
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12199-8
10.1016/j.heliyon.2024.e36168
e36168
Research Article
Dynamical property of interaction solutions to the Chafee-Infante equation via NMSE method
Hossain Mohammad Mobarak mobarak4074@gmail.com
ab⁎
Akter Sushika c
Roshid Md. Mamunur c
Roshid Harun-Or- ad
Sheikh Md. Abu Naim ab
a Department of Mathematics, Sunamgonj Science and Technology University, Bangladesh
b Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur, Bangladesh
c Department of Mathematics, Hamdard University Bangladesh, Bangladesh
d Department of Mathematics, Pabna University of Science and Technology, Bangladesh
⁎ Corresponding author. mobarak4074@gmail.com
13 8 2024
30 8 2024
13 8 2024
10 16 e3616831 5 2024
15 7 2024
12 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by/4.0/ This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

Keywords

Chafee-infante equation
New modified simple equation method
Conformable fractional derivative
Multi-soliton solution
Interaction solution
==== Body
pmc1 Introduction

Majority of modeling for natural serious nonlinear behaviors in this world can be illustrated by the nonlinear partial differential equations (NPDEs). To visualize the internal mechanism and properties of the models even natural happenings related to the models in both visible and non-visible effects, derivations of analytical solutions of the corresponding model are essential facts. The solutions to NPDEs are very much important due to their wide range of applications in physical sciences and engineering branches such as fluid mechanic, plasma physics, optical fibers, signal processing, mechanical engineering etc. A number of efficient and powerful techniques have been ascertained to extract analytical solution as well as to illustrate the dynamical behavior of the NPDEs. The (G′/G) -expansion method [1], Bright and dark optical solitons method [2], JEF method [3], SE method [4], the AD scheme [5], the MSE method [6], extension exponential rational function method [7], Hirota bilinear method [8], EMSE method [9] the Kudryashov's scheme [10,11], sine-Gordon expansion scheme [12], Lie–Bäcklund symmetries [13], Wronskian approach [14], the generalized exponential rational function [15], canonical-like transformation [16], exp-function method [17,18], Extended (G'/G) -Expansion method [19], improved (G′/G)-expansion approach [20], exp (−φ(ξ))-expansion method [21], Inverse-engineering structure [22], extended tanh method [23,24], EDAM method [25], IME tanh-function technique [[26], [27], [28]], ESE method [29], NMSE method [30], Hirota direct method [31], MEM method [32], and similar more are developed to extract solutions of NPDEs. Most of the above techniques can provide only single traveling wave solutions of nonlinear models, but the Wronskian, Hirota bilinear [33,34] and NMSE can derive multi-soliton and multi-wave solutions. The two techniques (Wronskian, Hirota bilinear) are very old and not suitable for the model which has no bilinear formation. Alongside this, NMSE technique is a recent innovative technique and can be applied on the model which has no bilinear formation. Owing to this fact, we choose the procedure to integrate nonlinear model to extract mutli-wave and multi-soliton solutions.

Based on the foregoing, the newly proposed solution of the Chafee-Infante equation (CI) equation is recovered using the NMSE method with conformable fractional derivative. As far as we are aware, this is the first time we have studied the analytical solution of the CI problem using this method. With the use of the new soliton solution, graphical representations, and thorough analysis, we examine the CI equation's dynamic behavior. The analytical answers demonstrate that the new approach can reconstruct the precise solution of the CI problem with a smaller sample size, faster convergence, and superior simulation results.

We will develop an algorithm for creating a multiple-wave soliton solution for in this chapter. Here we start the (1 + 1)-dimensional CI Model [35,36] as:(1) Dtζu−uxx−αu+αu3=0.

where 0<ζ≤1. The processes to find the solutions of this equation are the same as that has shown in the reference paper [13,15] and so on.

The main motivation of this study is investigated the multi soliton solutions and their interaction of fractional (1 + 1)-dimensional CI model by using new formation of modified simple equation method. The limitation of this technique is that, it cannot solve all nonlinear models through the system is integrable. The calculation is also complicated if the balance number is more than two.

The arrangement of the work is followed: In division two, some feature of fractional derivative and explore the work rule of NMSE technique; In division three, trail the NMSE technique to fractional CI model; In division four, discus the numerical form of the obtained solutions and show the fractional effect with three and density diagram; In section five, we compare this work with others existing work; In division six, the overview of this work.

2 Preliminaries and methodology

Fractional derivatives extend traditional calculus, offering enhanced modeling of nonlinear evolution equations (NLEEs). Unlike integer-order derivatives, they capture memory and hereditary effects, crucial for systems where the future state depends on past behaviors. This non-local property makes them ideal for describing complex dynamics in physical, biological, and financial systems [[37], [38], [39], [40]]. In anomalous diffusion, viscoelastic materials, and fractal-like phenomena, fractional derivatives outperform classical models by accommodating irregular patterns and scaling laws. They allow for greater flexibility in initial and boundary conditions, enhancing the fit to empirical data and improving predictive accuracy. This versatility spurs advancements in numerical techniques for solving differential equations, leading to more precise and stable solutions. Conformable derivative is one of the fractional derivatives, that's offering a more straightforward generalization, preserve essential properties like the chain rule, making them easier to apply while still capturing essential system behaviors. Moreover, conformable derivatives provide a robust framework for accurately modeling and analyzing NLEEs across various scientific and engineering fields [10,[41], [42], [43], [44]], capturing intricate system behaviors that traditional methods cannot.

2.1 Conformable derivative

Initially Khalil [45] and T. Abdeljawad [46] introduce such derivative as:

For F:(0,∞)→R, the fractional ζ order conformable derivatives defined by:Dtζf(t)=limh→0[F(t+εt1−ζ)−F(t)h]forall0<ζ≤1,0<t.

From the following theorem, conformable fractional derivatives [[41], [42], [43], [44]] are easily understood.Theorem 1 For ζ∈(0,1], G=G(t) and F=F(t),ζ− conformable differentiable [47] with t>0:(i) Dtζ(cF+dG)=cDtζF+dDtζG, for all d,c∈R.

(ii) Dtζ(tα)=αtα−ζ, ∀α∈R.

(iii) Dtζ(FG)=FDtζ(G)+GDtζ(F). :

(iv) Dtζ(F/G)=GDtζ(F)−FDtζ(G)G2. :

For differentiable F, Dtζ(F(t))=t1−ζdFdt.

Theorem 2 IfS:(0,∝)→Rbe real valued function asSbeζ-conformable differentiable [48] and H be a differentiable with same range, then:Dtζ(SoH)(t)=t1−ζH(t)ζ−1H′(t)Dtζ(S(t))t=H(t)

Where H′=dHdt:

2.2 NMSE method

In this subdivision, we explore the new modified simple equation (NMSE) technique to solve NLEEs. The main advantages are investigated the multi soliton solution and diverse type of interaction soliton solution. Most of the above techniques can provide only single traveling wave solutions of nonlinear models, but the Wronskian, Hirota bilinear [33,34] and NMSE can derive multi-soliton and multi-wave solutions. The two techniques (Wronskian, Hirota bilinear) are very old and not suitable for the model which has no bilinear formation.

For NMSE Technique [30], consider:(2) G(h,hxx,ht,hxt………)=0.

where h=h(x,t), G be polynomial of h and derivatives of h. The following stages can be used to explain the NMSE approach in its entirety.Step-01 : Trial solution of eq. (2) will be:(3) h=∑r=0M∑i+j=rM[αj(f2′(η2)f2(η2))j×αi(f1′(η1)f1(η1))i].

Where ηi=kix−ωit,i=1,2 and αi,αj with i,j=0,1,2…. are constant. αM≠0, the unknown functions f1(η1) and f2(η2) to be determine later.

Step-02 : To control M, we balance the highest-order derivatives and the nonlinear factor in eq. (2).

Step-03 : Substituting h(x,t) in eq. (2), we obtain a polynomial of f1i,f2j and f1−if2−j.

Step-04 : Comparing the co-efficient with same power of f1if2j and solving these, αi,αj,f1 and f2 can obtained.

RemarkIn the modified simple equation method [4,6], the solution of eq. (2) is considered the following form:u=∑i=0Mαi(f1′(η1)f1(η1))i

Where η1=kx−ωt and αi with i=0,1,2…. are arbitrary constant to be resolute later such that αM≠0 the functions f1(η1) is an unknown function.

3 Multi-soliton solution of CI model by NMSE technique

In this section, we inject the NMSE scheme on fractional CI [9] model as:(4) Dtζu−uxx−αu+αu3=0.

According to NMSE scheme, the solution of equation (4) becomes,(5) u(x,t)=a0+a1[f1′(η1)f1(η1)]+a2[f2′(η2)f2(η2)].

where η1=k1x−ω1tζζ, η2=k2x−ω2tζζ; k1,k2 are angular wave numbers and ω1,ω2 are the waves frequency, a0,a1,a2 are arbitrary constants such that a2,a1≠0, f1(η1) and f2(η2) are unknown function to be determine later. For simplification, we uniformly consider as:f2(η2)≡f2,f1(η1)≡f1andf′≡dfdη

Now we substitute the necessary derivative of u(x,t) in eq. (4). Then we get a polynomial.(6) −a1ω1f1″−a1k12f1‴−αa1f1′+3αa02a1f1′=0.

(7) −a1ω1f1″−a1k12f1‴−αa1f1′+3αa02a1f1′=0.

(8) −a2ω2f2′−a2k22f2‴−αa2f2′+3αa2a02f2′=0.

(9) a1ω1f1′2+3a1k12f1′f1″+3αa0a12f1′2=0.

(10) a2ω2f2′2+3a2k22f2′f2″+3αa0a22f2′2=0.

(11) −2a1k12f1′3+αa13f1′3=0.

(12) −2a2k22f2′3+αa23f2′3=0.

(13) 6αa0a1a2f1′f2′=0.

(14) 3αa1a22f1′f2′2=0.

(15) 3αa1a22f1′f2′2=0

From eq. (10) to eq. (15) we find thata0=0,1,−1;a1=0,±2αk1;a2=0,±2αk2

Case-I: when a0=0 then eq. (7) to eq. (9) gives us:(16) f1′=−3k12f1″ω1.

(17) f2′=−3k22f2″ω2.

Putting eq. (16) in eq. (6) and eq. (17) in eq. (7) then we get,(18) f1=C13k14ω12−3k12αeθη1+C2.

(19) f2=C33k24ω22−3k22αe∅η2+C4.

where θ=3k12α−ω12ω1k12 and ∅=3k22α−ω22ω2k22:

Now from eq. (5) with the help of eq. (18) and eq. (19) for a1=±2αk1, a2=±2αk2 we get,(20) u(x,t)=±2α3k13C1eθη1C13k14ω12−3k12αeθη1+C2∓2α3k23C3e∅η2C33k24ω22−3k22αe∅η2+C4.

where θ=3k12α−ω12ω1k12,∅=3k22α−ω22ω2k22 and ω1,k1,C1,C2,ω2,k2,C3,C4 are arbitrary constants.

With C1=±13ω12−αk12k14,C3=±13ω22−k22αk24,C2=C4=1 we get from eq. (20),(21) u(x,t)=ω12−3k122αk1ω1sech(θη12)eθη12+ω22−3k222αk2ω2sech(∅η22)e∅η22.

If C1=±13ω12−αk12k14,C3=±13ω22−k22αk24,C2=−1 and C4=1 we get from eq. (20) as,(22) u(x,t)=−ω12−3k122αk1ω1cosech(θη12)e−θη12+ω22−3k222αk2ω2sech(∅η22)e∅η22.

If C1=±13ω12−αk12k14,C3=±13ω22−k22αk24,C2=C4=−1 we get from eq. (20),(23) u(x,t)=−ω12−3k122αk1ω1cosech(θη12)e−θη12−ω22−3k222αk2ω2cosech(∅η22)e−∅η22.

Where θ=3k12α−ω12ω1k12,∅=3k22α−ω22ω2k22 and ω1,k1,ω2,k2 are arbitrary constants.

Case-II: For a0=1 then the remaining equations is gives us:(24) f1′=−3k12f1″ω1+3αa1.

(25) f2′=−3k22f2″ω2+3αa2.

Putting eq. (24) in eq. (6) and eq. (25) in eq. (7) then we get,(26) f1=C53k14ω12+3αa1ω1+6αk12emη1+C6.

(27) f2=C73k24ω22+3αa2ω2+6αk22enη2+C8.

where m=−ω12+3αa1ω1+6αk12k12(ω1+3αa1) and n=−ω22+3αa2ω2+6αk22k22(ω2+3αa2).

Now from eq. (5) with the help of eq. (26) and eq. (27) for a1=±2αk1, a2=±2αk2 we get,(28) u(x,t)=1∓2αC53k13ω1+3αa1emη1C53k14ω12+3αa1ω1+6αk12emη1+C6∓2αC73k23ω2+3αa2enη2C73k24ω22+3αa2ω2+6αk22enη2+C8.

where m=−ω12+3αa1ω1+6αk12k12(ω1+3αa1), n=−ω22+3αa2ω2+6αk22k22(ω2+3αa2) and ω1,k1,C5,C6,ω2,k2,C7, :

C8 are arbitrary constants.

Under the condition C5=13ω12+a1ω1α+2k12αk14,C6=1,C7=13ω22+a2ω2α+2k22αk24,C8=1 we get from eq. (28):(29) u(x,t)=1+ω12+3αa1ω1+6αk122αk1(ω1+3αa1)sech(mη12)emη12+ω22+3αa2ω2+6αk222αk2(ω2+3αa2)sech(nη22)enη22.

If C5=13ω12+a1ω1α+2k12αk14,C6=−1,C7=ω22+3a2ω2α+6k22α3k24,C8=1 then from eq. (28):(30) u(x,t)=1+ω12+3αa1ω1+6αk122αk1(ω1+3αa1)sech(mη12)emη12−ω22+3αa2ω2+6αk222αk2(ω2+3αa2)cosech(nη22)e−nη22.

If C5=13ω12+a1ω1α+2k12αk14,C6=−1,C7=13ω22+a2ω2α+2k22αk24,C8=−1 then from eq. (28):(31) u(x,t)=1−ω12+3αa1ω1+6αk122αk1(ω1+3αa1)cosech(mη12)e−mη12−ω22+3αa2ω2+6αk222αk2(ω2+3αa2)cosech(nη22)e−nη22.

Case-III: For a0=−1 then the remaining equations is gives us:(32) f1′=3k123αa1−ω1f1″.

(33) f2′=3k223αa2−ω2f2″.

where q=3αa2ω2−6αk22−ω22k22(ω2−3αa2) and q=3αa2ω2−6αk22−ω22k22(ω2−3αa2).

If we insert eq. (32) and eq. (33) in eq. (6) and eq. (7) then we get the following value respectively.(34) f1=C93k14ω12+6αk12−3αa1ω1epη1+C10.

(35) f2=C113k24ω22+6αk22−3αa2ω2epη2+C12.

Now from eq. (5), eq. (34) and eq. (35), with a1=±2αk1, a2=±2αk2 we get,(36) u(x,t)=−1±2α3k133αa1−ω1C9epη1C93k14ω12+6αk12−3αa1ω1epη1+C10∓2α3k233αa1−ω1C11epη2C113k24ω22+6αk22−3αa2ω2epη2+C12.

where p=3a1ω1α−6k12α−ω12k12(ω1−3a1α),q=3a2ω2α−6k22α−ω22k22(ω2−3a2α) and ω1,k1,C9,C10,ω2,k2,C11,C12 are arbitrary constants.

If C9=13ω12−αa1ω1+2αk12k14,C10=1,C11=13ω22−αa2ω2+2αk22k24,C12=1 we get from eq. (36),(37) u(x,t)=−1+ω12−3αa1ω1+6αk122αk1(3αa1−ω1)sech(pη12)epη12+ω22−3αa2ω2+6αk222αk2(3αa2−ω2)sech(qη22)eqη22.

If C9=13ω12−a1ω1α+2k12αk14,C10=−1,C11=13ω22−a2ω2α+2k22αk24,C12=1 then from eq. (36),(38) u(x,t)=−1−ω12−3αa1ω1+6αk122αk1(3αa1−ω1)cosech(pη12)e−pη12+ω22−3αa2ω2+6αk222αk2(3αa2−ω2)sech(qη22)eqη22.

If C9=13ω12−a1ω1α+2k12αk14,C10=−1,C11=13ω22−a2ω2α+2k22αk24,C12=−1 then,(39) u(x,t)=−1−ω12−3αa1ω1+6αk122αk1(3αa1−ω1)cosech(pη12)e−pη12−ω22−3αa2ω2+6αk222αk2(3αa2−ω2)cosech(qη22)e−qη22.

4 Numerical discussion

In this section, we discuss the behavior theoretically and graphically of the obtained solution with 3-D graph, corresponding density and contour plot for different values of ζ. Also deliberate the obtained solution under some conditions with numerical values. For the special value of free parameters under the conditions multi-kink soliton, multi-soliton solutions, interaction between kink and soliton, interaction between anti-kink and soliton, interaction of different soliton and double periodic soliton solutions are obtained. The interaction between a kink and a soliton solution derived via eq. (20) and its numerical presentation illustrated in Fig-1 for k1=−0.5,k2=5,w1=1,w2=0.5,c4=0.5,c3=1,c2=−3,c1=−1,α=3 within the interval −18≤x≤18,−18≤t≤18. Here, we illustrated the effect of changing fractionality in the traveling wave at the interaction wave solutions, the observation concluded that the classical traveling wave variable expressed the multi-solitons in a kink wave structure, but as the classical wave tend to fractional form the soliton going to diminish into the kink structured only. Interaction of two-kink soliton wave arises via the solution in eq. (21) for k1=0.5,k2=1,w1=1.5,w2=0.5,α=−0.25 in classical wave structured, which is completely elastic. But as traveling wave variable going to fractional form two-kink remain in the same structured and additionally few solitons visible in the physical structured, see Fig-2. Fig. 3 represented the interaction between kink and periodic solution with singularities for k1=7,k2=1,w1=1,w2=0.5,α=−4. The solutions eq. (28) and eq. (29) have the similar behaviors like Fig. 2 represented the multi-kink. Overhead view of density plot and the 2-D contour plot of eq. (28) are drawn for the parameters k1=0.5,k2=1,w1=1.5,w2=0.5,c5=2,c6=1,c7=−5,c8=−1,α=1 in Fig. 4 and the multi-kink soliton solution of eq. (29) for the value of the parameters k1=1,k2=11,w1=1,w2=0.5,α=1 in Fig. 5. Fig. 6, represents the multi-soliton wave of evolution of the imaginary slice of eq. (30) for k1=0.5,k2=0.5,w1=1,w2=0.5,α=0.25 with singularities. The lump-kink collision illustrated in Fig. 7 comes from eq. (31) with k1=k2=w1=w2=1,α=−0.25 in classical case, due to change in the fractionality the structure deformed with multi-peaked into the lump-kink structure. Fig. 8 represented interaction of bell and periodic rogue waves from eq. (36) for k1=0.5,k2=1,w1=1.5,w2=0.5,c9=c10=c11=c12=1,α=−0.25. In classical case the solutions exhibit such interaction of bell and periodic rogue waves, while fractionality reduced in comes to deform double bell wave. The interaction between the kink and soliton of eq. (36) for k1=10,k2=0.5,α=w1=1,w2=5 are presented in the Fig-9. Fig. 10 represents the novel multi-soliton with singularities via eq. (39) for k1=1,k2=−1,w1=1,w2=0.5,α=1. :Fig. 1 3-D, density and contour plots of eq. (20).

Fig. 1

Fig. 2 3-D, density and contour plots of eq. (21).

Fig. 2

Fig. 3 3-D plots of eq. (23) for ζ=0.1,ζ=0.5 and ζ=1:

Fig. 3

Fig. 4 3-D, density and contour plots of eq. (28).

Fig. 4

Fig. 5 3-D, density and contour plots of eq. (29).

Fig. 5

Fig. 6 3-D, density and contour plots of eq. (30).

Fig. 6

Fig. 7 3-D, density and contour plots of eq. (31).

Fig. 7

Fig. 8 3-D, density and contour plots of eq. (36).

Fig. 8

Fig. 9 3-D plots of eq. (36) for ζ=0.1,ζ=0.5 and ζ=1. :

Fig. 9

Fig. 10 3-D plots of eq. (39) for ζ=0.1,ζ=0.5 and ζ=1. 5. Remarks and Comparison.

Fig. 10

In this division, we compare this work with existing published work [15,49,50]. The mention article applied different methods and find various soliton solutions but we applied NMSE method and investigate multi-soliton solutions and their interaction. In the former research, Kumar et al. [15] claimed the analytic solutions that include solutions based on rational, exponential, trigonometric and hyperbolic functions of CI model in the form of contained bright and dark solitons, singular and combined singular soliton profiles, periodic oscillating nonlinear waves, single and mixed singular soliton profiles, as well as kink-wave profiles. Akbar et al. [49] applied the first integral scheme and they derived the kink and singular kink waves, bright-dark and singular bell waves for CI model and discussed physical significant of obtained solutions with different parameters. Demiray et al. [50] derived anti-bell, kink and singular-kink wave of CI model by S-G expansion technique. Mahmood et al. [51] applied modified Khater scheme on CI model and obtained bright-dark, kink and periodic-oscillating waves. The accurate kink type solution of CI model was discovered by Habiba et al. [36] by improved Kudryashov technique. Mao [16] used the canonical-like transformation scheme and trial equation scheme to arrive at the exact solution to the CI problem in terms of the elliptic function. They used various methods with changing auxiliary equations only with single traveling wave variable. Besides this our main innovation is that, we introduced here the double waves solutions with different arbitrary wave speed in conformable fractional mode of CI model. The injected double wave traveling variables η2 and η1 with different wave speed gives collisions of two distinct wave structures. As a result, various interacted waves such as double kink (kink-kink collisions), lump-kink, periodic lump-kink, parabolic singular (in Fig. 10 for ξ=0.5), singular double kink interaction wave and double periodic waves (Fig. 6) are providing which is exceptional to the previous any works.

5 Conclusion, limitations and future works

5.1 Conclusion

In this study, we successfully implemented the new formation of MSE technique to solve fractional Chafee-Infante Equation. The proposed method is integrated multi-soliton solutions with their interaction of Chafee-Infante equation. On the parametric condition, notably the obtained solution is expressed as hyperbolic, exponential, trigonometric functions, and combine of them. To the best of our knowledge, the recent modification in the simple equation approach cannot be used to answer this equation. To help visualize how the equations behave dynamically, some figures are presented. From the above information the new form of modified simple equation approach seems to be simpler, faster, and easier for a computer to manage. This will encourage the extensive use of the equations in a sensible way. The obtained solutions may be substantial and imperative for scrutinizing the nonlinear phenomena rising in pragmatic physical sciences.

5.2 6.2 limitations

In this article, we investigate the multi-soliton solutions with their interaction of CI model. The NMSE technique are applied to obtained our required solutions. This method gives us a system of algebraic equation to giant their parameters which sometimes becomes impossible to solve by computational computer software.

5.3 6.3 Possible future works

We apply NMSE scheme on (1 + 1)-dimensional CI model with fractional differential form to derive interacted wave pattern. In future, we will apply the cofe transformation with Hirota bilinear technique and investigate N-soliton solutions, lump wave, breather wave and diverse type of interaction.

Data availability statement

All the data associated this works are included in this manuscript.

Funding statement

No founding is received for this work.

CRediT authorship contribution statement

Mohammad Mobarak Hossain: Writing – review & editing, Writing – original draft, Software, Methodology, Formal analysis, Conceptualization. Sushika Akter: Validation, Methodology, Data curation, Conceptualization. Md. Mamunur Roshid: Writing – review & editing, Software, Methodology, Formal analysis. Harun-Or-Roshid: Writing – review & editing, Supervision, Methodology, Formal analysis, Data curation. Md. Abu Naim Sheikh: Supervision, Formal analysis, Data curation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
==== Refs
References

1 Bekir A. Application of the (G'/G) -expansion method for nonlinear evolution equations Phys. Lett. A. 372 2008 3400 3406
2 Wazwaz A.M. Bright and dark optical solitons for (3+ 1)-dimensional Schrödinger equation with cubic–quintic-septic nonlinearities Optik 255 2021 165752
3 Ali A.T. New generalized Jacobi elliptic function rational expansion method J. Comput. Appl. Math. 235 2011 4117 4127
4 Roshid M.M. Bashar H. Breather wave and kinky periodic wave solutions of one-dimensional Oskolkov equation Math. Model. Eng. Probl. 6 3 2019 460 466
5 Adomian G. Solving Frontier Problems of Physics: the Decomposition Method vol. 60 1994 Kluwer Academic Publishers Boston 22 68
6 Akter J. Akbar M.A. Exact solutions to the Benney–Luke equation and the Phi-4 equations by using modified simple equation method Results Phys. 5 2015 125 130
7 El-Rashidy K. New traveling wave solutions for the higher Sharma-Tasso-Olver equation by using extension exponential rational function method Results Phys. 17 2020 103066
8 Ma H. Cheng Q. Deng A. Solitons, breathers, and lump solutions to the (2 + 1)-dimensional generalized calogero–bogoyavlenskii–schiff equation Complexity 2021 2021 7264345
9 Sheikh M.A.N. Taher M.A. Hossain M.M. Akter S. Roshid H.O. Variable coefficient exact solution of Sharma–Tasso–Olver model by enhanced modified simple equation method, Part Differ. Equ. Appl. Math. 7 2023 100527
10 Hossain M.M. Sheikh M.A.N. Roshid M.M. Roshid H.O. Taher M.A. New soliton solutions and modulation instability analysis of the regularized long-wave equation in the conformable sense, Part Differ. Equ. Appl. Math. 9 2024 100615
11 Shehab M.F. El-Sheikh M.M.A. Mabrouk A.A.G. Ahmed H.M. Dynamical behavior of solitons with Kudryashov's quintuple power-law of refractive index having nonlinear chromatic dispersion using improved modified extended tanh-function method Optik 266 2022 169592
12 Ma W.-X. Osman M.S. Arshed S. Raza N. Srivastava H.M. Practical analytical approaches for finding novel optical solitons in the single-mode fibers Chin. J. Phys. 72 2021 475 486
13 Yusuf A. Sulaiman T. Abdeljabbar A. Alquran M. Breather waves, analytical solutions and conservation laws using Lie–Bäcklund symmetries to the (2 + 1)-dimensional Chaffee–Infante equation J. Ocean Eng. Sci. 8 2 2023 145 151
14 Abdeljabbar A. New double Wronskian solutions for a generalized (2+1)-dimensional Boussinesq system with variable coefficients, Part Differ. Equ. Appl. Math. 3 2021 100022
15 Kumar S. Almusawa H. Hamid I. Akbar M.A. Abdou M.A. Abundant analytical soliton solutions and evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium Results Phys. 30 2021 104866
16 Mao Y. Exact solution to (2+1)-dimensional Chafee-Infante equation J. Phys. 91 9 2018
17 Raza N. Jhangeer A. Arshed S. Butt A.R. Chu Y.-M. Dynamical analysis and phase portraits of two-mode waves in different media Results Phys. 19 2020 103650
18 Bekir A. Cevikel A.C. New solitons and periodic solutions for nonlinear physical models in mathematical physics Nonlinear Anal. R. World Appl. 11 2010 3275 3285
19 Roshid H.O. Kink type traveling wave solutions of right-handed non-commutative Burgers equations via extended (g'/g) -expansion method Phys. Sci. Int. J. 21 4 2019 1 6
20 He Y. Li S. Long Y. Exact solutions to the sharma-tasso-olver equation by using improved (G'/G) -expansion method J. Appl. Math. 2013 2013 247234
21 Hossain M.M. Roshid H.O. Sheikh M.A.N. Topological soliton, singular soliton and other exact travelling wave solution for Burger-Hexley Equation Asian J. Math. Comput. Res. 6 4 2015 312 331
22 Zubair A. Raza N. Mirzazadeh M. Liu W. Zhou Q. Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities Optik 173 2018 249 262
23 Roshid M.M. Karim M.F. Azad A.K. Rahman M.M. Sultana T. New solitonic and rogue wave solutions of a Klein–Gordon equation with quadratic nonlinearity, Part Differ. Equ. Appl. Math. 3 2021 100036
24 Samir I. Ahmed H.M. Mirzazadeh M. Triki H. Solutions for higher order Sasa–Satsuma equation by using the improved modified extended tanh scheme Optik 274 2023 170592
25 Raza N. Zubair A. Optical dark and singular solitons of generalized nonlinear Schödinger's equation with anti-cubic law of nonlinearity Mod. Phys. Lett. B 33 13 2019 1950158
26 El-shamy O. El-barkoki R. Ahmed H.M. Abbas W. Samir I. Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method Alex. Eng. J. 68 2023 611 618
27 Alhojilan Y. Ahmed H.M. Novel analytical solutions of stochastic Ginzburg-Landau equation driven by Wiener process via the improved modified extended tanh function method Alex. Eng. J. 72 2023 269 274
28 Samir I. Abd-Elmonem A. Ahmed H. General solitons for eighth-order dispersive nonlinear Schrödinger equation with ninth-power law nonlinearity using improved modified extended tanh method Opt. Quant. Electron. 55 5 2023 470
29 Rabie W.B. Ahmed H.M. Seadawy A.R. Althobaiti A. The higher-order nonlinear Schrödinger’s dynamicalequation with fourth-order dispersion and cubic-quinticnonlinearity via dispersive analytical soliton wave solutions Opt. Quant. Electron. 53 2021 668
30 Irshad A. Din S.T.M. Ahmed N. Khan U. A New Modification in Simple Equation Method and its applications on nonlinear equations of physical nature Results Phys. 7 2017 4232 4240
31 Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons Phys. Rev. Lett. 27 1971 1192
32 Alhojilan Y. Ahmed H.M. Rabie W.B. Stochastic solitons in birefringent fibers for biswas–arshed equation with multiplicative white noise via itô calculus by modified extended mapping method Symmetry 15 2023 207
33 Ren J. Ilhan O.A. Bulut H. Manafian J. Multiple rogue wave, dark, bright, and solitary wave solutions to the KP–BBM equation J. Geom. Phys. 164 2021 104159
34 Roshid H.O. Multi-soliton of the (2+ 1)-dimensional calogero-bogoyavlenskii-schiff equation and KdV equation Comput. methods differ. equ. 7 1 2019 86 95
35 Arshed S. Akram G. Sadaf M. Ain Q. Riaz M.B. Wojciechowski A. Solitary wave behavior of (2+1)-dimensional Chaffee-Infante equation PLoS One 18 1 2023 0276961
36 Habiba U. Salam M.A. Hossain M.B. Datta M. Solitary wave solution of chafee-infante equation and (2+1) –dimensional breaking soliton equation by the improved Kudryashov method Glov. J. Sci. Front. Res. 19 5 2019 34 42
37 Hossain M.M. Roshid M.M. Sheikh M.A.N. Taher M.A. Roshid H.O. Novel exact soliton solutions of Cahn–allen models with truncated M-fractional derivative Int. J. Theor. Appl. Mech. 8 6 2022 112 120
38 Wang K.-J. Wang G.-D. Shi F. The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets COMPEL - Int J Comput. 42 6 2023 1576 1593
39 Wang K.-J. Xu P. Generalized variational structure of the fractal modified kdv–zakharov–kuznetsov equation Fractals 31 7 2023 2350084
40 Eroğlu B.B.I. Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane MMNSA 3 3 2023 281 296
41 Soliman M. Samir I. Ahmed H.M. Badra N. Hashemi M.S. Bayram M. Dispersive perturbations of solitons for conformable fractional complex Ginzburg-Landau equation with polynomial law of nonlinearity using improved modified extended tanh-function method Opt. Quant. Electron. 56 2024 1084
42 Cevikel A.C. Traveling wave solutions of conformable Duffing model in shallow water waves Int. J. Mod. Phys. B 36 25 2022 2250164
43 Cevikel A.C. Bekir A. Guner O. Exploration of new solitons solutions for the Fitzhugh–Nagumo-type equations with conformable derivatives Int. J. Mod. Phys. B 37 23 2023 2350225
44 Cevikel A.C. Bekir A. Assorted hyperbolic and trigonometric function solutions of fractional equations with conformable derivative in shallow water Int. J. Mod. Phys. B 37 9 2023 2350084
45 Khalil R. Horani M.A. Yousef A. Sababheh M. A new deﬁnition of fractional derivative J. Comput. Appl. Math. 264 2014 65 70
46 Abdeljawad T. “On conformable fractional calculus J. Comput. Appl. Math. 279 2015 57 66
47 Noshad M. Pishkoo A. Darus M. Solving conformable fractional differential equations with “EJS” software and visualization of sub-diffusion process Eur. j. pure appl. 15 4 2022 1738 1749
48 Michal P. Škripková L.P. Sturm's theorems for conformable fractional differential equations Math. Commun. 21 2016 273 281
49 Akbar M.A. Ali N.H.M. Hussain J. Optical soliton solutions to the (2+1)-dimensional Chaffee–Infante equation and the dimensionless form of the Zakharov equation Adv. Differ. Equ. 2019 2019 446
50 Demiray S.T. Bayrakci U. Construction of soliton solutions for chaffee-infante equation AKU J. Sci. Eng. 21 2021 051301 1046-1051
51 Mahmood A. Abbas M. Akram G. Sadaf M. Riaz M.B. Abdeljawad T. Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method, Results Phys 48 2023 106416
