The dynamic behavior of Van der Waals gas system based on new extended (G'/G)-expansion method

Yulan Wang; Yulan Wang

doi:10.3934/math.2025845

AIMS Mathematics

2025, Volume 10, Issue 8: 18913-18928. doi: 10.3934/math.2025845

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The dynamic behavior of Van der Waals gas system based on new extended (G'/G)-expansion method

Yulan Wang ^,

Wuxi Vocational College of Science and Technology, Xinxi Road 8, Wuxi 214028, China

Received: 01 July 2025 Revised: 05 August 2025 Accepted: 11 August 2025 Published: 20 August 2025
MSC : 34C14, 37M05

In this study, I theoretically explored different aspects of the (1+1)-dimensional Van der Waals (VDW) gas system in the viscosity capillarity sense. To find the physical nature of the described system, a couple of new reliable integrating techniques such as the extended (N, R) expansion method and the new extended (G'/G)-expansion method were used, and a variety of soliton solutions were successfully derived. To enhance our understanding and visually represent these solutions, I created 2D and 3D plots, thereby utilizing appropriate parameter values that met the constraint conditions. Furthermore, I analyzed chaotic behavior, thereby observing its presence in the perturbed dynamical system and obtaining positive results that support chaotic motion. My computational analysis confirmed the effectiveness and versatility of my techniques in addressing nonlinear problems in mathematical science and engineering.
- Van der Waals gas system,
- chaotic analysis,
- extended (N,
- R)-expansion method,
- extended (G'/G)-expansion method,
- soliton dynamics
Citation: Yulan Wang. The dynamic behavior of Van der Waals gas system based on new extended (G'/G)-expansion method[J]. AIMS Mathematics, 2025, 10(8): 18913-18928. doi: 10.3934/math.2025845

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Abstract

In this study, I theoretically explored different aspects of the (1+1)-dimensional Van der Waals (VDW) gas system in the viscosity capillarity sense. To find the physical nature of the described system, a couple of new reliable integrating techniques such as the extended (N, R) expansion method and the new extended (G'/G)-expansion method were used, and a variety of soliton solutions were successfully derived. To enhance our understanding and visually represent these solutions, I created 2D and 3D plots, thereby utilizing appropriate parameter values that met the constraint conditions. Furthermore, I analyzed chaotic behavior, thereby observing its presence in the perturbed dynamical system and obtaining positive results that support chaotic motion. My computational analysis confirmed the effectiveness and versatility of my techniques in addressing nonlinear problems in mathematical science and engineering.

References

[1]	U. Younas, T. A. Sulaiman, J. Ren, On the study of optical soliton solutions to the three-component coupled nonlinear Schrodinger equation: Applications in fiber optics, Opt. Quant. Electron., 55 (2023), 72–81. https://doi.org/10.1007/s11082-022-04254-x doi: 10.1007/s11082-022-04254-x
[2]	W. Arendt, K. Urban, Partial differential equations: An introduction to analytical and numerical Methods, New York: Springer Cham, 2023. https://doi.org/10.1007/978-3-031-13379-4
[3]	W. Lefebvre, G. Loeper, H. Pham, Differential learning methods for solving fully nonlinear PDEs, Digit. Finance, 5 (2023), 183–229. https://doi.org/10.1007/s42521-023-00077-x doi: 10.1007/s42521-023-00077-x
[4]	P. G. LeFloch, J. M. Mercier, A class of mesh-free algorithms for some problems arising in finance and machine learning, J. Sci. Comput., 95 (2023), 75–89. https://doi.org/10.1007/s10915-023-02179-5 doi: 10.1007/s10915-023-02179-5
[5]	N. Raza, A. Deifalla, B. Rani, N. A. Shah, A. E. Ragab, Analyzing Soliton Solutions of the (n+1)-dimensional generalized Kadomtsev-Petviashvili equation: Comprehensive study of dark, bright, and periodic dynamics, Results Phys., 1 (2023), 107224. https://doi.org/10.1016/j.rinp.2023.107224 doi: 10.1016/j.rinp.2023.107224
[6]	R. Hadi, U. Najib, A. Lanre, A. Shah, H. Ahmad, Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov's method, Results Phys., 24 (2021), 104179. https://doi.org/10.1016/j.rinp.2021.104179 doi: 10.1016/j.rinp.2021.104179
[7]	A. Lanre, E. Francis, P. Valerio, O. Kayode, A study of nonlinear Riccati equation and its applications to multi-dimensional nonlinear evolution equations, Qual. Theor. Dyn. Syst., 23 (2024), 296–310. https://doi.org/10.1007/s12346-024-01137-2 doi: 10.1007/s12346-024-01137-2
[8]	M. Bilal, J. Iqbal, R. Ali, F. A. Awwad, E. A. A. Ismail, Establishing breather and N-soliton solutions for conformable Klein-Gordon equation, Open Phys., 22 (2024), 20240044. https://doi.org/10.1515/phys-2024-0044 doi: 10.1515/phys-2024-0044
[9]	R. Ali, S. Barak, A. Altalbe, Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations, Phys. Scripta, 99 (2024), 065235. https://doi.org/10.1088/1402-4896/ad4784 doi: 10.1088/1402-4896/ad4784
[10]	M. H. Rafiq, N. Jannat, M. N. Rafiq, Sensitivity analysis and analytical study of the three-component coupled NLS-type equations in fiber optics, Opt. Quant. Electron., 55 (2023), 637–648. https://doi.org/10.1007/s11082-023-04908-4 doi: 10.1007/s11082-023-04908-4
[11]	N. Raza, B. Rani, A. M. Wazwaz, A novel investigation of extended (3+1)-dimensional shallow water wave equation with constant coefficients utilizing bilinear form, Phys. Lett. A, 485 (2023), 129082. https://doi.org/10.1016/j.physleta.2023.129082 doi: 10.1016/j.physleta.2023.129082
[12]	M. N. Rafiq, H. Chen, M. H. Rafiq, Stability analysis and multi-wave structures of the ill-posed Boussinesq equation arising in nonlinear physical science, Opt. Quant. Electron., 55 (2023), 1243–1256. https://doi.org/10.1007/s11082-023-05537-7 doi: 10.1007/s11082-023-05537-7
[13]	M. H. Rafiq, N. Raza, A. Jhangeer, Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability, Chaos Soliton. Fract., 171 (2023), 113436. https://doi.org/10.1016/j.chaos.2023.113436 doi: 10.1016/j.chaos.2023.113436
[14]	N. Raza, B. Rani, Y. Chahlaoui, N. A. Shah, A variety of new rogue wave patterns for three coupled nonlinear Maccari's models in complex form, Nonlinear Dynam., 111 (2023), 18419–18437. https://doi.org/10.1007/s11071-023-08839-3 doi: 10.1007/s11071-023-08839-3
[15]	M. Djilali, A. L. I. Hakem, (G/G)-expansion method to seek traveling wave solutions for some fractional nonlinear PDEs arising in natural sciences, Adv. Theor. Nonlinear Anal. Appl., 7 (2023), 303–318. https://doi.org/10.31197/atnaa.1125691 doi: 10.31197/atnaa.1125691
[16]	V. Vinita, R. S. Saha, Quasi-self-adjointness, conservation laws, and symmetry reductions with analytical solutions using Lie symmetry analysis and geometric approach for the (3+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation, Phys. Fluids, 35 (2023), 1–15.
[17]	M. H. Rafiq, N. Raza, A. Jhangeer, Nonlinear dynamics of the generalized unstable nonlinear Schrodinger equation: A graphical perspective, Opt. Quant. Electron., 55 (2023), 628–634. https://doi.org/10.1007/s11082-023-04904-8 doi: 10.1007/s11082-023-04904-8
[18]	M. H. Rafiq, A. Jhangeer, N. Raza, Symmetry and complexity: A Lie symmetry approach to bifurcation, chaos, stability and travelling wave solutions of the (3+1)-dimensional Kadomtsev- Petviashvili equation, Phys. Scripta, 98 (2023), 115239. https://doi.org/10.1088/1402-4896/acff44 doi: 10.1088/1402-4896/acff44
[19]	W. X. Ma, Integrable nonlocal nonlinear Schrödinger hierarchies of type (-λ^, λ) and soliton solutions, Rep. Math. Phys.*, 92 (2023), 19–36. https://doi.org/10.1016/S0034-4877(23)00052-6 doi: 10.1016/S0034-4877(23)00052-6
[20]	D. Gao, X. Lü, M. S. Peng, Study on the (2+1)-dimensional extension of Hietarinta equation: Soliton solutions and Bäcklund transformation, Phys. Scripta, 98 (2023), 095225. https://doi.org/10.1088/1402-4896/ace8d0 doi: 10.1088/1402-4896/ace8d0
[21]	A. Hussain, Y. Chahlaoui, M. Usman, F. D. Zaman, C. Park, Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation, Sci. Rep., 13 (2023), 15383. https://doi.org/10.1038/s41598-023-42477-4 doi: 10.1038/s41598-023-42477-4
[22]	N. Raza, A. Batool, B. H. Mehmet, C. F. S. Vidal, A variety of soliton solutions of the extended Gerdjikov-Ivanov equation in the DWDMsystem, Int. J. Mod. Phys. B, 1 (2023), 2450075. https://doi.org/10.1142/S0217979224500759 doi: 10.1142/S0217979224500759
[23]	E. H. Zahran, A. Bekir, Optical soliton solutions to the perturbed biswas-milovic equation with kudryashov's law of refractive index, Opt. Quant. Electron., 55 (2023), 1211–1222. https://doi.org/10.1007/s11082-023-05453-w doi: 10.1007/s11082-023-05453-w
[24]	M. F. Alotaibi, N. Raza, M. H. Rafiq, A. Soltani, New solitary waves, bifurcation and chaotic patterns of Fokas system arising in monomode fiber communication system, Alex. Eng. J., 67 (2023), 583–595. https://doi.org/10.1016/j.aej.2022.12.069 doi: 10.1016/j.aej.2022.12.069
[25]	M. Iqbal, A. R. Seadawy, D. C. Lu, Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Mod. Phys. Lett. A, 33 (2018), 1850183. https://doi.org/10.1142/S0217732318501833 doi: 10.1142/S0217732318501833
[26]	D. C. Lu, A. R. Seadawy, M. Iqbal, Mathematical methods via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications, Results Phys., 11 (2018), 1161–1171. https://doi.org/10.1016/j.rinp.2018.11.014 doi: 10.1016/j.rinp.2018.11.014
[27]	A. H. Arnous, M. Mirzazadeh, M. S. Hashemi, N. A. Shah, J. D. Chung, Three different integration schemes for finding soliton solutions in the (1+1)-dimensional Van der Waals gas system, Results Phys., 55 (2023), 107178. https://doi.org/10.1016/j.rinp.2023.107178 doi: 10.1016/j.rinp.2023.107178
[28]	M. Malaver, H. D. Kasmaei, Analytical models for quark stars with van der Waals modified equation of state, Int. J. Astrophys. Space Sci., 7 (2019), 58–69. https://doi.org/10.11648/j.ijass.20190705.11 doi: 10.11648/j.ijass.20190705.11
[29]	L. Kipgen, R. Singh, D-shocks and vacuum states in the Riemann problem for isothermal van der Waals dusty gas under the flux approximation, Phys. Fluids, 35 (2023), 016116. https://doi.org/10.1063/5.0135491 doi: 10.1063/5.0135491
[30]	S. K. Mohanty, O. V. Kravchenko, M. K. Deka, A. N. Dev, D. V. Churikov, The exact solutions of the 2+1-dimensional Kadomtsev-Petviashvili equation with variable coefficients by extended generalized G'/G-expansion method, J. King Saud Univ. Sci., 35 (2023), 102358. https://doi.org/10.1016/j.jksus.2022.102358 doi: 10.1016/j.jksus.2022.102358
[31]	O. A. Ilhan, H. M. Baskonus, M. N. Islam, M. A. Akbar, D. Soybas, Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G'/G-expansion method, Int. J. Nonlin. Sci. Num., 24 (2021), 185–200. https://doi.org/10.1515/ijnsns-2020-0153 doi: 10.1515/ijnsns-2020-0153
[32]	M. Poonia, K. Singh, Exact solutions of nonlinear dynamics of microtubules equation using the methods of first integral and (G'/G) expansion, Asian-Eur. J. Math., 16 (2023), 2350007. https://doi.org/10.1142/s1793557123500079 doi: 10.1142/s1793557123500079
[33]	M. S. Uddin, M. Begum, H. O. Roshid, M. S. Ullah, A. Abdeljabbar, Soliton solutions of a (2+1)-dimensional nonlinear time-fractional Bogoyavlenskii equation model, Part. Differ. Equ. Appl. Math., 8 (2023), 100591. https://doi.org/10.1016/j.padiff.2023.100591 doi: 10.1016/j.padiff.2023.100591
[34]	N. Raza, M. H. Rafiq, T. A. Alrebdi, A. H. A. Aty, New solitary waves, bifurcation and chaotic patterns of coupled nonlinear Schrodinger system arising in fibre optics, Opt. Quant. Electron., 55 (2023), 853–866. https://doi.org/10.1007/s11082-023-05097-w doi: 10.1007/s11082-023-05097-w
[35]	M. H. Rafiq, N. Raza, A. Jhangeer, Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: A graphical perspective, Opt. Quant. Electron., 55 (2023), 628–634. https://doi.org/10.1007/s11082-023-04904-8 doi: 10.1007/s11082-023-04904-8
[36]	O. Z. E. R. Ahmet, A. K. I. N. Erhan, Tools for detecting chaos, Sakarya Univ. J. Sci., 9 (2005), 60–66.

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