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Full Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=tusc20Journal of Taibah University for ScienceISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/tusc20Phase characterization and exraction of new formsof solitons for the (3+1)-dimensional q-deformedSinh-Gordon equationHaifa I. Alrebdi, Nauman Raza, Farwa Salman, Norah A. M. Alsaif, Abdel-Haleem Abdel-Aty & H. EleuchTo cite this article: Haifa I. Alrebdi, Nauman Raza, Farwa Salman, Norah A. M. Alsaif, Abdel-Haleem Abdel-Aty & H. Eleuch (2024) Phase characterization and exraction of new formsof solitons for the (3+1)-dimensional q-deformed Sinh-Gordon equation, Journal of TaibahUniversity for Science, 18:1, 2321647, DOI: 10.1080/16583655.2024.2321647To link to this article: https://doi.org/10.1080/16583655.2024.2321647© 2024 The Author(s). 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Alrebdia, Nauman Razab, Farwa Salmanb, Norah A. M. Alsaifa, Abdel-Haleem Abdel-Aty c andH. Eleuchd,e,faDepartment of Physics, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia; bDepartment ofMathematics, University of the Punjab, Lahore, Pakistan; cDepartment of Physics, College of Sciences, University of Bisha, Bisha, SaudiArabia; dDepartment of Applied Physics and Astronomy, University of Sharjah, Sharjah, UAE; eCollege of Arts and Sciences, Abu DhabiUniversity, Abu Dhabi, UAE; f Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX, USAABSTRACTIn this article, the (3+1)-dimensional q-deformed Sinh-Gordon model is investigated to extractanalytical solutions using the unifiedmethod. This technique effectively extracts polynomial andrational function solutions. When the appropriate limiting constraints are given to the parame-ters, this technique successfully retrieves hyperbolic and trigonometric results. Some graphicalrepresentations of the solutions of the proposed equation are illustrated. Additionally, all feasi-ble phase portraits are shown, the planer dynamical system of the equation under discussion isbuilt using Galilean transformation, and sensitive inspection is used to verify the sensitivity ofthe equation under consideration. There aren’t many previous methods for solving this kind ofequation, either analytically or numerically. This work is highly valuable for the understandingof various symmetrical physical systems.ARTICLE HISTORYReceived 20 September 2023Revised 6 November 2023Accepted 8 January 2024KEYWORDSq-deformed Sinh-Gordonequation; travelling wavetransformation; unifiedmethod; soliton; quantitativeanalysis1. IntroductionThe study of optical solitons has gathered a great dealof interest and is currently one of the most hot top-ics of research. This topic plays an important role indescribing some phenomena in science, engineering,and technology, including quantum optics [1], opticalcommunications [2], nonlinear dynamics [3] andplasmaphysics [4]. Solitary wave (soliton) propagation is one ofthe most important phenomena that many researchershave been looking for to analyse and comprehend. Thesolitons have a specific direction onhow todevelop andspread the information. This form of wave is very com-mon in nature, and it has a lot of applications in opticalfibres [5]. The concept of soliton is defined as the wavepacket that keeps its shape while it travels at a steadyspeed.A delicate balance between dispersion and nonlin-earity can be used to describe this property. The solitonsolution plays a vital role in integrable models. Soli-tons are produced by a variety of famous nonlinearPDEs, including Sawada-Kotera equations [6,7], Biswas-Milovic equations [8], Schrödinger equations [9], cou-pled KDV equations [10] and BKP equations [11].Non-linear evolution equations (NLEEs) are nowusedto illustrate complex physical processes in several areasof research, including physics [12], chemistry [13], biol-ogy [14] and engineering [15]. Numerous schemes forextracting exact and numerical solutions for nonlinearevolution equations (NLEEs) are developed to obtainnecessary information for understanding real problemsin a variety of engineering fields and science, such asthe G’/G expansion method for exploring exact soli-tary wave solutions [16], Wronskian formulation [17],Painlevé approach [18], the Hirota bilinear method [19],linear superposition principle [20], inverse scatteringmethod [21] and the invariant method [22].In recent years, substantial progress has been achieved, and a number of reliable and efficient methodolo-gies for retrieving accurate NLEE solutions have beendevised [23–28]. The Sinh-Gordon equation has beenstudied using self-similar transformaion[29], the ana-lytic solutions have been extracted applying similar-ity transformation and Hirota’s bilinear method [30].Moreover, the two-soliton solution for this model hasbeen investigated in the literature [31]. While the underconsideration model and several methods of solv-ing it were well-known in the 19th century, its sig-nificance increased notably when it was discoveredthat the resulting solution contributes to the collidingfeatures of solitons (kink and antikink). Other practi-cal implications of the Sinh-Gordon equation includethe movement of a suspended pendulum tied to astretched wire, the transmission of flux in Josephsonjunctions (a connection between two superconductors)CONTACT Abdel-Haleem Abdel-Aty amabdelaty@ub.edu.sa Department of Physics, College of Sciences, University of Bisha, PO Box 344, Bisha61922, Saudi Arabia© 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in anymedium, provided the original work is properly cited. The terms onwhich this article has been published allow the posting of the AcceptedManuscript in a repository by the author(s) or with their consent.https://www.taibahu.edu.sa/Pages/EN/Home.aspxhttp://www.tandfonline.com/tusc20http://www.tandfonline.comhttps://crossmark.crossref.org/dialog/?doi=10.1080/16583655.2024.2321647&domain=pdf&date_stamp=2024-03-27http://orcid.org/0000-0002-6763-2569mailto:amabdelaty@ub.edu.sahttp://creativecommons.org/licenses/by/4.0/2 H. I. ALREBDI ET AL.and displacements in crystals. The governing problemis investigated using the unified approach [32], oneof the most evident, clear, and effective mathemati-cal techniques for obtaining accurate answers. Becauseit delivers exact solutions in the form of polynomialand rational functions, the applied technique is help-ful and efficient. Different wave solutions are repre-sented by polynomial solutions, whereas periodic andsoliton forms are described by rational function solu-tions. Moreover, the bifurcation of the dynamical sys-tem of the stated equation is also studied. Using abifurcation analysis of the planner dynamical system[33–35], all possibilities for parameter dependencyhavebeen examined to see the geometrical properties of thediscussed model.The rest of the article is organized as follows: Thegoverning model is discussed in Section 2. The princi-ples of mathematical analysis are covered in Section 3.Section 4 gives an insight into the technique in con-cern. The proposed procedure is discussed in Section 5,which explains how to generate soliton solutions. Thederived solutions to the governing problems are graph-ically represented in Section 6. The qualitative analysisis detailed in Section 7. Finally, the work’s conclusion isoffered in Section 8.2. GoverningmodelThe generalized (3+1) dimensional q-deformed Sinh-Gordon equation [36,37] (3+1)-dimensional Eleuchequation) is the subject of this paper, described as(∂2∂x2+ ∂2∂y2+ ∂2∂z2− ∂2∂t2)u = [sinhq(uγ )]p − δ,(1)Which can be written as�u =(� − ∂2∂t2)u = [sinhq(uγ )]p − δ, (2)where � is the d’Alembert operator where sinhq repre-sents the function of q-deformed characterized bysinhq(z) = ez − qe−z2where 0 < q ≤ 1. (3)With q = 1, conventional sinh functions are provided.coshq(z), tanhq(z) and their reciprocal are explained in[36]. The offered methodology will be used to investi-gate the optical soliton related to Equation (8).3. Themathematical examinationTo solve Equation (1) for travelling wave solutions con-sider the transformation [38] asξ = (x + y + z) − βt√1 − β2; (4)where the velocity of travelling wave is denoted by β .Equation (1) has taken the form of following ODE, usingEquation (4)d2y(ξ)dξ2= [sinhq(yγ (ξ))]p − δ. (5)For this article, For p = 1, γ = 1 and δ = 0, the afore-mentioned equation is investigated. In the next section,the travelling wave solutions for Equation (5) werederived using a recommended analytical method byapplying the transformation given below.f (γ ) = ey(γ ). (6)Using Equation (6), Equation (5) becomes− 2f ′2 + 2ff ′′ − f3 + qf = 0. (7)4. Overview of the proposedmethodUnified technique:Let the structure of q-deformedmodel be expressedlikeB(z, t, bx , bt , bxt , b2x , b2xt · · · , bcxt) = 0; c ≥ 0, (8)here B stands for the function. It is possible to changethe pattern ν = βz − wt using the travelling waveparameterB(b, b′, b′′, . . . , bc) = 0, (9)here b′ presents the differentiation for b including anew parameter ν. To extract the solution for (9), theUM technique has been used to find rational or poly-nomial function solutions. We are going to consider thepolynomial function solution. The method is discussedbelow:Polynomial form:(9) has a solution in polynomial form likeQ(μ) =m∑n=0riϒi(μ). rn �= 0, (10)ri are constants. The function ϒ(μ) will be found byconsidering the auxiliary equation:(ϒ ′(μ))σ =σ j∑n=0ciϒi(ν), σ = 1, 2, (11)where ci are parameters. Moreover, it’s important tolookout for numerical values related ton as examined interms related to j by substituting the hom*ogenous con-ditions among the highest derivatives along with high-est nonlinear terms involved in (9), whereas j is revealedby utilizing the consistency condition.where to solve (10), the Unified Method deals with(10) for the elliptic or elementary solution when σ = 1and σ = 2.JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 3Rational form:The basic part of the presented portion is to let (9)has the following solutionQ(ν) =∑nm=0 bmϒm(ν)∑rm=0 qmϒm(ν), n ≥ r, (12)with an auxiliary model(ϒ ′(ν))σ =σv∑i=0βiϒi(ν), σ = 1, 2. (13)In (12) and (13), bm, qm and βi are unspecified param-eters to be relieved, in such technique, the solutionformed through (12) fulfils (9). It’s vital for looking outnumerical interpretations of n, v are exerted employingbalancing criteria using the highest order for linear ornonlinear terms in (9). By applying the consistency crite-ria,we could foundunknownconstants among (12). TheUM technique will be employed for solving (12). Afterthat, we got a solution for σ = 1 or σ = 2, respectively.5. Extraction of solitonThis portion deals with the exploration of solutions con-cerning the governing Equation (1) with the help of theproposed technique.5.1. Exact solution using a unifiedmethodEquation (7) is investigated using a unified approach toretrieve soliton solutions as described below.Balancing f ′2 and f3 in Equation (7) produces balanc-ing number 2. The suggested solution is as follows:f (ξ) =1∑i=0riχi(ξ), r1 �= 0, (14)with the auxiliary equation(ϒ ′ (ξ))σ =2σ∑p=0ciϒi(ξ), σ = 1; 2. (15)5.1.1. Solitary wave solutionHere, we’ll retrieve the solitary solution regardingEquation (7).For this reason, we should substitute σ = 1 in theauxiliary Equation (15) ;therefore, we obtainedQ(ξ) = r0 + r1ϒ (ξ) + r2ϒ (ξ)2 ,ϒ ′(ξ) = c0 + c1ϒ (ξ) + c2ϒ2 (ξ) . (16)The system of nonlinear equations has revealed using(16) in (7). This system would solve by any softwareproducts such as Maple and Mathematica. The resultslisted below are obtainedc0 =((3 r12 + 8√5r2√n)r2 − r12r2)√580 R25/2,c1 = r1√510√r2, r2 =√5√r210,r0 = 3 r12 + 8√5r2√n12r2(17)By the auxiliary equation ϒ ′(ζ ) = c0 + c1ϒ(ζ) + b2ϒ2(ζ ) and put together with (17), we find that Equation (7)has a solitary solutionf (ξ) =(tanh(ξ2 4√1n))2√n−1. (18)Then the solution isu(x, y, z, t) = ln⎛⎜⎜⎜⎜⎝tanh2((x+y+z)−βt2√1−β2 4√1n)√1n⎞⎟⎟⎟⎟⎠ . (19)5.1.2. Soliton wave solutionHere, we extract new soliton solutions. To find the soli-ton solution, nowwe substitute σ = 2 in (15), we obtainf (ξ) = r0 + r1ϒ (ξ) + r2ϒ2 (ξ) ,ϒ ′(ξ) = ϒ(ξ)√c0 + c1ϒ (ξ) + c2ϒ2 (ξ). (20)Nonlinear equations are obtained by putting Equation(20) in Equation (7). That system would be solved uti-lizing symbolic computations and the following resultsare obtainedc0 = √n, c1 = r12, c2 = r2116√n,r2 = r21√n, r0 = √n. (21)Utilizing an auxiliary equation ϒ ′(ξ) = ϒ(ξ)√c0 + C1ϒ(ξ) + c2ϒ2(ξ) and using (17), we achieveda solution of Equation (7)f (ξ) =√n(r1 eξn1/4 + 1)2(r1 eξn1/4 − 1)2 . (22)By inserting (22), we getu(x, y, z, t) = ln⎛⎜⎜⎜⎜⎜⎝√q(r1e4√n((x+y+z)−βt)√1−β2 + 1)2(r1e4√n((x+y+z)−βt)√1−β2 − 1)2⎞⎟⎟⎟⎟⎟⎠ . (23)4 H. I. ALREBDI ET AL.5.2. Rational functionTo extract the solutions in the rational form, let us con-siderf (ν) =∑np=0 rmm(ν)∑jp=0 smm(ν), n ≥ j, (24)with the auxiliary model,( ′(ν))σ =σv∑i=0cii(ν), σ = 1, 2. (25)rm, sm, ci are parameters. By hom*ogenous balance,using (7), we find solutions when j = 1 and σ = 2. Solwe have 2 cases as5.2.1. Case 1: periodic formWe have the solution in the following form as shownbelow.f (ξ) = r0 + r1ϒ(ξ)s0 + s1ϒ(ξ),ϒ ′(ξ) =√c0 + c2ϒ2(ξ). (26)Utilizing Equation (26) in Equation (7), we obtain equa-tions in ϒ . Now utilizing some symbolic software,the constants are retrieved.The followin results areobtainedc0 =√−√ns0s1, c2 =√−√n,r0 = −√n, r1 = √n (27)Solving equationϒ ′(ξ) =√c0 + c2ϒ2(ξ), and utilizingthe values in Equation (27). The following result is thesolution for Equation (7)f (ξ) = −√n(√(tan(ξ√−√n))2 + 1+ tan(ξ√−√n))√(tan(ξ√−√n))2 + 1 − tan(ξ√−√n) .(28)We obtain the solutionu(x, y, z, t)= ln⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝√n(tan(√−√n((x+y+z)−βt)√1−β2)+ sec(√−√q((x+y+z)−βt)√1−β2))tan(√−√q((x+y+z)−βt)√1−β2)− sec(√−√q((x+y+z)−βt)√1−β2)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (29)5.2.2. Case 2: soliton solutionHerewe let the solution in the following formf (ξ) = r0 + r1ϒ(ξ)s0 + s1ϒ(ξ),ϒ ′(ξ) =√c0 + c1ϒ(ξ) + c2ϒ2(ξ). (30)Putting Equation (30) in Equation (7),we obtain equa-tions inϒ . Using any software, such as Maple or Mathe-matica, we get parameters. The outcomes listed beloware obtainedc0 = r0(c1s1 − r0)s21√n, c2 = √n, r1 = √n,s0 = c1s1 − r0√n(31)ϒ ′(ζ ) =√c0 + c1ϒ(ξ) + c2ϒ2(ξ) has been solved, aswell as putting values in Equation (31). Equation (7) hasthe following soliton solution:f (ξ) =(−4 4√nc1s12 + 4√neξ 4√ns12 + e−ξ 4√nc12s12−4 e−ξ 4√nc1r0s1 + 4 e−ξ 4√nr02 + 8 r0s1 4√n)√n4√neξ 4√ns12 + 4 4√nc1s12 + e−ξ 4√nc12c12−4 e−ξ 4√nc1r0s1 + 4 e−ξ 4√nr02 − 8 R0s1 4√n. (32)We obtain the solutionu(x, y, z, t)= ln⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝(−4 4√nc1s12 + 4√neξ 4√ns12+e−ξ 4√nc12s12 − 4 e−ξ 4√nc1r0s1+4 e−ξ 4√nr02 + 8 r0s1 4√n)√n4√neξ 4√ns12 + 4 4√nc1s12+e−ξ 4√nc12c12 − 4 e−ξ 4√nc1r0s1+4 e−ξ 4√nr02 − 8 r0s1 4√n⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (33)where ξ = (x+y+z)−βt√1−β2.6. Graphical depictionHere, we illustrate our solutions with a few two- andthree-dimensional representations. The graphical rep-resentation has been provided for the obtained solu-tions. Figure 1 represents the solitary solution using 3Dand 2D graphs. The graphical illustration of the soli-ton polynomial solution has been depicted in Figure 2.The solutions as rational periodic and soliton types areshown in Figures 3 and 4. There are records of brightsolitons, singular solitons and singular periodic solitons.7. Qualitative analysis of the q-deformedSinh-Gordon equationThis section includes the bifurcation behaviour for ourgoverning equation. To fulfil this, we transform theJOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 5Figure 1. Graphical representation of Equation (19) along z = 0.Figure 2. Graphical illustration of Equation (23) along z = 0. (a) 3D Plot3. (b) 2D Plot4.Figure 3. Graphical illustration of Equation (29) along z = 0. (a) 3D Plot5. (b) 2D Plot6.Figure 4. Graphical illustration of Equation (33) along z = 0. (a) 3D Plot7. (b) 2D Plot8.6 H. I. ALREBDI ET AL.model to ordinary differential Equation (7), through thetravelling wave transformation discussed before.By using (7), the planer dynamic system retrieved isshown belowf ′ = kk′ = k22+ f22− q2, (34)but the above system isn’t Hamiltonian. From (34), wegetdk2df= f3 − qf + 2k2f, (35)here f = 0 is singular point of the above equation ;therefore; just in exceptional circ*mstances could f hasa zero. Equation (35) has the following solutionk2 = f3 + qf + cf2 (36)Then, we get(f ′)2 − [f3 − qf + cf2] = 0, (37)c is the constant of integration. As q and c are constants,it’s feasible to obtain equivalent conserved quantityH(f , k) = k2 − [f3 − qf + cf2], (38)that is undoubtedly conserved quantity. Here, we per-form a qualitative examination based upon the govern-ing problem implementing the discrimination system.Equation (38) with its potential energy is shown belowF(f ) = −[f3 − qf + cf2], (39)and thus we getF′(f ) = −3f2 − q − 2cf . (40)J(f , k) is the coefficient matrix regarding the linearizedsystem at a fixed point (f , k)). That is termed as Joco-bian. The determinant of that matrix is as follows:∣∣∣∣ 0 1−6f − 2c 0∣∣∣∣ , (41)it is convenient to show that the eigenvalues at singularpoint (f , 0) areλ±(f , 0) = ±√−6f − c2, (42)therefore, (f , 0) is saddle if J(f , 0) is less than zero,if J(f , 0) is greater than zero, then its a centre pointwhereas it will be cusp if J(f , 0) is zero.Through discriminant of the polynomial� = −4c2 − 12q, (43)we get the following outcomes.Case I:let � = 0 and q>0, then we getF′(f ) = (f − s)2. (44)In this case, there exist only one equilibrium point (s,0),it will be cusp when n = 3, c = ±3, we have s = 1 forc = −3, whereas we get s = −1 for c = 3.Case II:If � > 0, c>0 and q<0, we getF′(f ) = (f − s)(f − t). (45)we obtain two equilibrium points. For c>0, we have(s, 0) that is a saddle but (t, 0) will be centre. Phaseportrait is shown taking n = −3 and c = −4. We gets = −3 and t = 0.333.Case III:If � > 0, q<0 and c<0, we getF′(f ) = (f − s)(f − l), (46)here, two equilibrium points are obtained. For c>0,(s, 0) a saddle point and (l, 0) is the centre. For n = −3and c = 4, we get s = −0.333 and l = 3 and the phaseportrait is plotted.Case IV:If � > 0, q = 0 and c>0, we getF′(f ) = f (f − s). (47)In the above equation, we get two equilibrium points(0, 0) with (s, 0). Regarding this case (0, 0) being a cen-tre, whereas (s, 0) is a saddle point. Phase portrait hasbeen plotted for c = 3, where we get s = −2.Case V:If � > 0, q = 0 and c<0, we getF′(f ) = f (f − p), (48)We get two equilibrium points (0, 0) with (s, 0). Here,(0, 0) and (p, 0) are saddle points. The phase depictionhas been plotted for c = −3, where we get s = 2. Thephase portrait is shown in Figures 5–8. The red lines rep-resent centre on (s, 0) and the lines in cyan colour showthe saddle behaviour of (0, 0) and the black line showsthe saddle behaviour on (0, 0) and centre towards (s, 0).7.1. Comparitive studyHere, we compare our work with the previous workdone on the governing model. The q-deformed Sinh-Gordon problem has been constructed in [36]. Quasi-periodic andchaoticbehaviourshavebeendocumentedwith some analytical solutions using different tech-niques in [39]. The generalized q-deformed Sinh-Gordonequationhas been solvednumerically using thefinite difference method and analytically using the newgeneral form of Kudryashov’s strategy in [40]. However,we applied a unified approach which helped to extractJOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 7Figure 5. Geometric visualization of Equation (38) regardingCase II.Figure 6. Geometric visualization of Equation (38) regardingCase III.Figure 7. Geometric visualization of Equation (38) regardingCase IV .Figure 8. Geometric visualization of Equation (38) regardingCase V .different solutions in many ways. Also, we used (3+1)dimensional model which is different from the gener-alized model, for qualitative analysis. Such work will beuseful for further study on such model.8. ConclusionThis article provided an investigation of the solitonsolutions in the (3+1)-dimensional q-deformed Sine-Gordonmodel and thepresenceof soliton solutions. Formodelling physical systems with violated symmetries,this newly established equation may be helpful. A uni-fied method has been applied to retrieve solutions tothe (3+1)-dimensional q-deformedSinhGordonmodel.Polynomial with rational solutions was retrieved in thearticle. When the appropriate limiting constraints aregiven to the parameters, this technique successfullyretrieves hyperbolic and trigonometric results. Somegraphical representations of solutions of the proposedequation are illustrated. Similarly, the impacts of theparameters on the expected non-singular solution aredepicted through 2D graphs. Moreover, the qualitativeanalysis i.e. Bifurcation is performed on the discussedmodel and sensitive inspection is used to verify the sen-sitivity of the equation under consideration. The systemis converted into a planer Hamiltonian dynamical sys-tem using Galilean transformation. The possible caseswere predicted and effectively shown in phase portraitsbased on discriminants. The obtained solutions werenovel and they were not shown previously.AcknowledgementThe authors extend their appreciation to the Deputyship forResearch & Innovation, Ministry of Education in Saudi Arabiafor funding this research work through the project numberRI-44-0077.8 H. I. ALREBDI ET AL.Disclosure statementNo potential conflict of interest was reported by the author(s).Data availability statementNo data were used to support this study.ORCIDAbdel-Haleem Abdel-Aty http://orcid.org/0000-0002-6763-2569References[1] Yusuf A, Sulaiman TA,Mirzazadeh M, et al. M-truncatedoptical solitons to a nonlinear Schrodinger equationdescribing the pulse propagation through a two-modeoptical fiber. Opt Quantum Electron. 2021;53(10):558.doi: 10.1007/s11082-021-03221-2[2] Haus HA, Wong WS. Solitons in optical communications.Rev Mod Phys. 1996;68(2):423–444. doi: 10.1103/RevModPhys.68.423[3] Raza N, Rani B, Chahlaoui Y, et al. A variety of newrogue wave patterns for three coupled nonlinear Mac-cari’s models in complex form. Nonlinear Dyn. 2023;111:18419–18437. doi: 10.1007/s11071-023-08839-3[4] Saha Ray S, Sagar B. Numerical soliton solutions offractional modified (2+1)-dimensional Konopelchenko-Dubrovsky equations in plasma physics. 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Nonlinear Eng.2023;12(1):20220255. doi: 10.1515/nleng-2022-0255https://doi.org/10.1016/j.aml.2014.07.015https://doi.org/10.1016/j.rinp.2021.103979https://doi.org/10.1016/j.rinp.2020.103650https://doi.org/10.1007/s11071-016-3183-5https://doi.org/10.1016/j.jksus.2022.102087https://doi.org/10.1016/j.joes.2022.05.034https://doi.org/10.3390/sym14112425https://doi.org/10.3390/sym15071324https://doi.org/10.1515/nleng-2022-02551. Introduction2. Governing model3. The mathematical examination4. Overview of the proposed method5. Extraction of soliton5.1. Exact solution using a unified method5.1.1. Solitary wave solution5.1.2. Soliton wave solution5.2. Rational function5.2.1. Case 1: periodic form5.2.2. Case 2: soliton solution6. Graphical depiction7. Qualitative analysis of the q-deformed Sinh-Gordon equation7.1. Comparitive study8. ConclusionAcknowledgementDisclosure statementORCIDData availability statementReferences<< /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /PageByPage /Binding /Left /CalGrayProfile () /CalRGBProfile (Adobe RGB \0501998\051) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.5 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /sRGB /DoThumbnails true /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 524288 /LockDistillerParams true /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments false /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings false /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Remove /UCRandBGInfo /Remove /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.40 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects true /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /Description << /ENU () >>>> setdistillerparams<< /HWResolution [600 600] /PageSize [595.276 841.890]>> setpagedevice