The exact solitary and periodic wave solutions of the loaded nonlinear evolution equations via the functional variable method
Abstract
In this article, we investigated new traveling wave solutions for the loaded modified Korteweg-de Vries-Kadomtsev-Petviashvili equation via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions of this equation have been examined and three-dimensional graphics of the obtained solutions have been drawn by using
the Matlab program. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions and solitary wave solutions of kink type. This method presents a wider applicability for handling nonlinear wave equations.
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References
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equations, Physics Letters A, Volume 353, pp.487-492.
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pp.59-67.
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Differential Equations, Volume 12, Issue 1, pp. 103-108.
36. Nakhushev A.M., Borisov V. N. (1977). Boundary value problems for loaded parabolic equations and their applications
to the prediction of ground water level, Differential Equations, Volume 13, Issue 1, pp. 105-110.
37. Nakhushev A.M. (1979). Boundary value problems for loaded integro-differential equations of hyperbolic type and some
of their applications to the prediction of ground moisture, Differential Equations, Volume 15, Issue 1, pp. 96-105.
38. Baltaeva U. I. (2012). 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, Volume 3, Issue 3, pp.
3-12.
39. Kozhanov A. I. (2004). A nonlinear loaded parabolic equation and a related inverse problem, Mathematical Notes,
Volume 76, Issue 5, pp. 784-795.
40. Hasanov A.B., Hoitmetov U.A. (2021). Integration of the general loaded Kortewegde Vries equation with an integral
source in the class of rapidly decreasing complex-valued functions, Russian Mathematics, Volume 7, pp. 52-66.
41. Hasanov A.B., Hoitmetov U.A. (2021). On integration of the loaded Korteweg-de Vries equation in the class of rapidly
decreasing functions, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of
Azerbaijan, Volume 47, Issue 2, pp. 250-261.
42. Khasanov A.B., Hoitmetov U.A. (2021). On integration of the loaded mKdV equation in the class of rapidly decreasing
functions, The bulletin of Irkutsk State University. Series Mathematics, Volume 38, pp. 19-35.
43. Urazboev G. U., Baltaeva I. I., Rakhimov I. D. (2021). Generalized G’/G - extension method for loaded Korteweg-de
Vries equation, Siberian Journal of Industrial Mathematics, Volume 24, Issue 4, pp. 72-76.
44. Yakhshimuratov, A.B., Matyokubov, M.M. (2016). Integration of a loaded Kortewegde Vries equation in a class of
periodic functions, Volume 60, pp. 72-76.
45. Zerarka A., Ouamane S., Attaf A. (2010). On the functional variable method for finding exact solutions to a class of
wave equations, Applied Mathematics and Computation, Volume 217, Issue 7, pp. 2897-2904.
46. Zerarka A., Ouamane S. (2010). Application of the functional variable method to a class of nonlinear wave equations,
World Journal of Modelling and Simulation, Volume 6, Issue 2, pp. 150-160.
Volume 4, p. 23.
2. Seadawy Aly R., Cheemaa Nadia. (2020). Some new families of spiky solitary waves of one-dimensional higher-order
KdV equation with power law nonlinearity in plasma physics, Indian Journal of Physics, Volume 94, Issue 1, pp. 117-126.
3. Seadawy Aly R., Cheemaa Nadia. (2019). Propagation of nonlinear complex waves for the coupled nonlinear Schr¨odinger
Equations in two core optical fibers, Physica A: Statistical Mechanics and its Applications, Volume 529, Issue 12, pp.
13-30.
4. Seadawy Aly R., Cheemaa Nadia. (2019). Applications of extended modified auxiliary equation mapping method for
high order dispersive extended nonlinear Schr¨odinger equation in nonlinear, Modern Physics Letters B, Volume 33,
Issue 18, pp. 1-11.
5. Cheemaa Nadia, Seadawy Aly R., Chen Sheng. (2018). More general families of exact solitary wave solutions of the
nonlinear Schr¨odinger equation with their applications in nonlinear optics, The European Physical Journal Plus, Volume
133, p. 547.
6. Tagare S.G., Singh S.V., Reddy R.V., Lakhina G.S. (2004). Electron-acoustic solitons in the Earth’s magnetotail,
Nonlinear Processes in Geophysics, Volume11, pp. 215-218.
7. Kakad A.P., Singh S.V., Reddy R.V., Lakhina G.S., Tagare S.G. (2009). Electron acoustic solitary waves in the earth’s
magnetotail region, Advances in Space Research, Volume 43, Issue 12, pp. 1945-1949.
8. Segur H., Finkel A. (1985). An analytical model of periodic waves in shallow water, Studies in Applied Mathematics,
Volume 73, pp. 183-220.
9. Hammack J., Scheffner N., Segur H. (1989). Two-dimensional periodic waves in shallow water, Journal of Fluid Mechanics,
Volume 209, pp. 567-589.
10. Hammack J., McCallister D., Scheffner N., Segur H. (1995). Two-dimensional periodic waves in shallow water. Part 2.
Asymmetric waves, Journal of Fluid Mechanics, Volume 285, pp. 95-122.
11. Infeld E., Rowlands G. (2001). Nonlinear waves, solitons and chaos, Cambridge University Press.
12. Tzvetkov N. (2000). Global low-regularity solutions for Kadomtsev-Petviashvili equations, Differential Integral Equations,
Volume 13, pp. 1289-1320.
13. Groves M.D., Sun S.M. (2008). Fully localised solitary-wave solutions of the three-dimensional gravity-capillary waterwave
problem, Archive for Rational Mechanics and Analysis, Volume 188, pp. 1-91.
14. De Bouard A., Saut J.C. (1997). Solitary waves of generalized Kadomtsev-Petviashvili equations, Annales Institut Henri
Poincare, Analyse Non Lineaire, Volume 14, pp. 211-236.
15. Ahmed M.T., Khan K., Akbar M.A. (2013). Study of nonlinear evolution equations to construct traveling wave solutions
via modified simple equation method, Physical Review and Research International, Volume 3, Issue 4, pp. 490-503.
16. Al- Fhaid A.S. (2012). New exact solutions for the modified KdV-KP equation using the extended F-expansion method,
Applied Mathematical Sciences, Volume 6, pp. 5315-5332.
17. Cevikel A.C., Bekir A., Akar M., San S. (2012). A procedure to construct exact solutions of nonlinear evolution
equations, Pramana, Volume 79, pp. 337-344.
18. Nazarzadeh A., Eslami M., Mirzazadeh M. (2013). Exact solutions of some nonlinear partial di erential equations using
functional variable method, Pramana, Volume 81, pp. 225-236.
19. Taghizadeh N., Mirzazadeh M., Farahrooz F. (2011). Exact soliton solutions of the modified KdV-KP equation and the
Burgers-KP equation by using the first integral method, Applied Mathematical Modelling, Volume 35, pp. 3991-3997.
20. Wazwaz A.M. (2008). Solitons and singular solitons for the Gardner–KP equation, Applied Mathematics and Computation,
Volume 204, pp. 162-169.
21. Kudryashov.N.A. (1990). Exact solutions of the generalized Kuram-Oto-Sivashinsky equation, Physics Letters, Volume
24, Issue 147, pp. 287-291.
22. Chen Y., Wang Q. (2005). Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi
elliptic functions solutions to (1+1)-dimensional dispersive long wave equation, Chaos Solitons Fractals, Volume 24, pp.
745-757.
23. Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations, American Journal of Physics, Volume 60, Issue
7, pp. 650-654.
NEW EXACT SOLITARY AND PERIODIC WAVE SOLUTIONS... 7
24. Ablowitz M.J., Clarkson P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge
University Press.
25. Hirota R. (1971). Exact solution of the KdV equation for multiple collisions of solutions, Physical Review Letters,
Volume 27, pp.1192-1194.
26. Rogers C., Shadwick W.F. (1982). Backlund Transformations and their applications, Academic Press, New York,
Mathematics in science and engineering, Volume 161, p.334.
27. He J.H., Wu X.H. (2006). Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, Volume 30, Issue
3, pp.700-708.
28. Kudryashov N.A. (1991). On types of nonlinear non-integrable equations with exact solutions, Physics Letters A,Volume
155, pp.269-275.
29. Abdou, M.A., Soliman, A.A. (2006). Modified extended tanh-function method and its application on nonlinear physical
equations, Physics Letters A, Volume 353, pp.487-492.
30. Zhao X., Tang D. (2002). A new note on a homogeneous balance method, Physics Letters A, Volume 297, Issue 1-2,
pp.59-67.
31. Kneser A. (1914). Belastete integralgleichungen. Rendiconti del Circolo Matematico di Palermo, Volume 37, Issue 1,
pp. 169-197.
32. Lichtenstein L. (1931). Vorlesungen ¨uber einige Klassen nichtlinearer Integralgleichungen und Integro-Differential-
Gleichungen nebst Anwendungen, Springer-Verlag.
33. Nakhushev A.M. (1995). Equations of mathematical biology, Visshaya Shkola, p. 301.
34. Nakhushev A.M. Loaded equations and their applications, Differential Equations, Volume 19, Issue 1, pp. 86-94.
35. Nakhushev A.M. (1976). The Darboux problem for a certain degenerate second order loaded integrodifferential equation,
Differential Equations, Volume 12, Issue 1, pp. 103-108.
36. Nakhushev A.M., Borisov V. N. (1977). Boundary value problems for loaded parabolic equations and their applications
to the prediction of ground water level, Differential Equations, Volume 13, Issue 1, pp. 105-110.
37. Nakhushev A.M. (1979). Boundary value problems for loaded integro-differential equations of hyperbolic type and some
of their applications to the prediction of ground moisture, Differential Equations, Volume 15, Issue 1, pp. 96-105.
38. Baltaeva U. I. (2012). 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, Volume 3, Issue 3, pp.
3-12.
39. Kozhanov A. I. (2004). A nonlinear loaded parabolic equation and a related inverse problem, Mathematical Notes,
Volume 76, Issue 5, pp. 784-795.
40. Hasanov A.B., Hoitmetov U.A. (2021). Integration of the general loaded Kortewegde Vries equation with an integral
source in the class of rapidly decreasing complex-valued functions, Russian Mathematics, Volume 7, pp. 52-66.
41. Hasanov A.B., Hoitmetov U.A. (2021). On integration of the loaded Korteweg-de Vries equation in the class of rapidly
decreasing functions, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of
Azerbaijan, Volume 47, Issue 2, pp. 250-261.
42. Khasanov A.B., Hoitmetov U.A. (2021). On integration of the loaded mKdV equation in the class of rapidly decreasing
functions, The bulletin of Irkutsk State University. Series Mathematics, Volume 38, pp. 19-35.
43. Urazboev G. U., Baltaeva I. I., Rakhimov I. D. (2021). Generalized G’/G - extension method for loaded Korteweg-de
Vries equation, Siberian Journal of Industrial Mathematics, Volume 24, Issue 4, pp. 72-76.
44. Yakhshimuratov, A.B., Matyokubov, M.M. (2016). Integration of a loaded Kortewegde Vries equation in a class of
periodic functions, Volume 60, pp. 72-76.
45. Zerarka A., Ouamane S., Attaf A. (2010). On the functional variable method for finding exact solutions to a class of
wave equations, Applied Mathematics and Computation, Volume 217, Issue 7, pp. 2897-2904.
46. Zerarka A., Ouamane S. (2010). Application of the functional variable method to a class of nonlinear wave equations,
World Journal of Modelling and Simulation, Volume 6, Issue 2, pp. 150-160.
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