Existence of periodic wave for a perturbed MEW equation

Minzhi Wei; Liping He; Minzhi Wei; Liping He

doi:10.3934/math.2023585

AIMS Mathematics

2023, Volume 8, Issue 5: 11557-11571. doi: 10.3934/math.2023585

Previous Article Next Article

Research article

Existence of periodic wave for a perturbed MEW equation

Minzhi Wei ¹,
Liping He ^{2
,
,}

1.
School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning 530003, China
2.
China-ASEAN Institute of Statistics, Guangxi University of Finance and Economics, Nanning 530003, China

Received: 06 January 2023 Revised: 03 March 2023 Accepted: 09 March 2023 Published: 15 March 2023
MSC : 34C25, 34C60, 37C27

A perturbed MEW equation including small backward diffusion, dissipation and nonlinear term is considered by the geometric singular perturbation theory. Based on the monotonicity of the ratio of Abelian integrals, we prove the existence of periodic wave on a manifold for perturbed MEW equation. By Chebyshev system criterion, the uniqueness of the periodic wave is obtained. Furthermore, the monotonicity of the wave speed is proved and the range of the wave speed is obtained. Additionally, the monotonicity of period is given by Picard-Fuchs equation.
- perturbed MEW equation,
- geometric singular perturbation theory,
- periodic wave,
- Abelian integral,
- Chebyshev system,
- Picard-Fuchs equation
Citation: Minzhi Wei, Liping He. Existence of periodic wave for a perturbed MEW equation[J]. AIMS Mathematics, 2023, 8(5): 11557-11571. doi: 10.3934/math.2023585

Related Papers:

Abstract

A perturbed MEW equation including small backward diffusion, dissipation and nonlinear term is considered by the geometric singular perturbation theory. Based on the monotonicity of the ratio of Abelian integrals, we prove the existence of periodic wave on a manifold for perturbed MEW equation. By Chebyshev system criterion, the uniqueness of the periodic wave is obtained. Furthermore, the monotonicity of the wave speed is proved and the range of the wave speed is obtained. Additionally, the monotonicity of period is given by Picard-Fuchs equation.

References

[1]	M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge: Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511623998
[2]	M. N. Islam, M. Asaduzzaman, M. S. Ali, Exact wave solutions to the simplified modified Camassa-Holm equation in mathematical physics, AIMS Math., 5 (2019), 26–41. https://doi.org/10.3934/math.2020003 doi: 10.3934/math.2020003
[3]	C. H. Gu, Soliton theory and its applications, Hangzhou: Zhejiang Science and Technology publishing House, Springer-Verlag, 1995.
[4]	Y. Chatibi, E. E. Kinani, A. Ouhadan, Lie symmetry analysis of conformable differential equations, AIMS Math., 4 (2019), 1133–1144. https://doi.org/10.3934/math.2019.4.1133 doi: 10.3934/math.2019.4.1133
[5]	V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Berlin: Springer, 1991.
[6]	W. X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Scr., 82 (2010), 065003. https://doi.org/10.1088/0031-8949/82/06/065003 doi: 10.1088/0031-8949/82/06/065003
[7]	R. Hirota, The direct method in soliton theory, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543043
[8]	A. M. Wazwaz, The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 148–160. https://doi.org/10.1016/j.cnsns.2004.07.001 doi: 10.1016/j.cnsns.2004.07.001
[9]	J. Li, Singular nonlinear traveling wave equations: bifurcation and exact solutions, Beijing: Science Press, 2013.
[10]	P. J. Morrison, J. D. Meiss, J. R. Cary, Scattering of regularized-long-wave solitary waves, Phys. D, 11 (1984), 324–336. https://doi.org/10.1016/0167-2789(84)90014-9 doi: 10.1016/0167-2789(84)90014-9
[11]	D. H. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815–827. https://doi.org/10.1017/S0022112067002605
[12]	L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, N. K. Amein, Simulations of the EW undular bore, Commun. Numer. Meth. Eng., 13 (1998), 583–592.
[13]	A. M. Wazwaz, The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 148–160. https://doi.org/10.1016/j.cnsns.2004.07.001 doi: 10.1016/j.cnsns.2004.07.001
[14]	A. Esen, A lumped Galerkin method for the numerical solution of the modified equal-width equation using quadratic B-splines, Int. J. Comput. Math., 83 (2006), 449–459. https://doi.org/10.1080/00207160600909918 doi: 10.1080/00207160600909918
[15]	A. Esen, S. Kutluay, Solitary wave solutions of the modified equal width wave equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1538–1546. https://doi.org/10.1016/j.cnsns.2006.09.018 doi: 10.1016/j.cnsns.2006.09.018
[16]	S. I. Zaki, Solitary wave interactions for the modified equal width equation, Comput. Phys. Commun., 126 (2000), 219–231. https://doi.org/10.1016/S0010-4655(99)00471-3 doi: 10.1016/S0010-4655(99)00471-3
[17]	B. Saka, Algorithms for numerical solution of the modified equal width wave equation using collocation method, Math. Comput. Model., 45 (2007), 1096–1117. https://doi.org/10.1016/j.mcm.2006.09.012 doi: 10.1016/j.mcm.2006.09.012
[18]	S. Haq, S. Islam, A. Ali, A numerical meshfree technique for the solution of the MEW equation, Comput. Model. Eng. Sci., 38 (2008), 1–23.
[19]	J. Lu, He's variational iteration method for the modified equal width equation, Chaos Solitons Fract., 39 (2009), 2102–2109. https://doi.org/10.1016/j.chaos.2007.06.104 doi: 10.1016/j.chaos.2007.06.104
[20]	R. J. Cheng, K. M. Liew, Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method, Eng. Anal. Bound. Elem., 36 (2012), 1322–1330. https://doi.org/10.1016/j.enganabound.2012.03.013 doi: 10.1016/j.enganabound.2012.03.013
[21]	T. Geyikli, S. B. G. Karakoç, Petrov-Galerkin method with cubic B-splines for solving the MEW equation, Bull. Belg. Math. Soc. Simon Stevin, 19 (2012), 215–227. https://doi.org/10.36045/bbms/1337864268 doi: 10.36045/bbms/1337864268
[22]	D. Shi, Y. Zhang, Diversity of exact solutions to the conformable space-time fractional MEW equation, Appl. Math. Lett., 99 (2020), 105994. https://doi.org/10.1016/j.aml.2019.07.025 doi: 10.1016/j.aml.2019.07.025
[23]	A. Saha, Bifurcation of travelling wave solutions for the generalized KP-MEW equations, Commum. Nonlinear Sci. Numer. Simul., 17 (2012), 3539–3551. https://doi.org/10.1016/j.cnsns.2012.01.005 doi: 10.1016/j.cnsns.2012.01.005
[24]	A. Saha, P. K. Prasad, A study on bifurcations of traveling wave solutions for the generalized Zakharov-Kuznetsov modified equal width equation, Int. J. Pure Appl. Math., 87 (2013), 795–808. http://dx.doi.org/10.12732/ijpam.v87i6.8 doi: 10.12732/ijpam.v87i6.8
[25]	M. Wei, S. Tang, H. Fu, G. Chen, Single peak solitary wave solutions for the generalized KP-MEW (2, 2) equation under boundary condition, Appl. Math. Comput., 219 (2013), 8979–8990. https://doi.org/10.1016/j.amc.2013.03.007 doi: 10.1016/j.amc.2013.03.007
[26]	A. M. Wazwaz, The tanh method and the sine-cosine method for solving the KP-MEW equation, Int. J. Comput. Math., 85 (2005), 235–246. https://doi.org/10.1080/00207160412331296706 doi: 10.1080/00207160412331296706
[27]	A. Saha, Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation, Comput. Math. Appl., 73 (2017), 1879–1885. https://doi.org/10.1016/j.camwa.2017.02.017 doi: 10.1016/j.camwa.2017.02.017
[28]	C. Normad, Y. Pomeau, M. G. Velarde, Convective instability: a physicist's approach, Rev. Mod. Phys., 49 (1977), 581–624. https://doi.org/10.1103/RevModPhys.49.581 doi: 10.1103/RevModPhys.49.581
[29]	P. L. Garcia-Ybarra, J. L. Castillo, M. G. Velarde, Bénard-Marangoni convection with a deformable interface and poorly conducting boundaries, Phys. Fluids, 30 (1987), 2655–2661. https://doi.org/10.1063/1.866109 doi: 10.1063/1.866109
[30]	T. Ogama, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401–422. https://doi.org/10.32917/hmj/1206128032 doi: 10.32917/hmj/1206128032
[31]	X. Fan, L. Tian, The existence of solitary waves of singularly perturbed mKdV-KS equation, Chaos Solitons Fract., 26 (2005), 1111–1118. https://doi.org/10.1016/j.chaos.2005.02.014 doi: 10.1016/j.chaos.2005.02.014
[32]	Y. Tang, W. Xu, J. Shen, L. Gao, Persistence of solitary wave solutions of singularly perturbed Gardner equation, Chaos Solitons Fract., 37 (2006), 532–538. https://doi.org/10.1016/j.chaos.2006.09.044 doi: 10.1016/j.chaos.2006.09.044
[33]	M. B. A. Mansour, A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation, J. Geo. Phys., 69 (2013), 116–122. https://doi.org/10.1016/j.geomphys.2013.03.004 doi: 10.1016/j.geomphys.2013.03.004
[34]	W. Yan, Z. Liu, Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), 537–555. https://doi.org/10.3846/13926292.2014.960016 doi: 10.3846/13926292.2014.960016
[35]	Z. Du, J. Li, X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988–1007. https://doi.org/10.1016/j.jfa.2018.05.005 doi: 10.1016/j.jfa.2018.05.005
[36]	Z. Du, J. Liu, Y. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differ. Equ., 270 (2021), 1019–1042. https://doi.org/10.1016/j.jde.2020.09.009 doi: 10.1016/j.jde.2020.09.009
[37]	Z. Du, J. Li, Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation, J. Differ. Equ., 306 (2022), 418–438. https://doi.org/10.1016/j.jde.2021.10.033 doi: 10.1016/j.jde.2021.10.033
[38]	J. Ge, R. Wu, Z. Du, Dynamics of traveling waves for the perturbed generalized KdV equation, Qual. Theory Dyn. Syst., 20 (2021), 42. https://doi.org/10.1007/s12346-021-00483-9 doi: 10.1007/s12346-021-00483-9
[39]	A. Chen, L. Guo, X. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differ. Equ., 261 (2016), 5324–5349. https://doi.org/10.1016/j.jde.2016.08.003 doi: 10.1016/j.jde.2016.08.003
[40]	A. Chen, L. Guo, W. Huang, Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation, Qual. Theory Dyn. Syst., 17 (2018), 495–517. https://doi.org/10.1007/s12346-017-0249-9 doi: 10.1007/s12346-017-0249-9
[41]	L. Guo, Y. Zhao, Existence of periodic waves for a perturbed quintic BBM euqation, Disc. Cont. Dyn. Syst., 40 (2020), 4689–4703. https://doi.org/10.3934/dcds.2020198 doi: 10.3934/dcds.2020198
[42]	X. Sun, P. Yu, Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms, Disc. Cont. Dyn. Syst. B, 24 (2019), 965–987. https://doi.org/10.3934/dcdsb.2018341 doi: 10.3934/dcdsb.2018341
[43]	X. Sun, W. Huang, J. Cai, Coexistence of the solitary and periodic waves in convecting shallow water fluid, Nonlinear Anal., 53 (2020), 103067. https://doi.org/10.1016/j.nonrwa.2019.103067 doi: 10.1016/j.nonrwa.2019.103067
[44]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53–98.
[45]	C. K. R. T. Jones, Geometric singular perturbation theory, In: R. Johnson, Dynamical systems, Lecture Notes in Mathematics, New York: Springer, 1609 (1995), 44–118. https://doi.org/10.1007/BFb0095239
[46]	M. Han, P. Yu, Normal forms, Melnikov functions and bifurcations of limit cycles, London: Springer Science, 2012. https://doi.org/10.1007/978-1-4471-2918-9
[47]	F. Ma$\tilde{n}$osas, J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equ., 251 (2011), 1656–1669. https://doi.org/10.1016/j.jde.2011.05.026 doi: 10.1016/j.jde.2011.05.026

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)