Simulation of Wave Solutions of a Mathematical Model Representing Communication Signals

Abstract

In this study, the Lonngren-wave equation is considered to be analyzed for its wave solutions. To implement this purpose the modified exponential function method is used and ultimately new hyperbolic, trigonometric and rational forms of the exact solutions are obtained. Furthermore, it was tested whether these forms satisfy the Lonngren-wave equation or not and it was seen that they verify the equation. Besides this, the two and three dimensional graphics together with the contour and density plots are presented.

Keywords

• Lonngren-wave equation
• the modified exponential function method
• exact wave solutions
• exact wave solutions
• nonlinear partial differential equation
• the hyperbolic
• trigonometric and rational functions

References

1. Akbar MA, Alam MdN, Hafez MdG, 2016. Application of the novel (G′/G)-expansion method to construct traveling wave solutions to the positive Gardner-KP equation. Indian Journal of Pure Applied Mathematics, 47(1):85-96.
2. Akcagil S, Aydemir T, 2016. Comparison between the (G′/G) -expansion method and the modified extended tanh method. Open Physics, 14(1):88-94.
3. Akcagil S, Aydemir T, 2018. A new application of the unified method. New Trends in Mathematical Sciences, 6(1):185-199.
4. Aktürk T, Gurefe Y, Pandır Y, 2017. An application of the new function method to the Zhiber-Shabat equation. An Interntional Journal of Optimization and Control: Theories & Applications, 7(3):271-274.
5. Arshed S, Biswas A, Zhou Q, Khan S, Adesynya S, Moshokoa SP, et al., 2019. Optical Solitons perturbation with Fokas-Lenells equation by exp(-ϕ(ξ))-expansion method. Optik, 179:341–345.
6. Baskonus HM, 2021. Dark and trigonometric soliton solutions in asymmetrical Nizhnik-Novikov-Veselov equation with (2+ 1)-dimensional. An International Journal of Optimization and Control: Theories & Applications, 11(1):92-99.
7. Biazar J, Badpeima F, Azimi F, 2009. Application of the homotopy perturbation method to Zakharov–Kuznetsov equations. Computers&Mathematics with Applications 58(11-12):2391-2394.
8. Bulut H, Baskonus HM, 2016. Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves in Random and Complex Media, 26(2):189-196.

9. Bulut H, Baskonus HM, 2016. New Complex Hyperbolic Function Solutions for the (2+1)-Dimensional Dispersive Long Water-Wave System. Mathematical and Computational Applications, 21(2): 6.
10. Chen G, Xin X, Liu H, 2019. The Improved exp⁡(-ϕ(ξ))-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics. Advances in Mathematical Physics, Article ID 4354310.
11. Demiray ST, Pandır Y, Bulut H, 2015. New solitary wave solutions of Maccari system. Ocean Engineering, 103:153-159.
12. Duran S, 2020. Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G'/G,1/G) -Expansion Method. Adıyaman University Journal of Science, 10(2):585-594.
13. Duran S, 2021. Dynamic interaction of behaviors of time-fractional shallow water wave equation system. Modern Physics Letters B, 35(22):2150353.
14. Duran S, 2021. Travelling wave solutions and simulation of the Lonngren wave equation for tunnel diode. Optical and Quantum Electronics, 53, Article number: 458.
15. Duran S, 2021. Breaking theory of solitary waves for the Riemann wave equation in fluid dynamics. International Journal of Modern Physics B, 35(9): 2150130.
16. Duran S, Karabulut B, 2021. Nematicons in liquid crystals with Kerr Law by sub-equation method. Alexandria Engineering Journal, https://doi.org/10.1016/j.aej.2021.06.077.
17. Elwakil SA, El-Labany SK, Zahran MA, Sabry R, 2005. Modified extended tanh-function method and its applications to nonlinear equations. Applied Mathematics and Computation, 161(2):403–412.
18. Fan E, 2000. Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4-5):212-218.
19. Gao W, Ismael H F, Mohammed SA, Baskonus HM, Bulut H, 2019. Complex and real optical soliton properties of the paraxial non-linear Schrödinger equation in Kerr media with M-fractional. Frontiers in Physics, https://doi.org/10.3389/fphy.2019.00197.
20. He JH, 2006a. Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 20(18): 2561-2568.
21. He JH, 2006b. Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1): 87-88.
22. He JH, 2008. Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis, 31(2):205-209.
23. Hietarinta J, 2005. Hirota’s bilinear method and soliton solutions. Physics AUC, 15(1):31-37.
24. Hirota R, Satsuma J, 1977. Nonlinear Evolution Equations Generated from the Bäcklund Transformation for the Boussinesq Equation. Progress of Theoretical Physics, 57(3):797-807.
25. Ismael HF, Bulut H, Baskonus HM, 2020. Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and (m + (G′ /G))-expansion method. Pramana-Journal of Physics, 94:35.
26. Kayum MdA, Ara S, Barman HK, Akbar MA, 2020. Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines. Results in Physics, 18:103269.
27. Kumar D, Manafian J, Hawlader F, Ranjbaran A, 2018. New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method. Optik, 160:159-167.
28. Lu D, Hong B, Tian L, 2006. Backlund transformation and N-soliton-like Solutions to the Combined KdV-Burgers Equation with Variable Coefficients. International. Journal of Nonlinear Sciences, 2(1):3–10.
29. Mahmud F, Samsuzzoha Md, Akbar MA, 2017. The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and Fisher equation. Results in Physics, 7:4296-4302.
30. Marwan A, Al-Khaled K, Ananbeh H, 2011. New Soliton Solutions for Systems of Nonlinear Evolution Equations by the Rational Sine-Cosine Method. Studies in Mathematical Sciences, 3(1): 1-9.
31. Misirli E, Gurefe Y, 2011. Exp-Function Method for Solving Nonlinear Evolution Equations. Mathematical and Computational Applications, 16(1): 258-266.
32. Naher H, 2012. The Basic ( G'/G )-Expansion Method for the Fourth Order Boussinesq Equation. Applied Mathematics, 3(10):1144-1152.
33. Naher H, Abdullah FA, 2013. New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation. American Institute of Physics Advances, 3(3):032116.
34. Parkes EJ, Duffy BR, Abbott PC, 2002. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations. Physics Letters A, 295(5-6):280-286.
35. Parkes EJ, 2010. Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations. Applied Mathematics and Computations, 217(4):1749-1754.
36. Qawasmeh A, Alquran M, 2014. Reliable Study of Some New Fifth-Order Nonlinear Equations by Means of G^'⁄G Expansion Method and Rational Sine-Cosine Method. Applied Mathematical Sciences, 8(120):5985-5994.
37. Radha R, Tang XY, Lou SY, 2007. Truncated Painleve Expansion – A Unified Approach to Exact Solutions and Dromion Interactions of (2+1)-Dimensional Nonlinear Systems. Zeitschrift für Naturforsch a, 62(3):107-116.
38. Rahman MM, Habib MA, Ali HMS, Miah MM, 2019. The Generalized Kudryashov Method: a Renewed Mechanism for Performing Exact Solitary Wave Solutions of Some NLEEs. Journal of Mechanics of Continua and Mathematical Sciences, 14(1):323-339.
39. Saleh R, Mabrouk SM, Wazwaz AM, 2021. Lie symmetry analysis of a stochastic gene evolution in double-chain deoxyribonucleic acid system. Waves in Random and Complex Media, https://doi.org/10.1080/17455030.2020.1871109.
40. Seadawy AR, Bilal M, Younis M, Rizvi STR, Althobaiti S, Makhlouf MM, 2021. Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique. Chaos, Solitons and Fractals, 144(17):110669.
41. Shrauner BA, 2019. Exact traveling wave solutions of nonlinear evolution equations: indeterminant homogeneous balance and linearizability. Mathematics and Statistics, 7(1):10–13.
42. Wang ML, Li X, Zhang JL, 2008. The G'/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4):417–423.
43. Wazwaz AM, 2007a. Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Applied Mathematics and Computation, 190(1):633–640.
44. Wazwaz AM, 2007b. The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Applied Mathematics and Computation, 188(2):1467-1475.
45. Wazwaz AM, 2016. The simplified Hirota’s method for studying three extended higher-order KdV-type equations. Jornal of Ocean Engineering and Sciences, 1(3):181-185.
46. Yang XF, Deng ZC, Wei Y, 2015. A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Advances in Difference Equations, 117(2015).
47. Yang Z, Hon YC, 2006. An Improved Modified Extended tanh-Function Method. Zeitschrift für Naturforsch, 61a:103-115.
48. Yel G, Baskonus HM, 2019. Solitons in conformable time-fractional Wu–Zhang system arising in coastal design. Pramana, 93(4):1-10.