Multiplicity of kink and hump structures in Van der Waals gas system

Rashid Ali; Rabia Imtiaz; Moin-ud-Din Junjua; Fuad A. Awwad; Emad A. A. Ismail; Ahmed S. Hendy; Rashid Ali; Rabia Imtiaz; Moin-ud-Din Junjua; Fuad A. Awwad; Emad A. A. Ismail; Ahmed S. Hendy

doi:10.3934/math.2025564

AIMS Mathematics

2025, Volume 10, Issue 5: 12493-12518. doi: 10.3934/math.2025564

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Research article Special Issues

Multiplicity of kink and hump structures in Van der Waals gas system

1.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
2.
Department of Mathematics, University of Engineering and Technology, Lahore 39161, Pakistan
3.
Department of Mathematics, Ghazi University, Dera Ghazi Khan 32200, Pakistan
4.
Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia
5.
Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia
6.
Department of Mechanics and Mathematics, Western Caspian University, Baku 1001, Azerbaijan

Received: 21 March 2025 Revised: 03 May 2025 Accepted: 09 May 2025 Published: 28 May 2025
MSC : 34G20, 35A20, 35A22, 35R11

In this work, numerous features of the nonlinear Van der Waals gas system in the sense of viscosity capillarity are analytically and theoretically investigated. A novel trustworthy integrating approach, namely the Riccati modified extended simple equation method (RMESEM), is utilized to determine the physical nature of the system. This technique successfully generates a novel range of kink, anti-kink, cuspon kink, and bell-shaped bright and dark hump solitary wave solutions in the form of exponential, periodic, hyperbolic, rational, and rational-hyperbolic functions. Using suitable parameter values that satisfy constraint criteria, we generate a set of 2D graphs to improve comprehension and graphically depict these solitary wave solutions. The efficiency and adaptability of our method in solving a range of nonlinear problems in mathematical science and engineering are validated by our computational research.

Keywords:

Citation: Rashid Ali, Rabia Imtiaz, Moin-ud-Din Junjua, Fuad A. Awwad, Emad A. A. Ismail, Ahmed S. Hendy. Multiplicity of kink and hump structures in Van der Waals gas system[J]. AIMS Mathematics, 2025, 10(5): 12493-12518. doi: 10.3934/math.2025564

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Abstract

1. Introduction

Of all the tourism segments, cultural tourism is, without a doubt, the one that has grown the most in the last decades. That is why this segment attracts not only the tourists' attention but that of the researchers too. Researchers have studied this segment from an economic, geographic and cultural point of view (Ferrari et al., 2018), considering the marked seasonality of the sector (Cuccia and Rizzo, 2011; Mondéjar-Jiménez et al., 2007). The tourist activity driven by the cultural and heritage elements (McKercher et al., 2005) provides an important source of income for their preservation. These market segments that have existed for years started to develop by the end of the last century (Ferrari et al., 2014). Doubtlessly, an important cultural and heritage offer means an improvement in the image of the destination and has a positive impact on the choice of the consumer (Castro et al., 2007, Pérez-Calderón et al., 2020). This has motivated the researchers to analyze this segment and to clearly define it in the scientific literature (Cetin and Bilgihan, 2016).

Many of these studies have focused on the cultural motivations of the tourists (Chhabra et al., 2003) to learn the profile and priorities when choosing a destination. However, they have particularly focused on assessing and knowing the main reasons that affect the choice of a destination and the opinion after its visit (Cordente-Rodríguez et al., 2011). This opinion can, doubtlessly, modify the initial idea and favor a return to the destination or its recommendation. Opinion has previously been studied in the literature (Yoon and Uysal, 2005). Nevertheless, there are few studies that manage to reflect, in a consistent manner, the previous motivations to the choice of destination. These motivations must be known in order to adapt the destination to the majority preferences of the visitors (McKercher, 2002; McKercher and du Cros, 2003) which, in many cases, are not managed properly (Garrod and Fyall, 2000).

Cultural tourism, although with different grades of motivation of the users (Tsiotsou and Vasaioti, 2006), encompasses several activities that, in many occasions, contribute to the main reason for visiting (McKercher and du Cros, 2003; Lavín et al., 2017). Segmenting the tourists according to new typologies can establish another research line to address in the future (Frochot, 2005; Poria et al., 2003).

2. Objectives and hypothesis

The main objective of this study is assess a theoretical model to identify the main factors that determine the choosing of a tourist destination with an important cultural offer. The motivations of the tourists (parking, restaurants, transport...), together with the services the destination provides and its cultural offer, must affect its choice. The two factors under study shall alter these motivations as well.

The model proposed intends to measure the effects in different variables and know if the relations among them are statistically significant. For that purpose, the following hypotheses are stated:

H1: Tourists' motivations have a positive effect on the choice of a destination. Although this connection has been widely studied and previously contrasted (Chen and Rahman, 2018; Chhabra, 2009; Mondéjar-Jiménez and Vargas-Vargas, 2009; Mondéjar-Jiménez et al., 2009; Vergori and Arima, 2020; Xu et al., 2020), its intensity is the key factor to determine potential visitors.

H2: Services offered by the destination have a positive effect on the choice of said destination and the motivations of the potential tourists have been object of analysis and determine the image and subsequent choice of a tourist destination (García-Pozo et al., 2019; Hossein et al., 2019; Salazar, 2012; Scheyvens and Biddulph, 2018; Ying and Zhou, 2007).

√H2.1: Services offered by the destination have a positive effect on the choice of said destination.

√H2.2: Services offered by the destination have a positive effect on the motivations of the potential tourists.

H3: Cultural offer of the destination has a positive effect on the choice of said destination and on the motivations of its potential tourists (Hou et al., 2005; Lynch et al., 2011; Mondéjar-Jiménez et al., 2012; Richards, 2018; Secondi et al., 2011; Stylianou-Lambert, 2011). Cultural and heritage offer is possibly the main factor to determine a cultural destination.

√H3.1: Cultural offer of the destination has a positive effect on the choice of said destination.

√H3.2: Cultural offer of the destination has a positive effect on the motivations of the potential tourists.

These hypotheses shall be contrasted by means of a structural model using Partial Least Squares (PLS) due to its predictive character and the assumption of normality of some the variables (Assaker et al., 2013; Do Valle and Assaker, 2016; Ali et al., 2018; Hair et al., 2019).

2.1. Methodology

PLS methodology based on variance structure is recommended for these initial models which, even though they are based on previous studies (Secondi et al., 2011), they also introduce some substantial changes. Models shall be adapted to the selected destination, in this case, the city of Malaga. This type of models shows a very solid performance (Henseler et al., 2009).

Partial Least Squares Path Modeling (PLS) technique analyzes the relation among the suggested latent variables which are measured through different items obtained in a survey of the city tourists. The relation among the latent variables can reflect direct and indirect effects. The sum of both effects is the total effect among latent variables. The number of observations is sufficient since they exceed the minimum needed for the model to function (Hair et al., 2019).

3. Empirical study

The city of Malaga has received more than 4.5 million visitors (tourists + hikers) with an economic impact of more than 3000 million euros, according to the city's tourist observatory. They have generated more than 2.5 million hotel overnight stays, with the United Kingdom, Germany and France being the main emitters of international tourists.

A study was conducted through interviews with visitors to the city of Malaga from June to December 2019. Given the great number of foreign visitors, the questionnaire was provided in Spanish and English. The questionnaire that was used was the one proposed in Mondéjar and Gómez (2009) which shows a breakdown by blocks of Motivations, Services and Cultural Offer of the visited destination.

Interviewed tourists were selected at random among the visitors of the main city monuments (the Cathedral of Malaga, the Alcazaba, Carmen Thyssen Museum and Picasso Museum) at the exit of their visit. A balance in places of origin, sex, age and nationality was pursued in order to obtain a sample as representative as possible. A total of 416 valid questionnaires have been used. Overall, 600 questionnaires were collected but many of them were not completely filled in. In some cases, because the tourist was in a hurry and could not wait to continue discovering the city; in other cases because it could be one of the first visited monuments and the tourist did not have a full idea of the services and cultural offer of the city.

A total of 18 indicators have been used to define 4 latent factors: Destination (2 indicators), Cultural Offer (3 indicators), Services (6 indicators) and Motivations (7 indicators). The Destination receives the direct effect of all the variables and the indirect effect of Services and Cultural Offer, which also has a direct effect on the Motivations of the tourists. Based on the block structure of the survey and the burden of each item, some have been suppressed from the original survey because they had a very low effect or did not meet the required minimum.

Table 1. Study report.

Latent factor	Indicators
Tourist Destination	Destination valuation (V1)	Comparative valuation (V2)
Cultural Offer	Kindness (R1)	Cultural offer (R2)	Heritage (R3)
City Services	Cleaning (S1)Safety (S4)	Preservation (S2)Green areas (S5)	Sign posts (S3)Access (S6)
Tourist Motivations	Visiting (M4)Nature (M7)Festivals (M10)	Languages (M5)Beach (M8)	Sport (M6)Relax (M9)

| Show Table

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4. Results

The SmartPLS 3 software has been used to estimate the model. The established relations among the latent variables can be observed in Figure 1.

Figure 1. Structural equation model.

DownLoad: Full-Size Img PowerPoint

Table 2 displays the results with the burdens for each indicator, all the recommended weight being sufficient in every latent factor except the factor Motivations. This is largely due to the heterogeneity of the sample and the different starting points of the tourists since, in some cases, the visitors already knew the destination.

Table 2. Cross loadings.

	Culture	Motivations	Destination	Services
M4	0.165	0.537	0.179	0.163
M5	0.291	0.579	0.131	0.270
M6	0.185	0.641	0.276	0.233
M7	0.233	0.521	0.123	0.134
M8	0.122	0.334	0.131	0.081
M9	0.100	0.313	0.045	0.056
M10	0.125	0.488	0.263	0.132
R1	0.764	0.342	0.315	0.446
R2	0.794	0.279	0.292	0.484
R3	0.765	0.198	0.323	0.544
S1	0.432	0.182	0.327	0.690
S2	0.502	0.215	0.309	0.642
S3	0.315	0.187	0.135	0.525
S4	0.458	0.238	0.310	0.783
S5	0.384	0.292	0.335	0.667
S6	0.390	0.177	0.175	0.631
V1	0.346	0.362	0.907	0.421
V2	0.360	0.242	0.846	0.311

| Show Table

DownLoad: CSV

The correlation matrix of latent variables is shown in the following table, the highest correlation being between the Cultural Offer and the Services provided by the destination.

Table 3. Matrix of correlation between latent variables.

	Culture	Motivations	Destination	Services
Culture	1
Motivations	0.3590	1
Destination	0.4004	0.3509	1
Services	0.6303	0.3325	0.4235	1

| Show Table

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Table 4 shows the different reliability measures of the model. The obtained R-Square values are high in both cases and the analysis of internal consistency is acceptable (Cronbach's Alpha) except for the latent factor Motivations, as might be expected in the light of the results in Table 2. It is the same with the composite reliability indices. It is in the Motivations variable where no value is higher than 0.8.

Table 4. Reliability measurements.

	AVE	CompositeReliability	R Square	Cronbach's Alpha	Communality	Redundancy
Culture	0.599	0.817		0.668	0.599
Motivations	0.250	0.689	0.147	0.520	0.250	0.029
Destination	0.768	0.869	0.245	0.702	0.768	0.082
Services	0.436	0.821		0.743	0.436

| Show Table

DownLoad: CSV

Table 5 shows the different reliability measures of the model. The obtained R-Square values are high in both cases and the analysis of internal consistency is acceptable (Cronbach's Alpha) except for the latent factor Motivations, as might be expected in the light of the results in Table 2. It is the same with the composite reliability indices. It is in the Motivations variable where no value is higher than 0.8. shows the hypothesis test for the direct effects, with a significance level of 95% in all cases. Consequently, all null hypotheses and the positive effects that the Motivations, Culture and Services variables have when choosing a destination can be confirmed. Even though Services has the greater intensity, the results align with previous studies. Indeed, the Services of the destination condition the choice together with the diverse Motivations of the tourist; to a lesser extent the Cultural Offer, since in many cases the tourist does not enjoy it fully, depending on the duration of the stay.

Table 5. Tests of hypotheses for direct effects between latent variables.

	Total Effects	T Statistics
Culture Motivations	0.247	2.838^*
Culture Destination	0.221	2.706^*
Motivations Destination	0.207	3.000^*
Services Motivations	0.176	1.947
Services Destination	0.247	3.423^*
Note: *Significant values at the 5% significance level.

| Show Table

DownLoad: CSV

5. Conclusions

The first of the conclusions is the validity and reliability of the proposed model which, given its predictive character, can be used by public and private agents for policy-making and destination improvement. Thus, some of the lowest-rated indicators, specially sign posts and safety, can be improved.

The second conclusion is related to the improvement of motivations of tourists when choosing a destination. The lowest rated are Beach and Relax. It is true that Malaga is not the main exponent of the Spanish Costa del Sol (one of the best valued sun-and-sand destinations globally) but the city of Malaga has magnificent unknown beaches that must be promoted. Furthermore, it is a destination where one can get high levels of relaxation, which is another motivation to choose it. It is certain that the exponential growth in tourists, caused mainly by the arrival of cruises, has turn Malaga into a mass destination. Nevertheless, this massification is concentrated in the historic center adjacent to the port and cruise terminals, leaving the rest of the city as a quiet place where one can enjoy.

The third conclusion is related to the significant increase in cultural offer in the city of Malaga in the last 20 years. Out of the many sporting, cultural and leisure events, we must highlight the approach of Malaga, together with Madrid, as a national referent of "City of Museums". In fact, it has one of the most important offers (at a national and European level) with a total of 40 museums, the most relevant being the following: Picasso Museum, Museum of the City of Malaga, Carmen Thyssen Malaga Museum, Malaga Pompidou Centre and Russian Museum Malaga.

In this paper, we showed the relative impact that the different tourist resources have on the election of a given destination. Undoubtedly, the strategic planning of the city must consider these results so as to improve them in future years. All the hypotheses and subhypotheses have been confirmed (influence of the Services in the Motivations, only at a 90%). Therefore, we must emphasize direct effects on the latent variable Destination. However, we should highlight that the effects have a similar result, with values around 0.25.

We must take into account that, among the future research lines to address, one would be the segmentation of the respondents by means of sex or place of residence (national/foreign). We believe that, in both cases, motivations shall be different, and the perception of services and cultural offer could suffer fluctuations.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	H. K. Ahmed, H. F. Ismael, Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics, Phys. Scr., 99 (2024), 105276. https://doi.org/10.1088/1402-4896/ad78c3 doi: 10.1088/1402-4896/ad78c3
[2]	M. A. E. Abdelrahman, H. A. Refaey, M. A. Alharthi, Investigation of new waves in chemical engineering, Phys. Scr., 96 (2021), 075218. https://doi.org/10.1088/1402-4896/abfb24 doi: 10.1088/1402-4896/abfb24
[3]	A. Rayal, System of fractal-fractional differential equations and Bernstein wavelets: A comprehensive study of environmental, epidemiological, and financial applications, Phys. Scr., 99 (2025), 025236. https://doi.org/10.1088/1402-4896/ada592 doi: 10.1088/1402-4896/ada592
[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. https://doi.org/10.1007/s10915-023-02179-5 doi: 10.1007/s10915-023-02179-5
[5]	R. Ali, M. M. Alam, S. Barak, Exploring chaotic behavior of optical solitons in complex structured conformable perturbed Radhakrishnan-Kundu-Lakshmanan model, Phys. Scr., 99 (2024), 095209. https://doi.org/10.1088/1402-4896/ad67b1 doi: 10.1088/1402-4896/ad67b1
[6]	E. M. E. Zayed, M. E. M. Alngar, A. Biswas, M. Asma, M. Ekici, A. K. Alzahrani, et al., Pure-cubic optical soliton perturbation with full nonlinearity by unified Riccati equation expansion, Optik, 223 (2020), 165445. https://doi.org/10.1016/j.ijleo.2020.165445 doi: 10.1016/j.ijleo.2020.165445
[7]	A. Farooq, T. Shafique, M. Abbas, A. Birhanu, Y. S. Hamed, Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics, Sci. Rep., 15 (2025), 1683. https://doi.org/10.1038/s41598-025-86177-7 doi: 10.1038/s41598-025-86177-7
[8]	M. Iqbal, W. A. Faridi, R. Ali, A. R. Seadawy, A. A. Rajhi, A. E. Anqi, et al., Dynamical study of optical soliton structure to the nonlinear Landau–Ginzburg–Higgs equation through computational simulation, Opt. Quant. Electron., 56 (2024), 1192. https://doi.org/10.1007/s11082-024-06401-y doi: 10.1007/s11082-024-06401-y
[9]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Probing families of optical soliton solutions in fractional perturbed Radhakrishnan-Kundu-Lakshmanan model with improved versions of extended direct algebraic method, Fractals Fract., 7 (2023), 512. https://doi.org/10.3390/fractalfract7070512 doi: 10.3390/fractalfract7070512
[10]	R. Conte, A. P. Fordy, A. Pickering, A perturbative Painlevé approach to nonlinear differential equations, Physica D, 69 (1993), 33–58. https://doi.org/10.1016/0167-2789(93)90179-5 doi: 10.1016/0167-2789(93)90179-5
[11]	Y. L. Xiao, S. Barak, M. Hleili, K. Shah, Exploring the dynamical behaviour of optical solitons in integrable kairat-Ⅱ and kairat-X equations, Phys. Scr., 99 (2024), 095261. https://doi.org/10.1088/1402-4896/ad6e34 doi: 10.1088/1402-4896/ad6e34
[12]	H. Khan, Shoaib, D. Baleanu, P. Kumam, J. F. Al-Zaidy, Families of travelling wave solutions for fractional-order extended shallow water wave equations, using an innovative analytical method, IEEE Access, 7 (2019), 107523–107532. https://doi.org/10.1109/ACCESS.2019.2933188 doi: 10.1109/ACCESS.2019.2933188
[13]	R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended $\left(\dfrac{G'}{G} \right)$ -Expansion methods for travelling wave solutions of fractional Maccari's system with complex structure, Alex. Eng. J., 79 (2023), 508–530. https://doi.org/10.1016/j.aej.2023.08.007 doi: 10.1016/j.aej.2023.08.007
[14]	H. Khan, R. Shah, J. F. Gómez-Aguilar, Shoaib, D. Baleanu, P. Kumam, Travelling waves solution for fractional-order biological population model, Math. Model. Nat. Phenom., 16 (2021), 32. https://doi.org/10.1051/mmnp/2021016 doi: 10.1051/mmnp/2021016
[15]	C. M. Khalique, A. Biswas, A Lie symmetry approach to nonlinear Schrödinger's equation with non-Kerr law nonlinearity, Commun. Nonlinear Sci., 14 (2009), 4033–4040. https://doi.org/10.1016/j.cnsns.2009.02.024 doi: 10.1016/j.cnsns.2009.02.024
[16]	N. Li, S. Y. Xu, Y. Z. Sun, Q. Chen, Bright and dark solitons under spatiotemporal modulation in (2+1)-dimensional spin-1 Bose-Einstein condensates, Nonlinear Dyn., 113 (2025), 767–782. https://doi.org/10.1007/s11071-024-10243-4 doi: 10.1007/s11071-024-10243-4
[17]	Y. Z. Sun, Z. H. Hu, H. Triki, M. Mirzazadeh, W. J. Liu, A. Biswas, et al., Analytical study of three-soliton interactions with different phases in nonlinear optics, Nonlinear Dyn., 111 (2023), 18391–18400. https://doi.org/10.1007/s11071-023-08786-z doi: 10.1007/s11071-023-08786-z
[18]	Q. Zhou, Y. Z. Sun, H. Triki, Y. Zhong, Z. L. Zeng, M. Mirzazadeh, Study on propagation properties of one-soliton in a multimode fiber with higher-order effects, Results Phys., 41 (2022), 105898. https://doi.org/10.1016/j.rinp.2022.105898 doi: 10.1016/j.rinp.2022.105898
[19]	L. W. Zeng, J. S. He, B. A. Malomed, J. B. Chen, X. Zhu, Spontaneous symmetry and antisymmetry breaking of two-component solitons in a combination of linear and nonlinear double-well potentials, Phys. Rev. E, 110 (2024), 064216. https://doi.org/10.1103/PhysRevE.110.064216 doi: 10.1103/PhysRevE.110.064216
[20]	L. W. Zeng, M. R. Belić, D. Mihalache, J. C. Shi, J. W. Li, S. Q. Li, et al., Families of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction, Nonlinear Dyn., 108 (2022), 1671–1680. https://doi.org/10.1007/s11071-022-07291-z doi: 10.1007/s11071-022-07291-z
[21]	J. B. Chen, D. Mihalache, M. R. Belić, J. C. Shi, D. F. Zhu, D. N. Deng, et al., Dark gap solitons in bichromatic optical superlattices under cubic-quintic nonlinearities, Chaos, 34 (2024), 113131. https://doi.org/10.1063/5.0232509 doi: 10.1063/5.0232509
[22]	M. A. S. Murad, H. F. Ismael, T. A. Sulaiman, Various exact solutions to the time-fractional nonlinear Schrödinger equation via the new modified Sardar sub-equation method, Phys. Scr., 99 (2024), 085252. https://doi.org/10.1088/1402-4896/ad62a6 doi: 10.1088/1402-4896/ad62a6
[23]	Q. Q. Li, J. J. Nie, W. Zhang, Multiple normalized solutions for fractional Schrödinger equations with lack of compactness, J. Geom. Anal., 35 (2025), 59. https://doi.org/10.1007/s12220-024-01897-y doi: 10.1007/s12220-024-01897-y
[24]	Q. Q. Li, J. J. Nie, W. Wang, J. W. Zhou, Normalized solutions for Sobolev critical fractional Schrödinger equation, Adv. Nonlinear Anal., 13 (2024), 20240027. https://doi.org/10.1515/anona-2024-0027 doi: 10.1515/anona-2024-0027
[25]	D. Gao, X. Lü, M.-S. Peng, Study on the (2+1)-dimensional extension of Hietarinta equation: soliton solutions and Bäcklund transformation, Phys. Scr., 98 (2023), 095225. https://doi.org/10.1088/1402-4896/ace8d0 doi: 10.1088/1402-4896/ace8d0
[26]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, R. Shah, Perturbed Gerdjikov-Ivanov equation: Soliton solutions via Backlund transformation, Optik, 298 (2024), 171576. https://doi.org/10.1016/j.ijleo.2023.171576 doi: 10.1016/j.ijleo.2023.171576
[27]	M. M. A. Khater, Analytical simulations of the Fokas system, extension (2+1)-dimensional nonlinear Schrödinger equation, Int. J. Mod. Phys. B, 35 (2021), 2150286. https://doi.org/10.1142/S0217979221502866 doi: 10.1142/S0217979221502866
[28]	J. Manafian, On the complex structures of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities, Eur. Phys. J. Plus, 130 (2015), 255. https://doi.org/10.1140/epjp/i2015-15255-5 doi: 10.1140/epjp/i2015-15255-5
[29]	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
[30]	M. Malaver, H. D. Kasmaei, Analytical models for quark stars with van der Waals modified equation of state, Astrophys. Space Sci., 7 (2019), 49–58. https://doi.org/10.11648/j.ijass.20190705.11 doi: 10.11648/j.ijass.20190705.11
[31]	L. Kipgen, R. Singh, $\delta$ -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
[32]	E. A. Az-Zo'bi, Solitary and periodic exact solutions of the viscosity-capillarity van der Waals gas equations, Appl. Appl. Math., 14 (2019), 349–358.
[33]	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
[34]	S. Bibi, N. Ahmed, U. Khan, S. T. Mohyud-Din, Some new exact solitary wave solutions of the van der Waals model arising in nature, Results Phys., 9 (2018), 648–655. https://doi.org/10.1016/j.rinp.2018.03.026 doi: 10.1016/j.rinp.2018.03.026
[35]	A. Zafar, B. Khalid, A. Fahand, H. Rezazadeh, A. Bekir, Analytical behaviour of travelling wave solutions to the Van der Waals model, $\mathrm{\overset\frown{In}t}$ . J. Appl. Comput. Math., 6 (2020), 131. https://doi.org/10.1007/s40819-020-00884-5
[36]	J.-L. Zhang, M.-L. Wang, Y.-M. Wang, Z.-D. Fang, The improved F-expansion method and its applications, Phys. Lett. A, 350 (2006), 103–109. https://doi.org/10.1016/j.physleta.2005.10.099 doi: 10.1016/j.physleta.2005.10.099
[37]	E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212–218. https://doi.org/10.1016/S0375-9601(00)00725-8 doi: 10.1016/S0375-9601(00)00725-8

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