Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods

Abdulghani R. Alharbi; Abdulghani R. Alharbi

doi:10.3934/math.2023062

AIMS Mathematics

2023, Volume 8, Issue 1: 1230-1250. doi: 10.3934/math.2023062

Previous Article Next Article

Research article

Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods

Abdulghani R. Alharbi ^,

Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

Received: 21 July 2022 Revised: 14 September 2022 Accepted: 23 September 2022 Published: 18 October 2022
MSC : 35A24, 35B35, 35Q51, 35Q92, 65N06, 65N40, 65N45, 65N50

In this article, we have achieved new solutions for the Novikov-Veselov system using several methods. The present solutions contain soliton solutions in the shape of hyperbolic, rational, and trigonometric function solutions. Magneto-sound and ion waves in plasma are examined by employing partial differential equations, such as, the Novikov-Veselov system. The Generalized Algebraic and the Modified F-expansion methods are employed to achieve various soliton solutions for the system. The finite difference method is well applied to convert the proposed system into numerical schemes. They are used to obtain the numerical simulations for NV. I also present a study of the stability and Error analysis of the numerical schemes. To verify the validity and accuracy of the exact solutions obtained using exact methods, we compare them with the numerical solutions analytically and graphically. The presented methods in this paper are suitable and acceptable and can be utilized for solving other types of non-linear evolution systems.
- Novikov-Veselov equations,
- solitary solutions,
- numerical solutions,
- magnetosound,
- electromagnetic
Citation: Abdulghani R. Alharbi. Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods[J]. AIMS Mathematics, 2023, 8(1): 1230-1250. doi: 10.3934/math.2023062

Related Papers:

Abstract

In this article, we have achieved new solutions for the Novikov-Veselov system using several methods. The present solutions contain soliton solutions in the shape of hyperbolic, rational, and trigonometric function solutions. Magneto-sound and ion waves in plasma are examined by employing partial differential equations, such as, the Novikov-Veselov system. The Generalized Algebraic and the Modified F-expansion methods are employed to achieve various soliton solutions for the system. The finite difference method is well applied to convert the proposed system into numerical schemes. They are used to obtain the numerical simulations for NV. I also present a study of the stability and Error analysis of the numerical schemes. To verify the validity and accuracy of the exact solutions obtained using exact methods, we compare them with the numerical solutions analytically and graphically. The presented methods in this paper are suitable and acceptable and can be utilized for solving other types of non-linear evolution systems.

References

[1]	A. Aasaraai, The application of modified F-expansion method solving the Maccari's system, Journal of Advances in Mathematics and Computer Science, 11 (2015), 1–14. http://dx.doi.org/10.9734/BJMCS/2015/19938 doi: 10.9734/BJMCS/2015/19938
[2]	C. Bai, C. Bai, H. Zhao, A new generalized algebraic method and its application in nonlinear evolution equations with variable coefficients, Z. Naturforsch. A, 60 (2005), 211–220. https://doi.org/10.1515/zna-2005-0401 doi: 10.1515/zna-2005-0401
[3]	A. Bekir, O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana-J. Phys., 79 (2012), 3–17. http://dx.doi.org/10.1007/s12043-012-0282-9 doi: 10.1007/s12043-012-0282-9
[4]	D. Kumar, A. Seadawy, A. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese J. Phys., 56 (2018), 75–85. http://dx.doi.org/10.1016/j.cjph.2017.11.020 doi: 10.1016/j.cjph.2017.11.020
[5]	G. Adomain, Solving frontier problems of physics: the decomposition method, Dordrecht: Springer, 1994. http://dx.doi.org/10.1007/978-94-015-8289-6
[6]	X. Feng, Exploratory approach to explicit solution of nonlinear evolutions equations, Int. J. Theor. Phys., 39 (2000), 207–222. http://dx.doi.org/10.1023/A:1003615705115 doi: 10.1023/A:1003615705115
[7]	A. Alharbi, M. Almatrafi, Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability, Results Phys., 16 (2020), 102870. http://dx.doi.org/10.1016/j.rinp.2019.102870 doi: 10.1016/j.rinp.2019.102870
[8]	A. Alharbi, M. Almatrafi, Riccati-Bernoulli sub-ODE approach on the partial differential equations and applications, Int. J. Math. Comput. Sci., 15 (2020), 367–388.
[9]	A. Alharbi, M. Almatrafi, New exact and numerical solutions with their stability for Ito integro-differential equation via Riccati–Bernoulli sub-ODE method, J. Taibah Univ. Sci., 14 (2020), 1447–1456. http://dx.doi.org/10.1080/16583655.2020.1827853 doi: 10.1080/16583655.2020.1827853
[10]	A. Alharbi, M. Almatrafi, Exact and numerical solitary wave structures to the variant Boussinesq system, Symmetry, 12 (2020), 1473. http://dx.doi.org/10.3390/sym12091473 doi: 10.3390/sym12091473
[11]	M. Almatrafi, A. Alharbi, C. Tunç, Constructions of the soliton solutions to the good Boussinesq equation, Adv. Differ. Equ., 2020 (2020), 629. http://dx.doi.org/10.1186/s13662-020-03089-8 doi: 10.1186/s13662-020-03089-8
[12]	A. Alharbi, M. Almatrafi, A. Seadawy, Construction of the numerical and analytical wave solutions of the Joseph-Egri dynamical equation for the long waves in nonlinear dispersive systems, Int. J. Mod. Phys. B, 34 (2020), 2050289. http://dx.doi.org/10.1142/S0217979220502896 doi: 10.1142/S0217979220502896
[13]	A. Alharbi, M. Almatrafi, Kh. Lotfy, Constructions of solitary travelling wave solutions for Ito integro-differential equation arising in plasma physics, Results Phys., 19 (2020), 103533. http://dx.doi.org/10.1016/j.rinp.2020.103533 doi: 10.1016/j.rinp.2020.103533
[14]	A. Alharbi, M. Almatrafi, Exact solitary wave and numerical solutions for geophysical KdV equation, J. King Saud Univ. Sci., 34 (2022), 102087. http://dx.doi.org/10.1016/j.jksus.2022.102087 doi: 10.1016/j.jksus.2022.102087
[15]	S. Tian, J. Tu, T. Zhang, Y. Chen, Integrable discretizations and soliton solutions of an Eckhaus-Kundu equation, Appl. Math. Lett., 122 (2021), 107507. http://dx.doi.org/10.1016/j.aml.2021.107507 doi: 10.1016/j.aml.2021.107507
[16]	S. Tian, M. Xu, T. Zhang, A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation, Proc. R. Soc. A., 477 (2021), 20210455. http://dx.doi.org/10.1098/rspa.2021.0455 doi: 10.1098/rspa.2021.0455
[17]	S. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett., 100 (2020), 106056. http://dx.doi.org/10.1016/j.aml.2019.106056 doi: 10.1016/j.aml.2019.106056
[18]	S. Tian, D. Guo, X. Wang, T. Zhang, Traveling wave, lump wave, rogue wave, multi-kink solitary wave and interaction solutions in a (3+1)-dimensional Kadomtsev-Petviashvili equation with Bäcklund transformation, J. Appl. Anal. Comput., 11 (2021), 45–58. http://dx.doi.org/10.11948/20190086 doi: 10.11948/20190086
[19]	J. Yang, S. Tian, Z. Li, Riemann-Hilbert problem for the focusing nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions, Physica D, 432 (2022), 133162. http://dx.doi.org/10.1016/j.physd.2022.133162 doi: 10.1016/j.physd.2022.133162
[20]	X. Gao, Y. Guo, W. Shan, Optical waves/modes in a multicomponent inhomogeneous optical fiber via a three-coupled variable-coefficient nonlinear Schrödinger system, Appl. Math. Lett., 120 (2021), 107161. http://dx.doi.org/10.1016/j.aml.2021.107161 doi: 10.1016/j.aml.2021.107161
[21]	X. Gao, Y. Guo, W. Shan, In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation, Acta Math. Sin.-English Ser., in press. http://dx.doi.org/10.1007/s10114-022-9778-5
[22]	X. Gao, Y. Guo, W. Shan, Similarity reductions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid dynamics, hinese J. Phys., 77 (2022), 2707–2712. http://dx.doi.org/10.1016/j.cjph.2022.04.014 doi: 10.1016/j.cjph.2022.04.014
[23]	X. Gao, Y. Guo, W. Shan, Taking into consideration an extended coupled (2+1)-dimensional Burgers system in oceanography, acoustics and hydrodynamics, Chaos Soliton. Fract., 161 (2022), 112293. http://dx.doi.org/10.1016/j.chaos.2022.112293 doi: 10.1016/j.chaos.2022.112293
[24]	C. Dai, Y. Wang, Y. Fan, J. Zhang, Interactions between exotic multi-valued solitons of the (2+1)-dimensional Korteweg-de Vries equation describing shallow water wave, Appl. Math. Model., 80 (2020), 506–515. http://dx.doi.org/10.1016/j.apm.2019.11.056 doi: 10.1016/j.apm.2019.11.056
[25]	J. Fang, D. Mou, H. Zhang, Y. Wang, Discrete fractional soliton dynamics of the fractional Ablowitz-Ladik model, Optik, 228 (2021), 166186. http://dx.doi.org/10.1016/j.ijleo.2020.166186 doi: 10.1016/j.ijleo.2020.166186
[26]	C. Dai, Y. Fan, Y. Wang, Three-dimensional optical solitons formed by the balance between different-order nonlinearities and high-order dispersion/diffraction in parity-time symmetric potentials, Nonlinear Dyn., 98 (2019), 489–499. http://dx.doi.org/10.1007/s11071-019-05206-z doi: 10.1007/s11071-019-05206-z
[27]	Y. Chen, X. Xiao, Vector bright-dark one-soliton and two-soliton of the coupled NLS model with the partially nonlocal nonlinearity in BEC, Optik, 257 (2022), 168708. http://dx.doi.org/10.1016/j.ijleo.2022.168708 doi: 10.1016/j.ijleo.2022.168708
[28]	Q. Cao, C. Dai, Symmetric and anti-symmetric solitons of the fractional second- and third-order nonlinear Schrödinger equation, Chinese Phys. Lett., 38 (2021), 090501. http://dx.doi.org/10.1088/0256-307x/38/9/090501 doi: 10.1088/0256-307x/38/9/090501
[29]	C. Dai, Y. Wang, Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals, Nonlinear Dyn., 102 (2020), 1733–1741. http://dx.doi.org/10.1007/s11071-020-05985-w doi: 10.1007/s11071-020-05985-w
[30]	X. Wen, R. Feng, J. Lin, W. Liu, F. Chen, Q. Yang, Distorted light bullet in a tapered graded-index waveguide with PT symmetric potentials, Optik, 248 (2021), 168092. http://dx.doi.org/10.1016/j.ijleo.2021.168092 doi: 10.1016/j.ijleo.2021.168092
[31]	Y. Fang, G. Wu, X. Wen, Y. Wang, C. Dai, Predicting certain vector optical solitons via the conservation-law deep-learning method, Opt. Laser Technol., 155 (2022), 108428. http://dx.doi.org/10.1016/j.optlastec.2022.108428 doi: 10.1016/j.optlastec.2022.108428
[32]	B. Li, J. Zhao, W. Liu, Analysis of interaction between two solitons based on computerized symbolic computation, Optik, 206 (2020), 164210. http://dx.doi.org/10.1016/j.ijleo.2020.164210 doi: 10.1016/j.ijleo.2020.164210
[33]	X. Wen, G. Wu, W. Liu, C. Dai, Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method, Nonlinear Dyn., 109 (2022), 3041–3050. http://dx.doi.org/10.1007/s11071-022-07583-4 doi: 10.1007/s11071-022-07583-4
[34]	G. Wu, L. Yu, Y. Wang, Fractional optical solitons of the space-time fractional nonlinear Schrödinger equation, Optik, 207 (2020), 164405. http://dx.doi.org/10.1016/j.ijleo.2020.164405 doi: 10.1016/j.ijleo.2020.164405
[35]	Z. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A, 292 (2001), 100–106. http://dx.doi.org/10.1016/S0375-9601(01)00772-1 doi: 10.1016/S0375-9601(01)00772-1
[36]	Z. Yan, Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations, Comput. Phys. Commun., 148 (2002), 30–42. http://dx.doi.org/10.1016/S0010-4655(02)00465-4 doi: 10.1016/S0010-4655(02)00465-4
[37]	G. Zhang, Z. Yan, The derivative nonlinear Schrödinger equation with Zero/Nonzero boundary conditions: inverse scattering transforms and N-Double-Pole solutions, J. Nonlinear Sci., 30 (2020), 3089–3127. http://dx.doi.org/10.1007/s00332-020-09645-6 doi: 10.1007/s00332-020-09645-6
[38]	Y. Chen, Z. Yan D. Mihalache, Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity, Phys. Rev. E., 102 (2020), 012216. http://dx.doi.org/10.1103/PhysRevE.102.012216 doi: 10.1103/PhysRevE.102.012216
[39]	X. Gao, Y. Guo, W. Shan, Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq-Burgers system for the shallow water waves in a lake or near an ocean beach, Commun. Theor. Phys., 72 (2020), 095002. http://dx.doi.org/10.1088/1572-9494/aba23d doi: 10.1088/1572-9494/aba23d
[40]	X. Gao, Y. Guo, W. Shan, T. Zhou, M. Wang, D. Yang, In the atmosphere and oceanic fluids: scaling transformations, bilinear forms, Bäcklund transformations and solitons for a generalized variable-coefficient Korteweg-de Vries-modified Korteweg-de Vries equation, China Ocean Eng., 35 (2021), 518–530. http://dx.doi.org/10.1007/s13344-021-0047-7 doi: 10.1007/s13344-021-0047-7
[41]	B. Boubir, H. Triki, A. Wazwaz, Bright solutions of the variants of the Novikov-Veselob equation with constant and variable coefficients, Appl. Math. Model., 37 (2013), 420–431. http://dx.doi.org/10.1016/j.apm.2012.03.012 doi: 10.1016/j.apm.2012.03.012
[42]	H. Barman, A. Seadawy, M. Akbar, D. Baleanu, Competent closed form soliton solutions to the Riemann wave equation and the Novikov-Veselov equation, Results Phys., 17 (2020), 103131. http://dx.doi.org/10.1016/j.rinp.2020.103131 doi: 10.1016/j.rinp.2020.103131
[43]	R. Croke, Investigation of the Novikov-Veselov equation, an: new solutions, stability and implications for the inverse Scattering transform, Ph. D thesis, Colorado State University, 2012.
[44]	M. Boiti, J. Leon, M. Manna, F. Pempinelli, On the spectral transform of a Korteweg-deVries equation in two spatial dimensions, Inverse Probl., 2 (1986), 271–279. http://dx.doi.org/10.1088/0266-5611/2/3/005 doi: 10.1088/0266-5611/2/3/005
[45]	A. Kazeykina, C. Klein, Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation, Nonlinearity, 30 (2017), 2566.
[46]	B. Sagar, S. Saha, Numerical soliton solutions of fractional (2+1)-dimensional Nizhnik-Novikov-Veselov equations in nonlinear optics, Int. J. Mod. Phys. B, 35 (2021), 2150090. http://dx.doi.org/10.1142/S0217979221500909 doi: 10.1142/S0217979221500909
[47]	L. Shampine, M. Reichelt, The matlab ode suite, SIAM J. Sci. Comput., 18 (1997), 1–22. http://dx.doi.org/10.1137/S1064827594276424
[48]	P. Brown, A. Hindmarsh, L. Petzold, Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput., 15 (1994), 1467–1488. http://dx.doi.org/10.1137/0915088 doi: 10.1137/0915088

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)