This article investigated the solitary wave solutions to the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation. The proposed model is particularly applicable to shallow water wave dynamics and may also extend to contexts such as acoustic wave propagation, nonlinear electric media, and oceanic wave phenomena. First, we constructed the ordinary differential equation form of the nonlinear partial differential equation with the help of the traveling wave transformation. After that, we utilized the generalized Arnous method and the modified sub-equation method to construct the solitary waves containing hyperbolic, exponential, trigonometric, and inverse functions. Using suitable parameter values, the graphical aspects of solutions are demonstrated by plotting a 3D surface plot (including a contour and density plot), a 2D surface plot, a streamline plot, and a polar plot. By utilizing these approaches, accurate analytical solutions for soliton waves were generated, which comprise kink, bright, and dark waves. We employed the generalized Arnous method and the modified sub-equation method to formulate a technique for addressing integrable systems, providing a valuable framework for examining nonlinear phenomena across various physical contexts. This study's outcomes enhance both nonlinear dynamical processes and solitary wave theory.
Citation: Khizar Farooq, Ejaz Hussain, Hamza Ali Abujabal, Fehaid Salem Alshammari. Propagation of nonlinear dispersive waves in shallow water and acoustic media in the framework of integrable Schwarz–Korteweg–de Vries equation[J]. AIMS Mathematics, 2025, 10(8): 17543-17566. doi: 10.3934/math.2025784
This article investigated the solitary wave solutions to the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation. The proposed model is particularly applicable to shallow water wave dynamics and may also extend to contexts such as acoustic wave propagation, nonlinear electric media, and oceanic wave phenomena. First, we constructed the ordinary differential equation form of the nonlinear partial differential equation with the help of the traveling wave transformation. After that, we utilized the generalized Arnous method and the modified sub-equation method to construct the solitary waves containing hyperbolic, exponential, trigonometric, and inverse functions. Using suitable parameter values, the graphical aspects of solutions are demonstrated by plotting a 3D surface plot (including a contour and density plot), a 2D surface plot, a streamline plot, and a polar plot. By utilizing these approaches, accurate analytical solutions for soliton waves were generated, which comprise kink, bright, and dark waves. We employed the generalized Arnous method and the modified sub-equation method to formulate a technique for addressing integrable systems, providing a valuable framework for examining nonlinear phenomena across various physical contexts. This study's outcomes enhance both nonlinear dynamical processes and solitary wave theory.
| [1] |
S. M. Rayhanul Islam, K. Khan, M. A. Akbar, Optical soliton solutions, bifurcation, and stability analysis of the Chen–Lee–Liu model, Results Phys., 51 (2023), 106620. https://doi.org/10.1016/j.rinp.2023.106620 doi: 10.1016/j.rinp.2023.106620
|
| [2] |
E. Hussain, Y. Arafat, S. Malik, F. S. Alshammari, The (2+1)-dimensional chiral nonlinear Schrödinger equation: extraction of soliton solutions and sensitivity analysis, Axioms, 14 (2025), 422. https://doi.org/10.3390/axioms14060422 doi: 10.3390/axioms14060422
|
| [3] |
J. T. Stuart, On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech., 4 (1958), 1–21. https://doi.org/10.1017/S0022112058000276 doi: 10.1017/S0022112058000276
|
| [4] |
K. Farooq, E. Hussain, U. Younas, H. Mukalazi, T. M. Khalaf, A. Mutlib, et al., Exploring the wave's structures to the nonlinear coupled system arising in surface geometry, Sci. Rep., 15 (2025), 11624. https://doi.org/10.1038/s41598-024-84657-w doi: 10.1038/s41598-024-84657-w
|
| [5] | W. F. Ames, Nonlinear partial differential equations in engineering, Academic Press, 1965. |
| [6] |
M. A. Mustafa, M. A. S. Murad, Soliton solutions to the time-fractional Kudryashov equation: applications of the new direct mapping method, Mod. Phys. Lett. B, 39 (2025), 2550147. https://doi.org/10.1142/S0217984925501477 doi: 10.1142/S0217984925501477
|
| [7] |
E. Hussain, A. H. Tedjani, K. Farooq, Beenish, Modeling and exploration of localized wave phenomena in optical fibers using the generalized Kundu–Eckhaus equation for femtosecond pulse transmission, Axioms, 14 (2025), 513. https://doi.org/10.3390/axioms14070513 doi: 10.3390/axioms14070513
|
| [8] |
W. B. Rabie, H. M. Ahmed, M. Mirzazadeh, A. Akbulut, M. S. Hashemi, Investigation of solitons and conservation laws in an inhomogeneous optical fiber through a generalized derivative nonlinear Schrödinger equation with quintic nonlinearity, Opt. Quant. Electron., 55 (2023), 825. https://doi.org/10.1007/s11082-023-05070-7 doi: 10.1007/s11082-023-05070-7
|
| [9] |
K. Farooq, A. H. Tedjani, Z. Li, E. Hussain, Soliton dynamics of the nonlinear Kodama equation with M-truncated derivative via two innovative schemes: the generalized Arnous method and the Kudryashov method, Fractal Fract., 9 (2025), 436. https://doi.org/10.3390/fractalfract9070436 doi: 10.3390/fractalfract9070436
|
| [10] |
M. A. Akbar, F. A. Abdullah, M. T. Islam, M. A. Al Sharif, M. S. Osman, New solutions of the soliton type of shallow water waves and superconductivity models, Results Phys., 44 (2023), 106180. https://doi.org/10.1016/j.rinp.2022.106180 doi: 10.1016/j.rinp.2022.106180
|
| [11] |
M. A. Abdou, A generalized auxiliary equation method and its applications, Nonlinear Dyn., 52 (2008), 95–102. https://doi.org/10.1007/s11071-007-9261-y doi: 10.1007/s11071-007-9261-y
|
| [12] |
G. L. Lamb Jr., Bäcklund transformations for certain nonlinear evolution equations, J. Math. Phys., 15 (1974), 2157–2165. https://doi.org/10.1063/1.1666595 doi: 10.1063/1.1666595
|
| [13] |
J.-H. He, X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700–708. https://doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
|
| [14] |
S. A. El-Wakil, M. A. Abdou, The extended Fan sub-equation method and its applications for a class of nonlinear evolution equations, Chaos Soliton. Fract., 36 (2008), 343–353. https://doi.org/10.1016/j.chaos.2006.06.065 doi: 10.1016/j.chaos.2006.06.065
|
| [15] |
J.-H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), 73–79. https://doi.org/10.1016/S0096-3003(01)00312-5 doi: 10.1016/S0096-3003(01)00312-5
|
| [16] | M. S. Islam, K. Khan, A. H. Arnous, Generalized Kudryashov method for solving some (3+1)-dimensional nonlinear evolution equations, New Trends in Mathematical Sciences, 3 (2015), 46–57. |
| [17] | J. Hietarinta, Introduction to the Hirota bilinear method, In: Integrability of nonlinear systems, Berlin, Heidelberg: Springer, 1997, 95–103. https://doi.org/10.1007/BFb0113694 |
| [18] |
Y. Gurefe, E. Misirli, A. Sonmezoglu, M. Ekici, Extended trial equation method to generalized nonlinear partial differential equations, Appl. Math. Comput., 219 (2013), 5253–5260. https://doi.org/10.1016/j.amc.2012.11.046 doi: 10.1016/j.amc.2012.11.046
|
| [19] | C. Gu, H. Hu, Z. Zhou, Darboux transformations in integrable systems: theory and their applications to geometry, Dordrecht: Springer, 2005. https://doi.org/10.1007/1-4020-3088-6 |
| [20] |
N. A. Kudryashov, Method for finding highly dispersive optical solitons of nonlinear differential equations, Optik, 206 (2020), 163550. https://doi.org/10.1016/j.ijleo.2019.163550 doi: 10.1016/j.ijleo.2019.163550
|
| [21] |
J. Zhang, X. Wei, Y. Lu, A generalized $G'/G$-expansion method and its applications, Phys. Lett. A, 372 (2008), 3653–3658. https://doi.org/10.1016/j.physleta.2008.02.027 doi: 10.1016/j.physleta.2008.02.027
|
| [22] |
S. M. Mirhosseini-Alizamini, H. Rezazadeh, M. Eslami, M. Mirzazadeh, A. Korkmaz, New extended direct algebraic method for the Tzitzica type evolution equations arising in nonlinear optics, Comput. Methods Differ., 8 (2020), 28–53. https://doi.org/10.22034/CMDE.2019.9472 doi: 10.22034/CMDE.2019.9472
|
| [23] |
M. A. E. Abdelrahman, E. H. M. Zahran, M. M. A. Khater, The Exp($-\phi(\Xi))$-expansion method and its application for solving nonlinear evolution equations, International Journal of Modern Nonlinear Theory and Application, 4 (2015), 37–47. https://doi.org/10.4236/ijmnta.2015.41004 doi: 10.4236/ijmnta.2015.41004
|
| [24] |
S. A. El-Wakil, M. A. Abdou, The extended mapping method and its applications for nonlinear evolution equations, Phys. Lett. A, 358 (2006), 275–282. https://doi.org/10.1016/j.physleta.2006.05.040 doi: 10.1016/j.physleta.2006.05.040
|
| [25] |
E. Hussain, Z. Li, S. A. A. Shah, E. A. Az-Zo'bi, M. Hussien, Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation, Opt. Quant. Electron., 55 (2023), 1274. https://doi.org/10.1007/s11082-023-05588-w doi: 10.1007/s11082-023-05588-w
|
| [26] |
Z. Li, E. Hussain, Qualitative analysis and optical solitons for the (1+1)-dimensional Biswas–Milovic equation with parabolic law and nonlocal nonlinearity, Results Phys., 56 (2024), 107304. https://doi.org/10.1016/j.rinp.2023.107304 doi: 10.1016/j.rinp.2023.107304
|
| [27] |
S. A. A. Shah, E. Hussain, W.-X. Ma, Z. Li, A. E. Ragab, T. M. Khalaf, Qualitative analysis and new variety of soliton profiles for the (1+1)-dimensional modified equal-width equation, Chaos Soliton. Fract., 187 (2024), 115353. https://doi.org/10.1016/j.chaos.2024.115353 doi: 10.1016/j.chaos.2024.115353
|
| [28] |
S. Malik, S. Kumar, A. Das, A (2+1)-dimensional combined KdV-mKdV equation: integrability, stability analysis and soliton solutions, Nonlinear Dyn., 107 (2022), 2689–2701. https://doi.org/10.1007/s11071-021-07075-x doi: 10.1007/s11071-021-07075-x
|
| [29] |
A. Zabihi, M. T. Shaayesteh, H. Rezazadeh, R. Ansari, N. Raza, A. Bekir, Solitons solutions to the high-order dispersive cubic–quintic Schrödinger equation in optical fibers, J. Nonlinear Opt. Phys., 32 (2023), 2350027. https://doi.org/10.1142/S0218863523500273 doi: 10.1142/S0218863523500273
|
| [30] |
Md. T. Islam, Md. A. Akbar, J. F. Gómez-Aguilar, E. Bonyah, G. Fernandez-Anaya, Assorted soliton structures of solutions for fractional nonlinear Schrödinger-type evolution equations, J. Ocean Eng. Sci., 7 (2022), 528–535. https://doi.org/10.1016/j.joes.2021.10.006 doi: 10.1016/j.joes.2021.10.006
|
| [31] |
Beenish, E. Hussain, U. Younas, R. Tapdigoglu, M. Garayev, Exploring bifurcation, quasi-periodic patterns, and wave dynamics in an extended Calogero-Bogoyavlenskii-Schiff model with sensitivity analysis, Int. J. Theor. Phys., 64 (2025), 146. https://doi.org/10.1007/s10773-025-06008-3 doi: 10.1007/s10773-025-06008-3
|
| [32] |
K.-J. Wang, Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field, Axioms, 11 (2022), 445. https://doi.org/10.3390/axioms11090445 doi: 10.3390/axioms11090445
|
| [33] |
S. Singh, S. S. Ray, Integrability and new periodic, kink-antikink and complex optical soliton solutions of (3+1)-dimensional variable coefficient DJKM equation for the propagation of nonlinear dispersive waves in inhomogeneous media, Chaos Soliton. Fract., 168 (2023), 113184. https://doi.org/10.1016/j.chaos.2023.113184 doi: 10.1016/j.chaos.2023.113184
|
| [34] |
E. H. Zahran, S. M. Mirhosseini-Alizamini, M. S. M. Shehata, H. Rezazadeh, H. Ahmad, Study on abundant explicit wave solutions of the thin-film Ferro-electric materials equation, Opt. Quant. Electron., 54 (2022), 48. https://doi.org/10.1007/s11082-021-03296-x doi: 10.1007/s11082-021-03296-x
|
| [35] |
X. Pu, W. Zhou, The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain, Nonlinear Anal., 251 (2025), 113688. https://doi.org/10.1016/j.na.2024.113688 doi: 10.1016/j.na.2024.113688
|
| [36] |
I. Ul. Haq, N. Ali, S. Ahmad, T. Akram, A hybrid interpolation method for fractional PDEs and its applications to fractional diffusion and Buckmaster equations, Math. Probl. Eng., 2022 (2022), 2517602. https://doi.org/10.1155/2022/2517602 doi: 10.1155/2022/2517602
|
| [37] |
W. A. Faridi, S. A. AlQahtani, The explicit power series solution formation and computation of Lie point infinitesimal generators: Lie symmetry approach, Phys. Scripta, 98 (2023), 125249. https://doi.org/10.1088/1402-4896/ad0948 doi: 10.1088/1402-4896/ad0948
|
| [38] |
Md. T. Islam, T. R. Sarkar, F. A. Abdullah, J. F. Gómez-Aguilar, Characteristics of dynamic waves in incompressible fluid regarding nonlinear Boiti-Leon-Manna-Pempinelli model, Phys. Scripta, 98 (2023), 085230. https://doi.org/10.1088/1402-4896/ace743 doi: 10.1088/1402-4896/ace743
|
| [39] |
H. Zheng, Y. Xia, Persistence of solitary-wave solutions for the delayed regularized long wave equation under Kuramoto–Sivashinsky perturbation and Marangoni effect, Chaos Soliton. Fract., 184 (2024), 115049. https://doi.org/10.1016/j.chaos.2024.115049 doi: 10.1016/j.chaos.2024.115049
|
| [40] | H. Zheng, Y. Xia, Bifurcation of the travelling-wave solutions in a perturbed (1+1)-dimensional dispersive long wave equation via a geometric approach, Proc. Roy. Soc. Edinb. A, in press. https://doi.org/10.1017/prm.2024.45 |
| [41] |
A.-M. Wazwaz, S. A. El-Tantawy, A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions, Nonlinear Dyn., 83 (2016), 1529–1534. https://doi.org/10.1007/s11071-015-2427-0 doi: 10.1007/s11071-015-2427-0
|
| [42] |
I. M. Krichever, S. P. Novikov, Holomorphic bundles over algebraic curves and non-linear equations, Russ. Math. Surv., 35 (1980), 53–79. https://doi.org/10.1070/RM1980v035n06ABEH001974 doi: 10.1070/RM1980v035n06ABEH001974
|
| [43] |
J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1983), 522–526. https://doi.org/10.1063/1.525721 doi: 10.1063/1.525721
|
| [44] |
J. Weiss, The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations, J. Math. Phys., 26 (1985), 258–269. https://doi.org/10.1063/1.526655 doi: 10.1063/1.526655
|
| [45] |
K. Toda, S.-J. Yu, The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+1) dimensions, J. Math. Phys., 41 (2000), 4747–4751. https://doi.org/10.1063/1.533374 doi: 10.1063/1.533374
|
| [46] |
J. Ramírez, J. L. Romero, M. S. Bruzón, M. L. Gandarias, Multiple solutions for the Schwarzian Korteweg–de Vries equation in (2+1) dimensions, Chaos Soliton. Fract., 32 (2007), 682–693. https://doi.org/10.1016/j.chaos.2005.11.019 doi: 10.1016/j.chaos.2005.11.019
|
| [47] |
A. Nakamura, The Miura transform and the existence of an infinite number of conservation laws of the cylindrical KdV equation, Phys. Lett. A, 82 (1981), 111–112. https://doi.org/10.1016/0375-9601(81)90924-5 doi: 10.1016/0375-9601(81)90924-5
|
| [48] |
J. Mas, E. Ramos, The constrained KP hierarchy and the generalised Miura transformation, Phys. Lett. B, 351 (1995), 194–199. https://doi.org/10.1016/0370-2693(95)00357-Q doi: 10.1016/0370-2693(95)00357-Q
|
| [49] |
M. S. Velan, M. Lakshmanan, Lie symmetries, Kac-Moody-Virasoro algebras and integrability of certain (2+1)-dimensional nonlinear evolution equations, J. Nonlinear Math. Phys., 5 (1998), 190–211. https://doi.org/10.2991/jnmp.1998.5.2.10 doi: 10.2991/jnmp.1998.5.2.10
|
| [50] |
A. R. Seadawy, M. Iqbal, D. Lu, Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev–Petviashvili modified equal-width dynamical equation, Comput. Math. Appl., 78 (2019), 3620–3632. https://doi.org/10.1016/j.camwa.2019.06.013 doi: 10.1016/j.camwa.2019.06.013
|
| [51] |
J. Wang, K. Shehzad, A. R. Seadawy, M. Arshad, F. Asmat, Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and coupled Zakharov–Kuznetsov systems with their stability, J. Taibah Univ. Sci., 17 (2023), 2163872. https://doi.org/10.1080/16583655.2022.2163872 doi: 10.1080/16583655.2022.2163872
|
| [52] |
X. Li, M. Zhang, Darboux transformation and soliton solutions of the (2+1)-dimensional Schwarz–Korteweg–de Vries equation, Mod. Phys. Lett. B, 34 (2020), 2050270. https://doi.org/10.1142/S021798492050270X doi: 10.1142/S021798492050270X
|
| [53] |
Z. Li, Diversity soliton excitations for the (2+1)-dimensional Schwarzian KdV equation, Therm. Sci., 22 (2018), 1781–1786. https://doi.org/10.2298/TSCI1804781L doi: 10.2298/TSCI1804781L
|
| [54] | M. A. S. Murad, M. A. Mustafa, Nonlinear conformable Schrödinger equation in weakly nonlocal media using a new generalized computational technique, International Journal of Computational Mathematics, in press. https://doi.org/10.1080/00207160.2025.2507677 |
| [55] |
M. L. Gandarias, M. S. Bruzón, J. Ramirez, Classical symmetry reductions of the Schwarz–Korteweg–de Vries equation in 2+1 dimensions, Theor. Math. Phys., 134 (2003), 62–71. https://doi.org/10.1023/A:1021867622943 doi: 10.1023/A:1021867622943
|
| [56] |
M. A. S. Murad, S. S. Mahmood, H. Emadifar, W. W. Mohammed, K. K. Ahmed, Optical soliton solution for dual-mode time-fractional nonlinear Schrödinger equation by generalized exponential rational function method, Results Eng., 27 (2025), 105591. https://doi.org/10.1016/j.rineng.2025.105591 doi: 10.1016/j.rineng.2025.105591
|
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Khizar Farooq, Ejaz Hussain, Hamza Ali Abujabal, Fehaid Salem Alshammari. Propagation of nonlinear dispersive waves in shallow water and acoustic media in the framework of integrable Schwarz–Korteweg–de Vries equation[J]. AIMS Mathematics, 2025, 10(8): 17543-17566. doi: 10.3934/math.2025784
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