SOLITON SOLUTIONS OF THE LOADED
MODIFIED CALOGERO-DEGASPERIS EQUATION
Bazar Babajanov1, Fakhriddin Abdikarimov2 1Department of Applied Mathematics and
Mathematical Physics, Urgench State University
Urgench, UZBEKISTAN 2Khorezm Mamun Academy
Khiva, UZBEKISTAN
In this article, we construct exact travelling wave solutions of the loaded modified Calogero-Degasperis equation by (G'/G) - expansion method. The efficiency of this method for finding these exact solutions has been demonstrated. We establish several classes of explicit solutions - hyperbolic and trigonometric solutions containing free parameters. The solitary wave solutions of this equation follow from the traveling wave solutions for certain values of the parameters. All calculations have been made with the aid of Matlab program. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1] A.M. Wazwaz, New solutions of distinct physical structures to highdimensional
nonlinear evolution equations, Applied Mathematics and Computation,
196, No 1 (2008), 363-370; doi: /10.1016/j.amc.2007.06.002.
[2] Y. Peng, New types of localized coherent structures in the BogoyavlenskiiSchiff equation, International Journal of Theoretical Physics, 45 (2006),
1764-1768; doi: 10.1007/s10773-006-9139-7.
[3] M.S. Bruzon, M.L. Gandarias, C. Muriel, J. Ramierez, S. Saez, F.R.
Romero, The Calogero-Bogoyavlenskii-Schff equation in (2+1) dimensions,
Journal of Theoretical and Mathematical Physics, 137 (2003), 1367-1377;
doi: 10.1023/A:1026040319977.
[4] A. Bansal, R.K. Gupta, Lie point symmetries and similarity solutions
of the time-dependent coefficients Calogero-Degasperis equation, Physica
Scripta, 86, No 3 (2012); doi: 10.1088/0031-8949/86/03/035005.
[5] A.M. Wazwaz, Multiple-soliton solutions for the Calogero-BogoyavlenskiiSchiff, Jimbo-Miwa and YTSF equation, Applied Mathematics and Computation,
203, No 2 (2008), 592-597; doi: 10.1016/j.amc.2008.05.004.
[6] R. Hirota, The Direct Method in Soliton Theory, Cambridge University
Press (2004); doi: 10.1017/CBO9780511543043.
[7] M.J. Ablowitz, P.A. Clarkson, Soliton Nonlinear Evolution Equations
and Inverse Scattering, Cambridge University Press (2010); doi:
10.1017/CBO9780511623998.
[8] E. Fan, Extended tanh-function method and its applications to nonlinear
equations, Physics Letters, A, No 277 (2000), 212-218; doi: 10.1016/S03759601(00)00725-8.
[9] M. Arshad, A.R. Seadawy, D. Lu, Modulation stability and optical soliton
solutions of nonlinear Schr¨odinger equation with higher order dispersion
and nonlinear terms and its applications, Superlattices and Microstructures,
113 (2017), 419-429; doi: 10.1016/j.spmi.2017.11.022.
[10] A. Kneser, Belastete integralgleichungen, Rendiconti del Cir-colo Matematico
di Palermo, 37, No 1 (1914), 169-197; doi: 10.1007/BF03014816.
[11] L. Lichtenstein, Vorlesungen ¨uber Einige Klassen Nichtlinearer Integralgleichungen
und Integro-differentialgleichungen: Nebst Anwendungen,
Springer (1931); doi: 10.1007/978-3-642-47600-6.
[12] A.M. Nakhushev, Equations of Mathematical Biology, Visshaya Shkola
(1995).
[13] A.M. Nakhushev, Loaded equations and their applications, Differential
Equations, 19, No 1 (1983), 86-94.
[14] A.M. Nakhushev, The Darboux problem for a certain degenerate second
order loaded integrodifferential equation, Differential Equations, 12, No 1
(1976), 103–108.
[15] A.M. Nakhushev, V.N. Borisov, Boundary value problems for loaded
parabolic equations and their applications to the prediction of ground water
level, Differential Equations, 13, No 1 (1977), 105–110.
[16] A.M. Nakhushev, Boundary value problems for loaded integro-differential
equations of hyperbolic type and some of their applications to the prediction
of ground moisture, Differential Equations, 15, No 1 (1979), 96–105.
[17] U.I. Baltaeva, On some boundary value problems for a third order loaded
integro-differential equation with real parameters, The Bulletin of Udmurt
University. Mathematics. Mechanics. Computer Science, 3, No 3 (2012),
3–12; doi: 10.20537/vm120301.
[18] A.I. Kozhanov, A nonlinear loaded parabolic equation and a related
inverse problem, Mathematical Notes, 76, No 5 (2004), 784–795; doi:
10.1023/B:MATN.0000049678.16540.a5.
[19] A.B. Hasanov, U.A. Hoitmetov, Integration of the general loaded
Korteweg-de Vries equation with an integral source in the class of rapidly
decreasing complex-valued functions, Russian Mathematics, 7 (2021), 52-
66; doi: 10.26907/0021-3446-2021-7-52-66.
[20] A.B. Hasanov, U.A. Hoitmetov, On integration of the loaded Korteweg-de
Vries equation in the class of rapidly decreasing functions, Proc. of the
Institute of Mathematics and Mechanics, National Academy of Sciences of
Azerbaijan, 47, No 2 (2021), 250-261; doi: 10.30546/2409-4994.47.2.250.
[21] A.B. Khasanov, U.A. Hoitmetov, On integration of the loaded mKdV
equation in the class of rapidly decreasing functions, The Bull. of Irkutsk
State University. Ser. Mathematics, 38 (2021), 19-35; doi: 10.26516/1997-
7670.2021.38.19.
[22] G.U. Urazboev, I.I. Baltaeva, I.D. Rakhimov, Generalized (G′/G) - extension
method for loaded Korteweg-de Vries equation, Siberian Journal
of Industrial Mathematics, 24, No 4 (2021), 139–147; doi: 10.33048/sibjim.
2021.24.410.
[23] A.B. Yakhshimuratov, M.M. Matyokubov, Integration of a loaded
Korteweg-de Vries equation in a class of periodic functions, Russian Mathematics,
60 (2016), 72–76; doi: 10.3103/S1066369X16020110.