Soliton structures of a Dullin-Gottwald-Holm model for the shallow water

Kalim U. Tariq; Mohammed Ahmed Alomair; Abdullah Mohammed Alomair; Arslan Ahmed; Kalim U. Tariq; Mohammed Ahmed Alomair; Abdullah Mohammed Alomair; Arslan Ahmed

doi:10.3934/math.20251027

AIMS Mathematics

2025, Volume 10, Issue 10: 23151-23168. doi: 10.3934/math.20251027

Previous Article Next Article

Research article Special Issues

Soliton structures of a Dullin-Gottwald-Holm model for the shallow water

1.
Department of Mathematics, Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 03 May 2025 Revised: 13 September 2025 Accepted: 18 September 2025 Published: 13 October 2025
MSC : 35Rxx, 35Qxx, 35Q51

This work analyzes the recently introduced nonlinear dispersive shallow water wave equation, the Dullin-Gottwald-Holm (DGH) model. By taking into account a wider range of wave events that frequently occur in real shallow water systems, such as dispersive shocks, bores, and solitary waves, the advanced nonlinear dispersive shallow water model has been devised to predominate the classical models. This is accomplished by combining linear and nonlinear dispersion. The several traveling wave solutions of the governing model are systematically arranged using several well-known analytical methods. The main focus is to investigate how several important parameters affect the dynamics and shape of the resulting wave formations. As a result, many wave solutions, including solitary, shock, singular, shock-singular, hyperbolic, singular periodic, and rational waves, can be constructed by symbolic computations. The parameter regimes guarantee that a careful analysis determines the generated solutions' validity and physical significance. The interpretation of the results is improved by displaying the solutions across a variety of parameter values using contour graphs, 2D plots, and 3D surface representations. These results confirm the efficiency and dependability of the used computational paradigm, which has a great deal of promise for further studies looking at more intricate nonlinear processes.
- the Dullin-Gottwald-Holm equation,
- nonlinear dynamics,
- analytical techniques,
- solitons,
- water waves
Citation: Kalim U. Tariq, Mohammed Ahmed Alomair, Abdullah Mohammed Alomair, Arslan Ahmed. Soliton structures of a Dullin-Gottwald-Holm model for the shallow water[J]. AIMS Mathematics, 2025, 10(10): 23151-23168. doi: 10.3934/math.20251027

Related Papers:

Abstract

This work analyzes the recently introduced nonlinear dispersive shallow water wave equation, the Dullin-Gottwald-Holm (DGH) model. By taking into account a wider range of wave events that frequently occur in real shallow water systems, such as dispersive shocks, bores, and solitary waves, the advanced nonlinear dispersive shallow water model has been devised to predominate the classical models. This is accomplished by combining linear and nonlinear dispersion. The several traveling wave solutions of the governing model are systematically arranged using several well-known analytical methods. The main focus is to investigate how several important parameters affect the dynamics and shape of the resulting wave formations. As a result, many wave solutions, including solitary, shock, singular, shock-singular, hyperbolic, singular periodic, and rational waves, can be constructed by symbolic computations. The parameter regimes guarantee that a careful analysis determines the generated solutions' validity and physical significance. The interpretation of the results is improved by displaying the solutions across a variety of parameter values using contour graphs, 2D plots, and 3D surface representations. These results confirm the efficiency and dependability of the used computational paradigm, which has a great deal of promise for further studies looking at more intricate nonlinear processes.

References

[1]	J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. phys. B, 20 (2006), 1141–1199. https://doi.org/10.1142/S0217979206033796 doi: 10.1142/S0217979206033796
[2]	Y. Wang, Z. Zhao, Y. Zhang, Numerical calculation of n-periodic wave solutions of the negative-order korteweg-de vries equations, Europhysics Lett., 146 (2024), 32002. https://doi.org/10.1209/0295-5075/ad3a10 doi: 10.1209/0295-5075/ad3a10
[3]	K. U. Tariq, R. N. Tufail, Lump and travelling wave solutions of a (3+1)-dimensional nonlinear evolution equation, J. Ocean Eng. Sci., 9 (2024), 164–172. https://doi.org/10.1016/j.joes.2022.04.018 doi: 10.1016/j.joes.2022.04.018
[4]	J. H. He, W. F. Hou, C. H. He, T. Saeed, T. Hayat, Variational approach to fractal solitary waves, Fractals, 29 (2021), 2150199. https://doi.org/10.1142/S0218348X21501991 doi: 10.1142/S0218348X21501991
[5]	S. Zheng, Y. Lou, S. Shen, J. Lu, Numerical investigation of fractal oscillator for a pendulum with a rolling wheel, Fractals, 2025. https://doi.org/10.1142/S0218348X2550077X
[6]	Z. Zhao, Y. Wang, $N$-periodic wave solutions of the (2+1)-dimensional integrable nonlocal nonlinear schrödinger equations, Wave Motion, 136 (2025), 103526. https://doi.org/10.1016/j.wavemoti.2025.103526 doi: 10.1016/j.wavemoti.2025.103526
[7]	M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge university press, 1991. https://doi.org/10.1017/CBO9780511623998
[8]	F. Badshah, K. U. Tariq, H. Ilyas, R. N. Tufail, Soliton, lumps, stability analysis and modulation instability for an extended (2+1)-dimensional Boussinesq model in shallow water, Chaos Soliton. Fract., 187 (2024), 115352. https://doi.org/10.1016/j.chaos.2024.115352 doi: 10.1016/j.chaos.2024.115352
[9]	S. Y. Lou, L. L. Chen, Formal variable separation approach for nonintegrable models, J. Math. Phys., 40 (1999), 6491–6500.
[10]	M. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 216 (1996), 67–75. https://doi.org/10.1016/0375-9601(96)00283-6 doi: 10.1016/0375-9601(96)00283-6
[11]	H. Qawaqneh, A. Altalbe, A. Bekir, K. U. Tariq, Investigation of soliton solutions to the truncated M-fractional (3+1)-dimensional Gross-Pitaevskii equation with periodic potential, AIMS Mathematics, 9 (2024), 23410–23433. https://doi.org/10.3934/math.20241138 doi: 10.3934/math.20241138
[12]	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.
[13]	E. Yaşar, Y. Yıldırım, A. R. Adem, Perturbed optical solitons with spatio-temporal dispersion in (2+1)-dimensions by extended kudryashov method, Optik, 158 (2018), 1–14. https://doi.org/10.1016/j.ijleo.2017.11.205 doi: 10.1016/j.ijleo.2017.11.205
[14]	T. A. Nofal, Simple equation method for nonlinear partial differential equations and its applications, J. Egyptian Math. Soc., 24 (2016), 204–209. https://doi.org/10.1016/j.joems.2015.05.006 doi: 10.1016/j.joems.2015.05.006
[15]	A. J. M. Jawad, M. D. Petković, A. Biswas, Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217 (2010), 869–877. https://doi.org/10.1016/j.amc.2010.06.030 doi: 10.1016/j.amc.2010.06.030
[16]	N. A. Kudryashov, N. B. Loguinova, Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205 (2008), 396–402. https://doi.org/10.1016/j.amc.2008.08.019 doi: 10.1016/j.amc.2008.08.019
[17]	H. O. Roshid, M. A. Akbar, M. N. Alam, M. F. Hoque, N. Rahman, et al., New extended (G?/G)-expansion method to solve nonlinear evolution equation: The (3+1)-dimensional potential-YTSF equation, SpringerPlus, 3 (2014), 122. https://doi.org/10.1186/2193-1801-3-122 doi: 10.1186/2193-1801-3-122
[18]	Y. Fang, H. Dong, Y. Hou, Y. Kong, Frobenius integrable decompositions of nonlinear evolution equations with modified term, Appl. Math. Comput., 226 (2014), 435–440. https://doi.org/10.1016/j.amc.2013.10.047 doi: 10.1016/j.amc.2013.10.047
[19]	A. Bekir, A. Boz, Exact solutions for nonlinear evolution equations using Exp-function method, Phys. Lett. A, 372 (2008), 1619–1625. https://doi.org/10.1016/j.physleta.2007.10.018 doi: 10.1016/j.physleta.2007.10.018
[20]	N. A. Kudryashov, M. B. Soukharev, Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation, Regul. Chaot. Dyn., 14 (2009), 407–419. https://doi.org/10.1134/S1560354709030046 doi: 10.1134/S1560354709030046
[21]	N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Soliton. Fract., 24 (2005), 1217–1231. https://doi.org/10.1016/j.chaos.2004.09.109 doi: 10.1016/j.chaos.2004.09.109
[22]	A. M. Wazwaz, The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput., 188 (2007), 1467–1475. https://doi.org/10.1016/j.amc.2006.11.013 doi: 10.1016/j.amc.2006.11.013
[23]	S. M. R. Islam, K. Khan, M. A. Akbar, Study of exp (-$\phi$ ($\xi$))-expansion method for solving nonlinear partial differential equations, J. Adv. Math. Comput. Sci., 5 (2015), 397–407. https://doi.org/10.973410.9734/BJMCS/2015/13387 doi: 10.973410.9734/BJMCS/2015/13387
[24]	E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277 (4-5) (2000) 212–218. https://doi.org/10.9734/BJMCS/2015/13387
[25]	M. Ali Akbar, N. H. M. Ali, The improved $F$-expansion method with Riccati equation and its applications in mathematical physics, Cogent Math., 4 (2017), 1282577. https://doi.org/10.1080/23311835.2017.1282577 doi: 10.1080/23311835.2017.1282577
[26]	K. U. Tariq, A. Jhangeer, M. N. Ali, H. Ilyas, R. N. Tufail, Lumps, solitons, modulation instability and stability analysis for the novel generalized (2+1)-dimensional nonlinear model arising in shallow water, Alex. Eng. J., 126 (2025), 45–52. https://doi.org/10.1016/j.aej.2025.03.110 doi: 10.1016/j.aej.2025.03.110
[27]	S. Bibi, N. Ahmed, U. Khan, S. T. Mohyud-Din, Auxiliary equation method for ill-posed Boussinesq equation, Phys. Scr., 94 (2019), 085213. https://doi.org/10.1088/1402-4896/ab1951 doi: 10.1088/1402-4896/ab1951
[28]	H. R. Dullin, G. A. Gottwald, D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501. https://doi.org/10.1103/PhysRevLett.87.194501 doi: 10.1103/PhysRevLett.87.194501
[29]	U. Younas, A. R. Seadawy, M. Younis, S. Rizvi, Dispersive of propagation wave structures to the dullin-Gottwald-Holm dynamical equation in a shallow water waves, Chinese J. Phys., 68 (2020), 348–364. https://doi.org/10.1016/j.cjph.2020.09.021 doi: 10.1016/j.cjph.2020.09.021
[30]	Y. Jawarneh, A. H. Hakami, A. A. Hassaballa, Generalized analysis of fractional Dullin-Gottwald-Holm equation: Soliton solutions and their applications in non-linear wave dynamics, Ain Shams Eng. J., 16 (2025), 103627. https://doi.org/10.1016/j.asej.2025.103627 doi: 10.1016/j.asej.2025.103627
[31]	H. R. Dullin, G. A. Gottwald, D. D. Holm, Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dyn. Res., 33 (2003), 73. https://doi.org/10.1016/S0169-5983(03)00046-7 doi: 10.1016/S0169-5983(03)00046-7
[32]	T. G. Ha, H. Liu, On traveling wave solutions of the $\theta$-equation of dispersive type, J. Math. Anal. Appl., 421 (2015), 399–414. https://doi.org/10.1016/j.jmaa.2014.06.058 doi: 10.1016/j.jmaa.2014.06.058
[33]	A. Biswas, A. H. Kara, 1-soliton solution and conservation laws of the generalized Dullin–Gottwald–Holm equation, Appl. Math. Comput., 217 (2010), 929–932. https://doi.org/10.1016/j.amc.2010.05.085 doi: 10.1016/j.amc.2010.05.085
[34]	Q. Meng, B. He, Y. Long, Z. Li, New exact periodic wave solutions for the Dullin–Gottwald–Holm equation, Appl. Math. Comput., 218 (2011), 4533–4537. https://doi.org/10.1016/j.amc.2011.10.035 doi: 10.1016/j.amc.2011.10.035
[35]	S. F. Tian, J. J. Yang, Z. Q. Li, Y. R. Chen, Blow-up phenomena of a weakly dissipative modified two-component Dullin–Gottwald–Holm system, Appl. Math. Lett., 106 (2020), 106378. https://doi.org/10.1016/j.aml.2020.106378 doi: 10.1016/j.aml.2020.106378
[36]	W. Cheng, T. Xu, Blow-up of solutions to a modified two-component Dullin–Gottwald–Holm system, Appl. Math. Lett., 105 (2020), 106307. https://doi.org/10.1016/j.aml.2020.106307 doi: 10.1016/j.aml.2020.106307
[37]	S. Demiray, Ö. Ünsal, A. Bekir, New exact solutions for Boussinesq type equations by using (G'/G, 1/G) and (1/G')-expansion methods, Acta Phys. Pol. A, 125 (2014), 1093–1098. https://doi.org/10.12693/aphyspola.125.1093 doi: 10.12693/aphyspola.125.1093
[38]	M. M. Miah, H. S. Ali, M. A. Akbar, A. M. Wazwaz, Some applications of the (G'/G, 1/g)-expansion method to find new exact solutions of NLEEs, Eur. Phys. J. Plus, 132 (2017), 252. https://doi.org/10.1140/epjp/i2017-11571-0 doi: 10.1140/epjp/i2017-11571-0

Reader Comments

Your name:*

Email:*
© 2025 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)