinear Sciences, 5(1), 237-248, 2020.
  • Iqbal M., Lu D., Seadawy A.R., Alomari F.A., Umurzakhova Z., Myrzakulov R., Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach, Modern Physics Letters B, 39(02), 2450396, 2025.
  • Murad M.A.S., Hamasalh F.K., Malik S., Arnous A.H., Iqbal M., Analysis of soliton solutions to the nonlinear conformable Schr¨odinger equation in weakly non-local media using two analytic algorithms, Nonlinear Dynamics, 113(10), 11881-11892, 2025.
  • Parkes E.J., Duffy B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications, 98(3), 288-300, 1996.
  • Kaya Sa˘glam F.N., New analytical wave structures for the (2+1)-dimensional Chaffee-Infante equation, Universal Journal of Mathematics and Applications, 8(1), 41-55, 2025.
  • Kopçasız B., Kaya Sağlam F.N., Exploration of soliton solutions for the Kaup–Newell model using two integration schemes in mathematical physics, Mathematical Methods in the Applied Sciences, 48(6), 6477-6487, 2025.
  • Kurt A., Ta¸sbozan O., Exact solution for nonlinear acoustics model with conformable derivative, Journal of Mathematical Sciences and Modelling, 8(1), 22-27, 2025.
  • Myrzakulova Z., Manukure S., Myrzakulov R., Nugmanova G., Integrability, geometry and wave solutions of some Kairat equations, arXiv:2307.00027, 2023.
  • Sahoo S., Ray S.S., Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations, Computers & Mathematics with Applications, 70(2), 158- 166, 2015.
  • Samina S., Munawar M., Ansari A.R., Jhangeer A., Wali S., Nonlinear optical dynamics and complex wave structures in nonlinear dispersive media, Scientific Reports, 15(1), 15562, 2025.
  • Sulaiman T.A., Yusuf A., Alshomrani A.S., Baleanu D., Lump collision phenomena to a nonlinear physical model in coastal engineering, Mathematics, 10(15), 2805, 2022.
  • Triki H., Wazwaz A.M., Soliton solution for an inhomogeneous highly dispersive media with a dualpower nonlinearity law, International Journal of Computer Mathematics, 87(5), 1178-1185, 2010.
  • ur Rahman M., Sun M., Boulaaras S., Baleanu, D., Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schr¨odinger equation, Boundary Value Problems, 2024(1), 15, 2024.
  • Wazwaz A.M., A sine-cosine method for handling nonlinear wave equations, Mathematical and Computer modelling, 40(5-6), 499-508, 2004.
  • Vaidya H., Tripathi D., Mebarek-Oudina F., Rajashekhar C., Baskonus H.M., Prasad K.V., Shivaleela, Scrutiny of MHD impact on Carreau Yasuda (CY) fluid flow over a heated wall of the uniform microchannel, Chinese Journal of Physics, 87, 766-781, 2024.
  • Younas U., Yao F., Nasreen N., Khan A., Abdeljawad T., Dynamics of M-truncated optical solitons and other solutions to the fractional Kudryashov’s equation, Results in Physics, 58, 107503, 2024.

Exploring the Novel Wave Structures of the Kairat-X Equation via Two Analytical Methods

Year 2025, Volume: 6 Issue: 2, 218 - 234, 30.07.2025

Fatma Nur Kaya Sağlam

https://doi.org/10.54974/fcmathsci.1652460

Abstract

This paper aims to investigate the Kairat-X equation in the context of the ferromagnetic materials, optical fibers, differential geometry of curves, and equivalence aspects. Two efficient techniques are used to obtain new solutions: the modified extended tanh expansion method and the ( G′/G2 )-expansion function method. By applying these methods, the nonlinear ordinary differential form of the analyzed equation is obtained using the appropriate wave transform. The effective application of the proposed approaches has yielded a substantial number of analytical solutions for the model, including hyperbolic, bright-dark soliton, W-shaped soliton, and mixed-type trigonometric, rational, and trigonometric solutions. These methods are advantageous in deriving a wide variety of exact solutions; however, they can also present limitations in terms of computational complexity and the scope of applicable equations. Various graphical representations are given to enhance the understanding of the obtained solutions. To the best of our knowledge, all derived solutions are novel. Furthermore, the correctness of each solution has been verified using Maple software.

Keywords

Kairat-X equation the modified extended tanh expansion method the $\left(\frac{G soliton solutions

Ethical Statement

The author declares that the materials and methods used in her study do not require ethical committee and/or legal special permission.

References

  • Ala V., Demirbilek U., Mamedov K.R., An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation, Aims Mathematics, 5(4), 3751- 3761, 2020.
  • Ahmad J., Mustafa Z., Anwar M., Kouki M., Shah N.A., Exploring solitonic wave dynamics in the context of nonlinear conformable Kairat-X equation via unified method, AIMS Mathematics, 10(5), 10898-10916, 2025.
  • Ali S., Ahmad S., Ullah A., Ahmad S., Analyzing abundant optical soliton solutions for coupled nonlinear Helmholtz system arising in optics communication, Qualitative Theory of Dynamical Systems, 24(3), 115, 2025.
  • Aljoufi L.S., Almatrafi M.B., Seadawy A.R., Dynamical analysis of discrete time equations with a generalized order, Alexandria Engineering Journal, 64, 937-945, 2023.
  • Alquran M., Al-Khaled K., The tanh and sine–cosine methods for higher order equations of Korteweg– de Vries type, Physica Scripta, 84(2), 025010, 2011.
  • Behera S., Analysis of traveling wave solutions of two space-time nonlinear fractional differential equations by the first-integral method, Modern Physics Letters B, 38(04), 2350247, 2024.
  • Bibi I., Muhammad S., Shakeel M., Ceesay B., Optical soliton structure solutions, sensitivity, and modulation stability analysis in the chiral nonlinear Schr¨odinger equation with Bohm potential, Advances in Mathematical Physics, 2025(1), 9185387, 2025.
  • Biswas, A., Temporal 1-soliton solution of the complex Ginzburg-Landau equation with power law nonlinearity, Progress in Electromagnetics Research, 96, 1-7, 2009.
  • Debnath L., Nonlinear Partial Differential Equations for Scientists and Engineers, Birkh¨auser, 2005.
  • Dokuyucu M.A., Celik E., Bulut H., Baskonus H.M., Cancer treatment model with the Caputo-Fabrizio fractional derivative, The European Physical Journal Plus, 133, 92, 2018.
  • Durur H., Exact solutions of the (3+1)-dimensional Khokhlov-Zabolotskaya-Kuznetsov equation via the modified sub-equation method, Modern Physics Letters B, 2450502, 2024.
  • Fan E., Zhang J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics Letters A, 305(6), 383-392, 2002.
  • Faridi W.A., Tipu G.H., Riaz M.B., Mostafa A.M., Al Qahtani S.A., Myrzakulov R., Umurzakhova Z., Analyzing optical soliton solutions in Kairat-X equation via new auxiliary equation method, Optical and Quantum Electronics, 56(8), 1317, 2024.
  • Gündoğdu H., Gözükızıl Ö.F., Solving Benjamin-Bona-Mahony equation by using the sn-ns method and the tanh-coth method, Mathematica Moravica, 21(1), 95-103, 2017.
  • Hussain A., Chahlaoui Y., Zaman F.D., Parveen T., Hassan A.M., The Jacobi elliptic function method and its application for the stochastic NNV system, Alexandria Engineering Journal, 81, 347-359, 2023.
  • Isah M.A., Külah¸cı M.A., Special curves according to bishop frame in minkowski 3-space, Applied Mathematics and Nonlinear Sciences, 5(1), 237-248, 2020.
  • Iqbal M., Lu D., Seadawy A.R., Alomari F.A., Umurzakhova Z., Myrzakulov R., Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach, Modern Physics Letters B, 39(02), 2450396, 2025.
  • Murad M.A.S., Hamasalh F.K., Malik S., Arnous A.H., Iqbal M., Analysis of soliton solutions to the nonlinear conformable Schr¨odinger equation in weakly non-local media using two analytic algorithms, Nonlinear Dynamics, 113(10), 11881-11892, 2025.
  • Parkes E.J., Duffy B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications, 98(3), 288-300, 1996.
  • Kaya Sa˘glam F.N., New analytical wave structures for the (2+1)-dimensional Chaffee-Infante equation, Universal Journal of Mathematics and Applications, 8(1), 41-55, 2025.
  • Kopçasız B., Kaya Sağlam F.N., Exploration of soliton solutions for the Kaup–Newell model using two integration schemes in mathematical physics, Mathematical Methods in the Applied Sciences, 48(6), 6477-6487, 2025.
  • Kurt A., Ta¸sbozan O., Exact solution for nonlinear acoustics model with conformable derivative, Journal of Mathematical Sciences and Modelling, 8(1), 22-27, 2025.
  • Myrzakulova Z., Manukure S., Myrzakulov R., Nugmanova G., Integrability, geometry and wave solutions of some Kairat equations, arXiv:2307.00027, 2023.
  • Sahoo S., Ray S.S., Improved fractional sub-equation method for (3+1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations, Computers & Mathematics with Applications, 70(2), 158- 166, 2015.
  • Samina S., Munawar M., Ansari A.R., Jhangeer A., Wali S., Nonlinear optical dynamics and complex wave structures in nonlinear dispersive media, Scientific Reports, 15(1), 15562, 2025.
  • Sulaiman T.A., Yusuf A., Alshomrani A.S., Baleanu D., Lump collision phenomena to a nonlinear physical model in coastal engineering, Mathematics, 10(15), 2805, 2022.
  • Triki H., Wazwaz A.M., Soliton solution for an inhomogeneous highly dispersive media with a dualpower nonlinearity law, International Journal of Computer Mathematics, 87(5), 1178-1185, 2010.
  • ur Rahman M., Sun M., Boulaaras S., Baleanu, D., Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schr¨odinger equation, Boundary Value Problems, 2024(1), 15, 2024.
  • Wazwaz A.M., A sine-cosine method for handling nonlinear wave equations, Mathematical and Computer modelling, 40(5-6), 499-508, 2004.
  • Vaidya H., Tripathi D., Mebarek-Oudina F., Rajashekhar C., Baskonus H.M., Prasad K.V., Shivaleela, Scrutiny of MHD impact on Carreau Yasuda (CY) fluid flow over a heated wall of the uniform microchannel, Chinese Journal of Physics, 87, 766-781, 2024.
  • Younas U., Yao F., Nasreen N., Khan A., Abdeljawad T., Dynamics of M-truncated optical solitons and other solutions to the fractional Kudryashov’s equation, Results in Physics, 58, 107503, 2024.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions, Applied Mathematics (Other)
Journal Section Research Articles
Authors

Fatma Nur Kaya Sağlam 0000-0001-7488-3254

Publication Date July 30, 2025
Submission Date March 6, 2025
Acceptance Date June 27, 2025
Published in Issue Year 2025 Volume: 6 Issue: 2

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