Reliable analysis for obtaining exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation

http://www.aimspress.com/journal/Math AIMS Mathematics, 9(6): 16666–16686. DOI: 10.3934/math.2024808 Received: 08 March 2024 Revised: 23 April 2024 Accepted: 30 April 2024 Published: 14 May 2024 Research article Reliable analysis for obtaining exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation Naveed Iqbal1, Muhammad Bilal Riaz2,3,∗, Meshari Alesemi4, Taher S. Hassan1,5,6, Ali M. Mahnashi7and Ahmad Shafee8 1Deparment of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia 2IT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic 3Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon 4Department of Mathematics, College of Science, University of Bisha, P.O. Box 511, Bisha 61922, Saudi Arabia 5Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt 6Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy 7Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia 8PAAET, College of Technological Studies, Laboratory Technology Department, Shuwaikh 70654, Kuwait *Correspondence: Email: [email protected]. Abstract: The (2+1)-dimensional Chaffee-Infante equation (CIE) is a significant model of the ionacoustic waves in plasma. The primary objective of this paper was to establish and examine closedform soliton solutions to the CIE using the modified extended direct algebraic method (m-EDAM), a mathematical technique. By using a variable transformation to convert CIE into a nonlinear ordinary differential equation (NODE), which was then reduced to a system of nonlinear algebraic equations with the assumption of a closed-form solution, the strategic m-EDAM was implemented. When the resulting problem was solved using the Maple tool, many soliton solutions in the shapes of rational, exponential, trigonometric, and hyperbolic functions were produced. By using illustrated 3D and density plots to evaluate several soliton solutions for the provided definite values of the parameters, it was possible to determine if the soliton solutions produced for CIE are cuspon or kink solitons. Additionally, it has been shown that the m-EDAM is a robust, useful, and user-friendly instrument that provides extra generic wave solutions for nonlinear models in mathematical physics and engineering. 16667 Keywords: nonlinear partial differential equations; (2+1)-dimensional Chaffee-Infante equation; modified extended direct algebraic method; kink soliton; cuspons Mathematics Subject Classification: 34G20, 35A20, 35A22, 35R11 1. Introduction The intriguing quality of nonlinearity in nature has led many scientists to view nonlinear science as the most significant area of research for a fundamental understanding of nature. The study of a variety of classes of NODEs and nonlinear partial differential equations (NPDEs) is crucial to the mathematical modelling of complex processes that vary over time. These models come from a wide range of disciplines, including optical fibres, economics, solid state physics, plasma physics, infectious disease epidemiology, elasticity, neural networks, and population ecology. Due to such applications, the examination of soliton solutions of the before stated phenomena has been an exciting and extremely dynamic topic of research for the past several decades, with the related issue of the construction of closed-form wave solutions to a broad class of nonlinear differential equations, particularlyNPDEs. For such situations, closed-form soliton solutions offer improved internal information. Thus, a great deal of work has been done by mathematicians and physical scientists to obtain closed-form soliton solutions of such NPDEs. A number of effective and powerful methods have been developed, including the first integral method [1], B¨ acklund transformation method [2], the modified simple equation method [3], the (G0/G)-expansion method [4–6], the exp-function method [7], the sinecosine method [8], the homogeneous balance method [9], the modified Kudryashov method [10], the F-expansion method [11], the bilinear method [12], the tanh-function method [13], the projective Riccit equation method [14], and the (G0/G,1/G)-expansion method [15] Riccati-Bernoulli SubODE [16], Poincar´ e-Lighthill-Kuo method [17], sub-equation method [18], Khater method [19], Sardar sub-equation method [20], EDAM [21–25], Hirota bilinear method [26], planar dynamic system method [27] and Bifurcation analysis method [28], among others [29, 30]. The computation of complex standard eigenvalue problems for laminar-turbulent transition prediction, while the investigates dynamically controlling terahertz wavefronts. In the third reference, structural reliability analysis is performed using an improved wolf pack algorithm, while the fourth reference introduces a class of digital integrators based on trigonometric quadrature rules [31–33]. Lastly, the fifth reference focuses on opinion formation analysis in group decision-making systems. These references collectively showcase the breadth and depth of research across various domains, highlighting the importance of interdisciplinary approaches in addressing complex scientific and engineering challenges [34,35]. One of the most significant, straightforward, and efficient algebraic techniques in the aforementioned analytical methods for obtaining soliton solutions to NPDEs is the m-EDAM. The strategic m-EDAM comprises converting CIE into a NODE by variable transformation, which is then reduced to a set of nonlinear algebraic equations with the assumption of a closed-form solution [36–38]. Many soliton solutions in terms of exponential, rational, trigonometric, and hyperbolic functions are produced when the resultant system is solved using the Maple tool. A soliton is a single, selfreinforcing wave packet that moves across a medium without altering its shape or velocity. Since soliton solutions to nonlinear fractional partial differential equations (NFPDEs) provide a higher level AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16668 of detail and coverage than traditional solutions, their study is still important from an academic standpoint [39–41]. Their resilience and fermion stability make them valuable in many areas of engineering and science. They make it possible for nonlinear systems to maintain long-distance coherence and convey information efficiently [42,43]. Numerous researchers have used the proposed m-EDAM to construct a new plethora of soliton solutions for many NPDEs andNFPDEs. For example, Saima et al. in [44] constructed soliton solutions for the (2+1)-dimensional Nizhnik-Novikov-Veselov model (NNVM) using a novel variant of m-EDAM called r+mEDAM. Ikram et al. employed the suggested mEDAM to analyse front events and solitons in the fractional Kolmogorov-Petrovskii-Piskunov equation [45]. Bilal et al. explored the soliton phenomena in the nonlinear spatiotemporal fractional Higgs system analytically, using an improved version of the ground-breaking analytical technique m-EDAM [46]. Lastly, Rashid et al. investigated soliton solutions in fractional quantum mechanical equations of nonlinear systems using the effective m-EDAM [47]. Overall, the m-EDAM research revealed that it is a reliable, practical, and easy-to-use tool that offers more general wave solutions for nonlinear models in mathematical physics and engineering. Motivated by the ongoing research on soliton solutions, this research aims to extract and analyse soliton solutions for (2+1)-dimensional CIE using the m-EDAM. This model is articulated as [48]: ztx +(λz3−zxx −λz)x+σzyy =0,(1.1) where z=z(x,y,t), σis the degradation coefficient, while λis the diffusion coefficient. In a physical environment, the diffusion of a gas in a homogeneous medium is a significant phenomenon, and the CIE offers a helpful model to investigate such phenomena. One famous response to the Duffing equation in the physical sciences is the (2+1)-dimensional CIE [49]. Numerous scholars have addressed the CIE by employing various analytical approaches. Using the Hirota bilinear approach, for example, Sualiman et al. produced lump soliton solutions for CIE with variable coefficients [50]. To derive the kink and cuspon soliton solutions for CIE, Akbar et al. used the first integral equation in [48]. Similarly, a number of new generalised solitonary solutions for the CIE with unique physical structures have been constructed by Sakthivel and Chun [49] using the expfunction method. Lastly, Demiray and Bayrakci created soliton solutions for CIE in [51] by applying the sine-Gordon expansion approach. However, in this study, we aim to address CIE to construct soliton solutions for it using m-EDAM. Our study reveals that the soliton solutions obtained for CIE are either cuspon and kink soliton solutions, such as dark kink, bright kink, and bright-dark kink. A kink soliton is a kind of solitary wave that is distinguished by an abrupt change or discontinuity in the field or variable it represents. It is also referred to as a topological soliton or domain wall. Kink solitons usually occur in systems that experience a phase transition of a scalar field. Conversely, a cuspon soliton, also known as a cusped soliton, is a different kind of solitary wave that has a cusp, or sharp peak, at the crest. It is a solution to a specific class of nonlinear wave equations in which the propagating solitary wave keeps its shape. The format of this article is as follows: Section 1 provides an introduction. Section 2 presents the operational methodology of m-EDAM. In Section 3, we build some new families of soliton solutions for CIE, and in Section 4, several illustrations and a graphical description are shown and analysed. Finally, Section 5 provides the conclusion. AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16669 2. The operational procedure of m-EDAM We describe the operating methods of the m-EDAM in this portion of the paper, focusing on addressing the subsequent general NPDE: R(z,zt,zy1,zy2,zzy1, . . .)=0,(2.1) where z=z(t,y1,y2,y3,...,yj). The strategy taken to solve Eq (2.1) is given as follows: (1). Commencing with a variable transformation of the framework z(t,y1,y2,y3,...,yj)=Z(ϑ), where ϑcan be stated in a number of ways, Eq (2.1) proceeds through a transformation, resulting in the post-NODE: Q(Z,Z0Z,Z0, . . . )=0,(2.2) where Z0=dZ dϑ. On some occasions, the NODE becomes vulnerable to the homogeneous balancing principle due to the integration of Eq (2.2). (2). Next, we suggest a series-based solution to the NODE in Eq (2.1) using the Riccati ODE, which is as follows: Z(ϑ)= N X l=−N ρl(ζ(ϑ))l.(2.3) In this context, ρl(l=−N, ..., N) denotes the unknown parameters, and ζ(ϑ) is the solution to the first-order Riccati ODE stated as: ζ0(ϑ)=R(ζ(ϑ))2+Qζ(ϑ)+P,(2.4) where Q,R, and Pare invariables. (3). Applying a homogeneous balancing strategy to Eq (2.2) that involves balancing the greatest nonlinear component and the highest-order derivative will yield the positive integer Nutilised in (2.3). (4). Following that, we place (2.3) into (2.2) or the equation that arises from integrating (2.2), organising terms ζ(ϑ) with identical orders. An expression in ζ(ϑ) is produced by this process. A set of algebraic equations representing the variables ρl(where l=−N,...,N) and other pertinent parameters is then expressed after the coefficients in this expression are equated to zero. (5). This system of algebraic equations is then solved using Maple tool. (6). Then, by calculating and entering the unknown values into Eq (2.3) together with the ζ(ϑ) (the solution to (2.4)), analytical soliton solutions for (2.2) are obtained. We can get the resulting families of soliton solutions by using the generic solution of Eq (2.4): Family 1. For Φ<0 & R,0, ζ1(ϑ)=−Q 2R+ √−Φtan 1 2√−Φϑ 2R, ζ2(ϑ)=−Q 2R− √−Φcot 1 2√−Φϑ 2R, ζ3(ϑ)=−Q 2R+ √−Φtan √−Φϑ+sec √−Φϑ 2R, AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16670 ζ4(ϑ)=−Q 2R− √−Φcot √−Φϑ+csc √−Φϑ 2R, and ζ5(ϑ)=−Q 2R+ √−Φtan 1 4√−Φϑ−cot 1 4√−Φϑ 4R. Family 2. For Φ>0 & R,0, ζ6(ϑ)=−Q 2R− √Φtanh 1 2√Φϑ 2R, ζ7(ϑ)=−Q 2R− √Φcoth 1 2√Φϑ 2R, ζ8(ϑ)=−Q 2R− √Ztanh √Φϑ+isech √Φϑ 2R, ζ9(ϑ)=−Q 2R− √Φcoth √Φϑ+csch √Φϑ 2R, and ζ10(ϑ)=−Q 2R− √Φtanh 1 4√Φϑ−coth 1 4√Φϑ 4R. Family 3. For QP >0 & Q=0, ζ11(ϑ)=rP Rtan √RPϑ, ζ12(ϑ)=−rP Rcot √RPϑ, ζ13(ϑ)=rP Rtan 2√RPϑ+sec 2√RPϑ, ζ14(ϑ)=−rP Rcot 2√RPϑ+csc 2√RPϑ, and ζ15(ϑ)=1 2rP R tan 1 2√RPϑ!−cot 1 2√RPϑ!!. Family 4. For RP <0 and Q=0, ζ16(ϑ)=−r−P Rtanh √−RPϑ, ζ17(ϑ)=−r−P Rcoth √−RPϑ, AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16671 ζ18(ϑ)=−r−P Rtanh 2√−RPϑ+isech 2√−RPϑ, ζ19(ϑ)=−r−P Rcoth 2√−RPϑ+csch 2√−RPϑ, and ζ20(ϑ)=−1 2r−P R tanh 1 2√−RPϑ!+coth 1 2√−RPϑ!!. Family 5. For P=R&Q=0, ζ21(ϑ)=tan (Pϑ), ζ22(ϑ)=−cot (Pϑ), ζ23(ϑ)=tan (2Pϑ)+(sec (2Pϑ)) , ζ24(ϑ)=−cot (2Pϑ)+(csc (2Pϑ)) , and ζ25(ϑ)=1 2tan 1 2Pϑ!−1 2cot 1 2Pϑ!, Family 6. For R=−P&Q=0, ζ26(ϑ)=−tanh (Pϑ), ζ27(ϑ)=−coth (Pϑ), ζ28(ϑ)=−tanh (2Pϑ)+(isech (2Pϑ)) , ζ29(ϑ)=−coth (2Pϑ)+(csch (2Pϑ)) , and ζ30(ϑ)=−1 2tanh 1 2Pϑ!−1 2coth 1 2Pϑ!. Family 7. For Φ = 0, ζ31(ϑ)=−2P(2+Qϑ) Q2ϑ. Family 8. For Q=µ,P=pµ(p,0) & R=0, ζ32(ϑ)=eµϑ −p. Family 9. For Q=R=0, ζ33(ϑ)=Pϑ . Family 10. For Q=P=0, ζ34(ϑ)=−1 Rϑ. Family 11. For P=0, R,0 & Q,0, ζ35(ϑ)=−Q R(cosh (Qϑ)−sinh (Qϑ)+1), AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16672 and ζ36(ϑ)=−Q(cosh (Qϑ)+sinh (Qϑ)) R(1+sinh (Qϑ)+cosh (Qϑ)). Family 12. For Q=µ,R=pµ(p,0) & P=0, ζ37(ϑ)=eµϑ 1−peµϑ , where Φ = Q2−4RP. 3. The execution of m-EDAM In this section of the study, soliton solutions for the CIE, as specified in Eq (1.1), are generated using the mEDAM approach. First, apply the following variable transformation as part of the method: z(x,y,t)=Z(ϑ), ϑ =y+x−ωt.(3.1) The following NODE is obtained by applying this transformation to (1.1): −ωZ00 +(λ(Z3−Z)−Z00)0+σZ00 =0.(3.2) The following outcome can be obtained from a single integration on (3.2) with a zero integration constant: (σ−ω)Z0+λ(Z3−Z)−Z00 =0.(3.3) After creating a homogeneous balancing condition between Z00 and Z3, it is determined that N=1. Substituting N=1 into Eq (2.3), we obtain the ensuing closed-form solution for (3.3): Z(ϑ)= 1 X l=−1 ρl(ζ(ϑ))l.(3.4) We acquire an expression based on the terms ζ(ϑ) via including (3.4) in (3.3) and bringing together terms with powers alike ζ(ϑ). By setting the coefficients to zero, this expression can be reduced to a set of algebraic nonlinear equations. After using Maple to solve this system, the following two sets of solutions are obtained: Case 1. ρ0=Q √Φ , ρ1=2R √Φ , ρ−1=0, ω =σ, λ = Φ, σ =σ. (3.5) Case 2. ρ0=1 2 Q √Φ +1 2, ρ1=0, ρ−1=P √Φ , ω =σ−3√Φ, λ =2Φ, σ =σ. (3.6) By considering Case 1 and using (3.1), (3.4), and the corresponding general solution to (2.4), we construct the ensuing families of soliton solutions for CIE given in (1.1): Family 1.1. When Φ<0,R,0, z1,1(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R+1 2 √−Φtan 1 2√−Φϑ R        ,(3.7) AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16673 z1,2(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √−Φcot 1 2√−Φϑ R        ,(3.8) z1,3(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R+1 2 √−Φtan √−Φϑ+sec √−Φϑ R        ,(3.9) z1,4(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √−Φcot √−Φϑ+csc √−Φϑ R        ,(3.10) and z1,5(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R+1 4 √−Φtan 1 4√−Φϑ−cot 1 4√−Φϑ R        .(3.11) Family 1.2. When Φ>0,R,0, z1,6(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √Φtanh 1 2√Φϑ R        ,(3.12) z1,7(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √Φcoth 1 2√Φϑ R        ,(3.13) z1,8(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √Φtanh √Φϑ+isech √Φϑ R        ,(3.14) z1,9(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 2 √Φcoth √Φϑ+csch √Φϑ R        ,(3.15) and z1,10(x,y,t)=Q √Φ +2R √Φ       −1 2 Q R−1 4 √Φtanh 1 4√Φϑ−coth 1 4√Φϑ R        .(3.16) Family 1.3. When RP >0 and Q=0, z1,11(x,y,t)=−itan √RPϑ,(3.17) z1,12(x,y,t)=icot √RPϑ,(3.18) z1,13(x,y,t)=−itan 2√RPϑ+sec 2√RPϑ,(3.19) z1,14(x,y,t)=icot 2√RPϑ+csc 2√RPϑ,(3.20) and z1,15(x,y,t)=1 2i tan 1 2√RPϑ!−cot 1 2√RPϑ!!.(3.21) AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16674 Family 1.4. When RP <0 and Q=0, z1,16(x,y,t)=−tanh √−RPϑ,(3.22) z1,17(x,y,t)=−coth √−RPϑ,(3.23) z1,18(x,y,t)=−tanh 2√−RPϑ+isech 2√−RPϑ,(3.24) z1,19(x,y,t)=−coth 2√−RPϑ+csch 2√−RPϑ,(3.25) and z1,20(x,y,t)=−1 2 tanh 1 2√−RPϑ!+coth 1 2√−RPϑ!!.(3.26) Family 1.5. When R=Pand Q=0, z1,21(x,y,t)=−itan (Pϑ),(3.27) z1,22(x,y,t)=icot (Pϑ),(3.28) z1,23(x,y,t)=−i(tan (2Pϑ)+sec (2Pϑ)),(3.29) z1,24(x,y,t)=−i(−cot (2Pϑ)−csc (2Pϑ)) ,(3.30) and z1,25(x,y,t)=−i 1 2tan 1 2Pϑ!−1 2cot 1 2Pϑ!!.(3.31) Family 1.6. When R=−Pand Q=0, z1,26(x,y,t)=tanh (Pϑ),(3.32) z1,27(x,y,t)=coth (Pϑ),(3.33) z1,28(x,y,t)=tanh (2Pϑ)+isech (2Pϑ),(3.34) z1,29(x,y,t)=coth (2Pϑ)+csch (2Pϑ),(3.35) and z1,30(x,y,t)=1 2tanh 1 2Pϑ!+coth 1 2Pϑ!.(3.36) Family 1.7. When P=0, Q,0 and R,0, z1,31(x,y,t)=1−2 cosh (Qϑ)−sinh (Qϑ)+1,(3.37) AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16681 Figure 8. The 3D and contour plots of the kink soliton solution z2,19 stated in (3.58) are plotted for P=5,Q=0,R=−5, σ =4,t=15. Figure 9. The 3D and contour plots of the bright-dark soliton solution z2,23 stated in (3.62) are plotted for P=6,Q=0,R=6, σ =100,t=0. Figure 10. The 3D and contour plots of the kink soliton solution z2,31 stated in (3.70) are plotted for P=10, µ =5,p=2,Q=5,R=0, σ =0,t=2. AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16682 5. Conclusions In conclusion, our study has thoroughly investigated the propagation of solitons in the CIE, a noteworthy model in plasma sciences, by employing the upgraded m-EDAM. By converting the CIE into a NODE and assuming a closed-form solution, numerous cuspon and kink soliton solutions have been identified, including hyperbolic, trigonometric, exponential, and rational functions. Density charts and 3D models that illustrate the propagating behaviour of specific soliton solutions provide important insights into the behaviour of soliton propagation processes, with direct applicability to domains related to the CIE. The results shown here are remarkable and demonstrate the novelty of using m-EDAM to the CIE, as they have not been covered in academic literature before. These discoveries advance our understanding of temporal evolution processes and nonlinear dynamics, which has significant implications for our comprehension of related physical phenomena. Moreover, the efficacy and uniformity of the employed techniques in this study underscore their broader significance for nonlinear problems across other scientific domains. While the utilisation of m-EDAM has significantly enhanced our understanding of soliton dynamics and their impact on the CIE, it is imperative to acknowledge the constraints of the methodology, particularly with the fair handling of the highest derivative and nonlinear variables. Despite this limitation, the work shows that our understanding of soliton behaviour and nonlinear dynamics will continue to expand and opens up new directions for future research in related fields. That is, the future scope of the current study lies in extending the proposed m-EDAM to study soliton events in nonlinear stochastic models, which will shed further light on how mechanistic and stochastic dynamics interact. Author contributions Naveed Iqbal: Conceptualization, Investigation and Writing-review & editing; Muhammad Bilal Riaz: Methodology, Conceptualization, Formal analysis and Administration and Funding acquisition; Meshari Alesemi: Methodology; Taher S. Hassan: Software and Investigation; Ali M. Mahnashi: Validation and Resources; Ahmad Shafee: Formal analysis and Resources. All authors have read and agreed to the published version of the manuscript. Use of AI tools declaration The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article. Acknowledgments This article has been produced with the financial support of the European Union under the REFRESH (Research Excellence For Region Sustainability and High-tech Industries) project number CZ.10.03.01/00/22 003/0000048 via the Operational Programme Just Transition. The authors are thankful to the Deanship of Graduate Studies and Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program. AIMS Mathematics Volume 9, Issue 6, 16666–16686. 16683 Conflict of interest The authors declare that they have no conflicts of interest. References 1. A. Seadawy, A. Sayed, Soliton solutions of cubic-quintic nonlinear Schr¨ odinger and variant Boussinesq equations by the first integral method, Filomat,31 (2017), 4199–4208. http://dx.doi.org/10.2298/FIL1713199S 2. H. Yasmin, A. Alshehry, A. Ganie, A. Mahnashi, R. 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