A generalized $(G'/G)$-expansion method for the loaded
Sibirskij žurnal industrialʹnoj matematiki, Tome 24 (2021) no. 4, pp. 139-147.

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The article is devoted to find the solutions of the loaded Korteweg—de Vries equation. It is shown that to find the solutions using the $(G'/G)$-expansion method is one of the most effective methods for finding solutions to integrable nonlinear evolution equations due to the convenience of using known software packages in comparison with other known methods such as the method Hirota, Darboux transforms, the inverse scattering method, etc.
Keywords: loaded Korteweg—de Vries equations, nonlinear evolution equations, expansion method, direct methods, traveling wave.
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G. U. Urazboev; I. I. Baltaeva; I. D. Rakhimov. A generalized $(G'/G)$-expansion method for the loaded. Sibirskij žurnal industrialʹnoj matematiki, Tome 24 (2021) no. 4, pp. 139-147. http://geodesic.mathdoc.fr/item/SJIM_2021_24_4_a9/

[1] N. A. Kudryashov, I. L. Chernyavskii, “Nonlinear waves in fluid flow through a viscoelastic tube”, Fluid Dynamics, 41:1 (2006), 49–62 | DOI | Zbl

[2] H. Demiray, “Variable coefficient modified KdV equation in fluid-filled elastic tubes with stenosis: Solitary waves”, Chaos Soliton Fract., 42:1 (2009), 358–364 | DOI | Zbl

[3] B. B. Kadomtsev, V. I. Karpman, “Nonlinear waves”, Sov. Phys. Usp., 14 (1971), 40–60 | DOI

[4] A. M. Nakhushev, “On nonlocal problems with shift and their connection with loaded equations”, Differents. Uravn., 21:1 (1985), 92–101 | Zbl

[5] J. R. Cannon, H. M. Yin, “On a class of nonlinear nonclassical parabolic problems”, J. Different. Equat., 79 (1989), 266–288 | DOI | Zbl

[6] J. M. Chadam, A. Peirce, H. M. Yin, “The blowup property of solutions to some diffusion equations with localized nonlinear reactions”, J. Math. Analys. Appl., 169:2 (1992), 313–328 | DOI | Zbl

[7] U. Baltaeva, P. J. Torres, “Analog of the Gellerstedt problem for a loaded equation of the third order”, Mathematical Methods in the Applied Sciences, 42 (2011), 3865–3876 | DOI

[8] R. Hirota, “Exact solution of the Korteweg de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27 (1971), 1192–1194 | DOI | Zbl

[9] G. U. Urazboev, I. I. Baltayeva, “On integration of the general loaded Korteweg de Vries equation with a self-consistent source”, Instruments and Systems: Monitoring, Control, and Diagnostics, 10 (2019), 7–9

[10] A. B. Khasanov, G. U. Urazboev, “On the sine-Gordon equation with a self-consistent source”, Mat. Trudy, 11:1 (2008), 153–166 | Zbl

[11] C. Rogers, W. F. Shadwick, Backlund Transformations, Acad. Press, N. Y., 1982 | Zbl

[12] M. Wadati, H. Shanuki, K. Konno, “Relationships among inverse method, Backlund transformation and an infinite number of conservative laws”, Prog. Theor. Phys., 53 (1975), 419–436 | DOI | Zbl

[13] V. A. Matveev, M. A. Salle, Darboux Transformation and Solitons, Springer-Verl., Berlin, 1991

[14] Z.-L. Li, “Constructing of new exact solutions to the GKdV-mKdV equation with any-order nonlinear terms by $(G'/G)$-expansion method”, Appl. Math. Comput., 217 (2010), 1398–1403 | Zbl

[15] E. M. E. Zayed, “The $(G'/G)$-expansion method and its applications to some nonlinear evolution equations in the mathematical physics”, J. Appl. Math. Comput., 30:1 (2009), 89–103 | DOI | Zbl

[16] E. M. Zayed, “The $(G'/G)$-expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs”, J. Appl. Math. Inform., 29 (2011), 351–367 | Zbl

[17] E. M. Zayed, KA. Alurrfi, “Extended generalized $(G'/G)$-expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation”, Ricerche Mat., 65 (2016), 235–254 | DOI | Zbl

[18] N. Shang, B. Zheng, “Exact solutions for three fractional partial differential equations by the $(G'/G)$-expansion method”, Int. J. Appl. Math., 43 (2013), 3

[19] A. Bekir, O. Guner, “Exact solutions of nonlinear fractional differential equations by $(G'/G)$-expansion method”, Chin. Phys. B, 22:11 (2013), 110202 | DOI

[20] A. Bekir, “Application of the $(G'/G)$-expansion method for nonlinear evolution equations”, Phys. Lett. A, 372 (2008), 3400–3406 | DOI | Zbl

[21] S. Zhang, J. L. Tong, W. Wang, “A generalized $(G'/G)$-expansion method for the mKdV equation with variable coefficients”, Phys. Lett. A, 372 (2008), 2254–2257 | DOI | Zbl

[22] E. M. E. Zayed, K. A. Gepreel, “The $(G'/G)$ expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics”, J. Math. Phys., 50:1 (2009), 12 | DOI

[23] M. Wang, X. Li, J. Zhang, “The $(G'/G)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics”, Phys. Lett. A, 372 (2008), 417–423 | DOI | Zbl