
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39289413
65218
10.1038/s41598-024-65218-7
Article
A plethora of novel solitary wave solutions related to van der Waals equation: a comparative study
Butt Asma Rashid 1
Jhangeer Adil 23
Akgül Ali aliakgul00727@gmail.com

34
Hassani Murad Khan mhassani@gu.edu.af

5
1 grid.444938.6 0000 0004 0609 0078 Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
2 grid.440850.d 0000 0000 9643 2828 IT4Innovations, VSB Technical University of Ostrava, Ostrava, Czech Republic
3 https://ror.org/00hqkan37 grid.411323.6 0000 0001 2324 5973 Department of Computer Science and Mathematics, Lebanese American University, Beirut, 11022801 Lebanon
4 https://ror.org/05ptwtz25 grid.449212.8 0000 0004 0399 6093 Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey
5 https://ror.org/0075h8406 grid.448871.6 0000 0004 7386 4766 Department of Mathematics, Ghazni University, Ghazni, Afghanistan
17 9 2024
17 9 2024
2024
14 216651 10 2023
18 6 2024
© The Author(s) 2024
2024
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In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: β, and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

Keywords

Van der Waals equation
Soliton solutions
M-truncated derivative
Beta-derivative
Fractional wave transform
New extended direct algebraic method
Subject terms

Applied mathematics
Computational science
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Solitary wave solutions to the nonlinear partial differential equations (NLPDEs) have been gaining attention due to their potential to provide insights into a broad spectrum of physical phenomena in a wide range of domains. Fluid dynamics, thermodynamics, hydrodynamics, optical fibres, and ocean engineering are some of engineering and scientific disciplines where nonlinear wave phenomena occur and have fundamental importance. The search for analytical and numerical solutions to NLPDEs has received a lot of attention from physicists and mathematicians. To develop the traveling wave solutions to NLPDEs, they used a variety of direct and effective methods1–8 some of them are modified simple equation method9, new extended algebraic method10, Backlund transform method11, and many others. For calculating exact solutions of the (2+1) dimensional Kundu–MukherjeeNaskar (KMN) problem12, the new extended direct algebraic method, a recently developed methodology, is adopted by the researchers. In addition, for a two-dimensional modified Zakharov–Kuznetsov equation13, numerous analytical and numerical solutions are found that had stability properties. Similarly,14–18 contains several more efficient strategies to consider.

Nonlinear fractional differential equations (NFDEs) play a significant role in a wide range of phenomena, including acoustic waves, hydromagnetic waves, fractal dynamics, and plenty of others, where systems display complex factors including power-law distributions, fractal patterns, and non-Markovian motion. A real order fractional derivative has undergone much development during the past few decades. Research on fractional derivative operators is often discussed in academic circles. Nonlinear fractional derivatives are used in signal processing to process and evaluate signals that have non-Gaussian characteristics and nonlinear dynamics. When attempting to extract features from complicated signals, as those seen in chaotic systems, non-stationary processes, or nonlinear dynamics, they are especially helpful. In recent years, a great deal of effort has gone into this area, and many discoveries have been made, some of which are included in19–23.

Granular materials are used in a diverse variety of engineering and science applications. By combining distinct solid and macroscopic particles, granular matter is formed. During their collisions, the particles interact and lose energy. The vdW equation35,36 is given by1 ∂2u∂t2+∂2∂x2(∂2u∂x2-η∂u∂t-u3-ϱu)=0,

where u signifies the correction to critical average vertical density, x denotes the granular system’s horizontal direction, η is the efficient viscosity, and ϱ is the factor of bifurcation, which is corresponding to the compressibility coefficient. Accounting for the pressure surrounding the critical average vertical density are the first two terms on the left-hand side. An example of the interface tension is given by the term with a high spatial derivative.

The vdW equation for fluidized granular matter has been solved by most researchers in recent years using various methodologies24–26. The goal of this paper is to compare two distinct interpretations of fractional derivatives for the discovered soliton figures, in addition to discovering new families of soliton solutions for the vdW equation by using extended direct algebraic methods. This type of analysis has never been done before for the model in question, therefore it’s worth mentioning here.

The paper’s layout is seen here: Sect. “Basic definitions” contains some fundamental definitions collected from the literature. The examination of the governing model using various definitions is described in Sect. “Governing equation”. The extended direct algebraic method’s general technique and its implementation are presented in Sect. “Soliton solutions”. In Sect. “Study of solutions with various fractional derivatives in comparison, a thorough comparison” study is carried out. The conclusion is provided at the end of the article.

Basic definitions

Beta derivative

Definition 2.1

The following is the definition of beta derivative19:2 0ATxα(G(x))=limε→∞G(x+ε(x+1Γ(α)))-G(x)ε,

with these properties.

Theorem 2.2

If 0<α<1, a,b∈R,F,C and differentiable of α order at a point>0, then: 0ATxα(aF(x)+bG(x))=a0ATxαF(x)+b0ATxαG(x);

0ATxα(k)=0, where k represent the constant term.

0ATxα(F(x)∗G(x))=G(x)0ATxαF(x)+F(x)0ATxαG(x);

0ATxα(F(x)G(x))=G(x)0ATxαF(x)-F(x)0ATxαG(x)G2(x);

0ATxα(F(t)G(x))=tF′(t).

The proof of given relations are mentioned in27

M-truncated derivative

Definition 2.3

The one-parameter shortened Mittag-Leffler function20 is described as given:3 ιEβ(z)=∑k=1ι(zkΓ(kβ+1),

in which β>0 and z∈C.

Definition 2.4

Assume that g:[0,∞)→R and α∈(0,1), The definition of the M-truncated derivatives of g with degree α is:4 ιTMα,βg(t)=limε→∞g(t+Eβ(εt-α))-g(t)ε,

for t>0 and Eβ(.), β>0.

Theorem 2.5

If α∈(0,1], β>0, g and h are differentiable upto α order for t>0, we have: ιTMα,β(pg+qh)=pιTα,βM(g)+qιTMα,β(h);

ιTMα,β(tv)=vtv-α,v∈R;

ιTMα,β(gh)=gιTMα,β(h)+hιTMα,β(g);

ιTMα,β(gh)=gιTMα,β(h)-hιTMα,β(g)h2;

ιTMα,β(g(t)=t1-αΓ(β+1)g′(t);

ιTMα,β(g.h)=f′(h(t))ιTMα,βh(t).

Governing equation

The vdW equation is presented in this section with respect to several types of derivatives. (i) The discussed equation can be stated as follows when the beta derivative is considered: 5 0ADt2αu+0ADx2α(0ADx2αu-η0ADtαu-u3-ϱu)=0,

where 0ADtα and 0ADxα represent beta derivatives of t and x respectively.

(ii) When using the M-truncated derivative formulation, the equation becomes: 6 0ADM,t2α,βu+0ADM,x2α,β(0ADM,x2α,βu-η0ADM,tα,βu-u3-ϱu)=0,

where 0ADM,tα,β and 0ADM,xα,β are M-truncated derivatives of t and x respectively.

Mathematical analysis

We’ll use the following transformation to solve Eqs. (5) and (6):7 u(x,t)=U(τ),

where the soliton’s pulse shape is indicated by u(x, t) and τ is defined in various ways: In the case of the beta derivative, τ is calculated as8 τ=jα(x+1Γ(α))α-wα(t+1Γ(α))α,

Whereas, in the sense of an M-truncated derivative, we have9 τ=Γ(β+1)α(jxα-wtα),

where j and w stand for the wave number and speed of solitons. By using the transformation (7) in Eqs. (5) and (6), we obtain the ordinary differential equation as follows:10 w2-ϱj2j2U+j2U′′+ηwU′-U3=0,

Soliton solutions

In this section, we find solitons solutions for the vdW problem with beta and M-truncated derivative using an expanded direct algebraic approach.

Method description

The description of the new expanded direct algebraic approach28–34 is explained in this part. This technique can be used to solve several additional non-linear partial differential equations that arise in scientific research and is more efficient and useful to find several solitary wave solutions for a variety of non-linear issues. The researchers claim that the new extended algebraic approach gives a more powerful mathematical framework for non-linear partial differential equations than any other strategy. This is done with the aid of symbolic calculation. Suppose the non-linear partial differential equation is in the form:11 r(z,zt,zx,ztt,zxx,...)=0,

By u(x,t)=U(τ), we obtain:12 R(Z,Z′,Z′′,...)=0.

Suppose that:13 U(τ)=a0+∑i=1N[aiZ(τ)i-1],

where14 Z′(τ)=ln(ρ)ν+κZ(τ)+λZ2(τ).

We have

μ1=(β2-4αγ). For μ1<0 and γ≠0,Z1(τ)=-κ2λ+-μ12λtanρ-μ12τ,Z2(τ)=-κ2λ--μ12λcotρ-μ12τ,Z3(τ)=-κ2λ+-μ12λtanρ-μ1τ±pqsecρ-μ1τ,Z4(τ)=-κ2λ+-μ12λcotρ-μ1τ±pqcscρ-μ1τ,Z5(τ)=-κ2λ+-μ14λtanρ-μ14τ-cotρ-μ14τ.

For μ1>0 and γ≠0,Z6(τ)=-κ2λ-μ12λtanhρμ12ξ,Z7(τ)=-κ2λ-μ12λcothρμ12ξ,Z8(τ)=-κ2λ+μ12λ-tanhρμ1τ±ιpqsechρμ1τ,Z9(τ)=-κ2λ+μ12λ-cothρμ1τ±pqcschρμ1τ,Z10(τ)=-κ2λ-μ14λtanhρμ14τ+cothρμ14τ.

For αγ>0 and β=0,Z11(τ)=νλtanρνλτ,Z12(τ)=-νλcotρνλτ,Z13(τ)=νλtanρ2νλτ±pqsecρ2νλτ,Z14(τ)=νλ-cotρ2νλτ±pqcscρ2νλτ,Z15(τ)=12νλtanρνλ2ξ-cotκ(νγ)2ξ.

When αγ<0 and β=0,Z16(τ)=--νλtanhρ-νλτ,Z17(τ)=--νλcothρ-νλτ,Z18(τ)=-νλ-tanhρ2-νλτ±ipqsechρ2-νλτ,Z19(τ)=-νλ-cothρ2-νλτ±pqcschρ2-νλτ,Z20(τ)=-12-νλtanhρ-νλ2τ+cothρ-νλ2τ.

(5) For β=0 and α=γ,Z21(τ)=tanρντ,Z22(τ)=-cotρντ,Z23(τ)=tanρ2ντ±pqsecρ2ντ,Z24(τ)=-cotρ2ντ±pqcscρ2ντ,Z25(τ)=12tanρν2τ-cotρν2τ.

For β=0 and γ=-α,Z26(τ)=-tanhρντ,Z27(τ)=-cothρντ,Z28(τ)=-tanhρ2ντ±ipqsechρ2ντ,Z29(τ)=-cothρ2ντ±pqcschρ2ντ,Z30(τ)=-12(tanhρν2τ+cothρν2τ).

For β2=4αγ, Z31(τ)=-2ν(κτlnρ+2)κ2τlnρ.

When β=e,α=ef,(f≠0) and γ=0, Z32(τ)=ρeτ-f.

When β=γ=0, Z33(τ)=ντlnρ.

For β=α=0, Z34(τ)=-1λτlnρ.

For α=0 and β≠0, Z35(τ)=-pκλcoshρκτ-sinhρκτ+p.Z36(τ)=-κsinhρκτ+coshρκτλsinhρκτ+coshρκτ+q.

For β=e,γ=ef,(f≠0 and α=0), Z37(τ)=-pρeτp-fqρeτ.sinhρ(τ)=mρτ-qρ-τ2,coshκ(ξ)=pρτ+nρ-τ2.tanhρ(τ)=pρτ-qρ-τpρτ+nρ-τ,cothρ(τ)=pρτ+qρ-τpρτ-qρ-τ.sechρ(τ)=2pρτ+qρ-τ,cschρ(τ)=2pρτ-nρ-τ.sinρ(τ)=pρiτ-qρ-iτ2i,cosρ(τ)=pρiτ+qρ-iτ2.tanρ(τ)=-ipρiτ-qρ-iτpρiτ+qρ-iτ,cotρ(τ)=ipρiτ+qρ-iτpρiτ-qρ-iτ.secρ(τ)=2pρτ+qρ-τ,cscρ(τ)=2ipρτ-qρ-τ.

Implimentation of the method

The purpose of the given section is to identify the solitary solutions for the provided framework, including various derivatives. To do so, first solve Eq. (12). After balancing the highest-order derivative terms with the nonlinear terms in Eq. (12), we will get N = 1. So the solution will be of the following kind:15 U(τ)=a0+a1Z(τ),

where Z(τ) satisfies Z′(τ)=ln(p)γ+kZ(τ)+rZ2(τ). Then, we obtain Z0(τ)  : -a0ϱ+a0w2j2+j2a1kln(p)2γ+wηγa1ln(p)-a03=0, Z1(τ)  : -a1ϱ+a1w2j2+j2a1k2ln(p)2+2j2a1rγln(p)2+a1kwηln(p)-3a02a1=0, Z2(τ)  : 3j2a1krln(p)2+rwa1ηln(p)-3a0a12=0, Z3(τ)  : 2j2a1r2ln(p)2-a13=0. Thus, we obtain

       a0=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2, a1=k2η2ϱc+ba02kη2ϱc, w=3kη3ϱ2(k2-4rγblnp, j=ηϱ-clnp2, whereb=k2η4(9-2η2)2ϱ2(k2-4rγ)3,c=(-9+2η2)(k2-4rγ).

Let l=k2-4rγ. Now, we will discuss all the cases for the proposed method: Case 1. If l<0 and r≠0, then:16 U1=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+-l2rtanp-l2τ,

17 U2=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r--l2rcotp-l2τ,

18 U3=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+-l2rtanp-lτ±gh.secp-lτ,

19 U4=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+-l2r-cotp-lτ±gh.cscp-lτ,

20 U5=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+-l4rtanp-l4τ-cotp-l4τ.

Case 2. If l>0 and r≠0, then:21 U6=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r-l2rtanhpl2τ,

22 U7=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r-l2rcothpl2τ,

23 U8=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+l2r-tanhplτ±ιgh.sechplτ,

24 U9=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+l2r-cothplτ±gh.cschplτ,

25 U10=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-k2r+l4rtanhpl4τ+cothpl4τ.

Case 3. If γ.r>0 and k=0, then :26 U11=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcγrtanpγ.rτ,

27 U12=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-γrcotpγ.rτ,

28 U13=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcγrtanp2γ.rτ±gh.secp2γ.rτ,

29 U14=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcγr-cotp2γ.rτ±gh.cscp2γ.rτ,

30 U15=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc12γrtanpγ.r2τ-cotpγ.r2τ.

Case 4. If γ.r<0 and k=0, then:31 U16=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc--γrtanhp-γ.rτ,

32 U17=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc--γrcothp-γ.rτ,

33 U18=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-γr-tanhp2-γ.rτ±ιgh.sechp2-γ.rτ,

34 U19=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-γr-cothp2-γ.rτ±ιgh.cschp2-γ.rτ,

35 U20=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-12-γrtanhp-γ.r2τ+cotp-γ.r2τ.

Case 5. If k=0 and γ=r, then:36 U21=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱctanp(γτ),

37 U22=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-cotp(γτ)),

38 U23=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(tanp(2γτ)±gh.secp(2γ.τ)),

39 U24=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-cotp(2γτ)±gh.cscp(2γ.τ))

40 U25=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc12)tanpγ2τ-cotpγ2τ.

Case 6. If =0 and r=-γ, then:41 U26sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-tanhp(γτ)),

42 U27sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-cothp(γτ)),

43 U28=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-tanhp(2γτ)±ιgh.sechp(2γ.τ)),

44 U29=sqrtb-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(-cothp(2γτ)±gh.cschp(2γ.τ)),

45 U30=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-12)tanhpγ2τ+cothpγ2τ.

(7): For k2=4γr, then:46 U31=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-2γ(kτln(p)+2)k2τln(p).

(8): For k=d,γ=Nd,(N≠0) and r=0, then:47 U32=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcpdτ-N.

(9): For k=r=0, then:48 U33=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcγτln(p).

(10): For =γ=0, then:49 U34=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc-1τln(p)r.

(11): For γ=0 and k≠0, then:50 U35=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcgkcoshpkτ-sinhpkτ+g.

51 U36=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱcksinhpkτ+coshpkτsinhpkτ+coshpkτ+h.

(12): For k=d,r=Nd,(N≠0 and γ=0), then:52 U37=b-η2ϱc(k2-2rγ)(9-2η2)2(k2-4rγ)2-k2η2ϱc+ba02kη2ϱc(g.pd.τh-Ngpd.τ).

where g,h,d>0 are arbitrary constant.

Study of solutions with various fractional derivatives in comparison

For different values of the fractional parameter α, two solutions, U2 and U6, are considered in this section in terms of two derivatives and shown in Figs. 1, 2,3, 4, 5, 6, 7, 8, 9 and 10. For U2 space, we used the parameters as ϱ=1,p=e,η=1,k=-1,γ=1,r=-0.5, β=1, and α=0.3. In Figure 1 a depicts the graph of the U2 space with the β-derivative, whereas b illustrates the behavior of the U2 space with the M-truncated derivative, and c represents a 2D depiction of both β and M-truncated derivatives of U2 at t=1. In Fig. 2a exhibits its graph with the β-derivative with α=0.5, b shows its behavior using the M-truncated derivative with α=0.5, and c displays a 2D graph of U2 at t=1. In Fig.  3, with α=0.7, a presents its graph with the β- derivative, b displays its behavior with an M-truncated derivative at β=1, c offers a 2D depiction of U2 with both fractional derivatives at t=1. In Fig. 4a provides its graph with the beta derivative at α=0.9, b illustrates its behavior with the M-truncated derivative at β=1, and c represents a 2D graph of U2 at time t=1. In Fig. 5a displays its graph with the β-derivative for various values of “α”, whereas b displays its behavior with the M-truncated derivative. Figure 6 displays its graph together with the β-derivative and an M-truncated for α=1 and β=0.Figure 1 (a) 3D depiction of β- derivative with α=0.3 for U2, (b) 3D depiction of M-truncated with α=0.3 for U2, (c) U2 plot in 2D for both derivatives with t=1.

Figure 2 (a) 3D depiction of U2 for β- derivative with α=0.5, (b) 3D depiction of M-truncated derivative with α=0.5 for U2, (c) U2 plot in 2D with t=1.

Figure 3 (a) 3D depiction of U2 for β-derivative with α=0.7 , (b) 3D depiction of M-truncated derivative with α=0.7 for U2, (c) U2 depiction in 2D for both derivatives with t=1.

Figure 4 (a) 3D depiction of U2 for β-derivative with α=0.9, (b) 3D depiction of M-truncated derivative with α=0.9 for U2, (c) U2 plot in 2D with t=1.

Figure 5 2D-depiction of U2 for given derivatives at t=1 for multiple values of α.

Figure 6 2D-representation of U2 with α=1 and β=0 for different derivatives.

We have used the parameters η=1,p=e,r=1,k=-0.01,ϱ=1/6,γ=0.02, and β=2 in the equation U6. In Fig. 7a provides the graph of U6 with the beta derivative for α=0.3, whereas b illustrates its behavior with the M-truncated derivative with α=0.3 and c represents U6 at t=1. In Fig. 8a exhibits its graph with the beta derivative with α=0.5, b illustrates its behavior with the M-truncated derivative with β=2, and c represents a 2D graph of U6 at t=1. In Fig. 9a exhibits its graph with α=0.7, b demonstrates its behavior with β=1 M-truncated derivative, and c represents a 2D graph of U6 at t=1. In Fig.  10a provides its graph with a beta derivative with α=0.9, b illustrates its behavior with an M-truncated derivative with β=1, and c represents a 2D graph of U6 at time t=1. In Fig. 11a depicts its graph with the beta derivative for various values of α, whereas b displays its behavior with the M-truncated derivative. Fig. 12 displays its graph together with the beta derivative and an M-truncated for α=1 and β=0.Figure 7 (a) 3D depiction of U6 with β-derivative for α=0.3, (b) 3D depiction for M-truncated derivative with α=0.3 for U6, (c) U6 plot in 2D for two provided derivatives with t=1.

Figure 8 (a) 3D depiction of U6 with β-derivative for α=0.5, (b) 3D representation of M-truncated derivative with α=0.5 for U2, (c) U2 plot in 2D with t=1.

Figure 9 (a) 3D representation of U6 with β-derivative for α=0.7, (b) 3D depiction of M-truncated derivative with α=0.7 for U6, (c) U6 plot in 2D with various derivatives for t=1.

Figure 10 (a) 3D figure of U6 with β-derivative for α=0.9, (b) 3D representation of M-truncated derivative with α=0.9 for U6 , (c) U6 plot in 2D with t=1.

Figure 11 2D-depictiin of U6 for multiple derivatives at t=1 for multiple values of α.

Figure 12 2D-representation with α=1 and β=0 of U6 for multiple derivatives.

Conclusion

In this article, the vdW equation is studied using beta and M-truncated derivatives. This equation has established solitary wave solutions that exhibit decaying behavior and become unstable when the viscosity η is considered. These soliton solutions were obtained via a new extended algebraic technique along with two different definitions that have taken into account some physical characteristics. Numerous soliton solutions, such as dark solitons, dark singular soliton, dark–bright soliton, and singular solutions of types 1 and 2, have been observed in the described model, along with some constraint conditions. For different values of α, solitary wave solutions with beta formulation behaved differently from the M-truncated derivative, which is found in shape and structure. A 2-dimensional and 3-dimensional plots were also utilized to graphically explain the derived solitons. In comparison to other strategies in the literature, the applied technique was straightforward, short, simple, and easy to implement. It is also very skillful and well developed in terms of generating novel accurate solutions to nonlinear dispersive equations that appear in science and engineering.

Acknowledgements

This article has been produced with the financial support of the European Union under the REFRESH-Research Excellence For Regional Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048

Author contributions

All authors reviewed the manuscript.

Data availibility

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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