Exploring travelling wave solutions, bifurcation, chaos, and sensitivity analysis in the (3+1)-dimensional gKdV-ZK model: A comprehensive study using Lie symmetry methodology

Resul s in Enginee ing 22 (2024) 102194 A ailable online 6 May 2024 2590-1230/© 2024 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by- nc-nd/4.0/). Con en s lis s a ailable a ScienceDi ec Resul s in Enginee ing jou nal homepage: www.sciencedi ec .com/jou nal/ esul s-in-enginee ing Resea ch pape Explo ing a elling wa e solu ions, bi u ca ion, chaos, and sensi i i y analysis in he (3+1)-dimensional gKdV-ZK model: A comp ehensi e s udy using Lie symme y me hodology Adil Jhangee a,b,∗, Tahi a Jamalc, Abdallah M. Tala had, Muhammad Bilal Riaz a,e aIT4Inno a ions, VSB – Technical Uni e si y o Os a a, Os a a, Czech Republic bDepa men o Ma hema ics, Namal Uni e si y, 30KM Talagang Road, Mianwali 42250, Pakis an cDepa men o Ma hema ics, Uni e si y o he Punjab, Pakis an dDepa men o Ma hema ics and Na u al Sciences, P ince Mohammad bin Fahd Uni e si y, Al-Khoba , Saudi A abia eDepa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon A R T I C L E I N F O A B S T R A C T Keywo ds: The (3+1)-dimensional gKdV-ZK equa ion Lie symme y analysis Analy ical solu ions Modified auxilia y equa ion me hod Bi u ca ion analysis Examina ion o chao ic dynamics Sensi i i y analysis This a icle p esen s a s udy on he gene alized Ko eweg-de V ies-Zakha o -Kuzne so (gKdV-ZK) model, which is a nonlinea sys em ha demons a es he effec o magne ic fields on weak ion-acous ic wa es in plasma consis ing o cold and ho elec ons. The esea ch en ails in es iga ing he educ ion o symme y h ough Lie g oup analysis, sc u inizing he cha ac e is ics o he dynamic s uc u e using bi u ca ion phase diag ams, and examining he dynamic beha iou o he pe u bed dynamical sys em employing chaos heo y. Me hods such as 3D and 2D phase po ai s, ime se ies analysis, Poinca é maps, explo a ion o mul is abili y in he au onomous s uc u e ac oss a ious ini ial condi ions, Lyapuno exponen s, and bi u ca ion diag ams a e exe cised o demons a e chao ic beha iou . Addi ionally, he esea ch es ablishes gene al o ms o soli a y wa e solu ions, encompassing hype bolic, igonome ic, and a ional soli on solu ions, h ough he u iliza ion o a modified auxilia y equa ion app oach o analy ically add ess he examined p oblem. These findings a e isually depic ed as 2D and 3D g aphs wi h ca e ully selec ed pa ame e s, accompanied by hei co esponding cons ain condi ions. Fu he mo e, he sensi i i y analysis o he s udied equa ion is delibe a ed upon and isually illus a ed. The unco e ed findings a e cap i a ing, inno a i e, and po en ially beneficial o comp ehending a ious physical phenomena in enginee ing and science. 1. In oduc ion Nonlinea pa ial diffe en ial equa ions (NLPDEs) play nume ous c ucial oles in enginee ing and science, as hey simula e a ious physical phenomena in he eal wo ld [1–4]. These equa ions cons i u e he mos essen ial models needed o in es iga e nonlinea p ocesses, which a e p e alen in fields such as ae ospace enginee ing, ma ine science, clima ology, nonlinea mechanics, biology, popula ion ecology, plasma physics, and fluid mechanics. The significance o sol ing he ma hema ical models go e ning hese physical p ocesses canno be o e s a ed. Howe e , i is widely acknowledged ha he e is no sys ema ic me hod o ob aining a closed- o m solu ion o nonlinea pa ial diffe en ial equa ions. Despi e his challenge, esea che s ha e de eloped se e al efficien me hods o ob aining easible and unique solu ions o NLPDEs. Fo ins ance, hese me hods include he gene alized Kud yasho echnique [5], he (𝐺′ 𝐺)expansion app oach [6], he ex ended di ec algeb aic me hod [7], he modified ans o med a ional unc ion app oach [8], he soli on ansa z echnique [9], as well as app oaches such as he Da boux ans o ma ion, Bäcklund ans o ma ion, Cole-Hop ans o ma ion, a ious Jacobi ellip ic unc ion echniques, a ious Tanh echniques, a iable sepa a ion me hod, Painle é p ocedu e, simila i y educ ion echnique, and homogeneous balance p ocedu e, among o he s [10–14]. * Co esponding au ho . E-mail add esses: [email p o ec ed] (A. Jhangee ), [email p o ec ed] (T. Jamal), [email p o ec ed] (A.M. Tala ha), [email p o ec ed] (M.B. Riaz). h ps://doi.o g/10.1016/j. ineng.2024.102194 Recei ed 16 Feb ua y 2024; Recei ed in e ised o m 8 Ap il 2024; Accep ed 27 Ap il 2024 Resul s in Enginee ing 22 (2024) 102194 2 A. Jhangee , T. Jamal, A.M. Tala ha e al. The KdV equa ion was de eloped o simula e wa es in shallow wa e wi h weak nonlinea i y. T a elling wa e solu ions exis o he KdV equa ion, wi h one no able solu ion known as a soli on. Soli ons find applica ions in a ious fields, including wa e diffusion in nonlinea op ics, oceanog aphy, wa e s o age con aine s, and plasma simula ions [15]. Obse a ions o elec on-posi on (EP) plasma in a ious space condi ions ha e been conduc ed in he ea lie uni e se. Such plasma is ound in acc e ion discs, neu on s a s, pulsa magne osphe es, ac i e galac ic nuclei, cosmic sola fla es, and black hole magne osphe es. EP plasmas exhibi dis inc physical cha ac e is ics compa ed o elec on-ion plasmas. Resea che s ha e in es iga ed he modula ional ins abili y o he wa e equa ion o a highly in ense, linea ly pola ized lase pulse p opaga ing h ough EP plasma. By compa ing he magne o-hyd odynamics o elec on-posi on plasma wi h ha o elec on-ion plasma, no el physics conclusions can be d awn [16]. The gKdV-ZK equa ions se e as essen ial models o a wide ange o physical phenomena, encompassing shallow and s a ified in e nal wa es, LC ci cui wa es, ion-acous ic wa es in plasma physics, space-based applica ions, nonlinea op ics, hyd odynamics, and a ious o he s. In plasma physics, he highe -dimensional gKdV-ZK equa ion is u ilized o elucida e he influence o magne ic fields on weak ions and acous ic wa es [17]. A simple model in ol es ion-acous ic soli ons, whe e he coole ions a e ea ed as a fluid wi h adiaba ic p essu es, while he ho e iso he mal elec ons a e desc ibed by a Bol zmann dis ibu ion [18]. To in es iga e he s uc u es o ion-acous ic wa es in plasma physics, he (3+1)-dimensional gene alized Ko eweg-de V ies-Zakha o -Kuzne so equa ion is conside ed [18–20]: 𝑢𝑡+𝑎𝑢2𝑢𝑥+𝑏𝑢𝑥𝑥𝑥 +𝑑(𝑢𝑦𝑦 +𝑢𝑧𝑧)𝑥=0.(1) He e, he wa e p ofile is ep esen ed by he spa io empo al a iables 𝑢(𝑥, 𝑦, 𝑧, 𝑡)in liquid ions, and he eal cons an s 𝑎, 𝑏, and 𝑑go e n he oblique dis ibu ion o nonlinea elec os a ic modes, delinea ing combina ions o wa m iso he mal, ho adiaba ic liquid, and s a iona y backg ound cons i uen s [18]. Recognizing he inhe en symme ies wi hin a diffe en ial equa ion and me hodically u ilizing hem esul s in p ecise ou comes [21,22]. Fluid dynamics can g ea ly benefi om employing Lie poin symme y analysis [23–25], especially when ackling he in ica e Na ie -S okes equa ions. Such equa ions ha answe he mo ion o iscous ma e ials ha e challenges because o issues o hei complexi y and nonlinea p ope ies. Lie poin symme y is applicable in his field, which would enable simplifica ion o he s uc u e o equa ions wi h p oduc ion o essen ial s able esul s o unde s anding o wa e mo emen . The field o chaos heo y and bi u ca ion ha e become an in e disciplina y one wi h applica ions in s udy a ea such as communica ions, engi- nee ing, economics and ecology [26]. When pe iodic ex e nal dis u bances induce significan chao ic beha iou in diffe en ial equa ions, i can be disce ned h ough he dynamic cha ac e is ics o hese equa ions. Dynamical and quasipe iodic p ope ies o pe iodic nonlinea wa es ypically o m he ocal poin o s udies in ol ing diffe en ial equa ions. Bi u ca ion analysis, which in ol es ma hema ically e alua ing he quali a i e changes in ou comes wi hin a se o diffe en ial equa ions, se es as a well-es ablished echnique o unde s anding dynamic s uc u es. The me hodology o bi u ca ion analysis has gained p ominence in ecen decades o comp ehending diffe en ial p oblems, as e idenced by he ci ed li e a u e which highligh s significan con ibu ions and con empo a y s udies on he subjec [27,28]. Addi ionally, he in es iga ion in o bi u ca ion heo y emains a opic ha has no been ex ensi ely explo ed p e iously. Rehman e al. [29] applied modified 𝐺′ 𝐺2 o ob ain pe iodic ype soli ons in he chi ped o m o nonlinea Sch ödinge equa ion. The au ho s also s udied bi u ca ion, sensi i i y analysis, chao ic and mul is abili y phenomenon. Paul e al. [30] in es iga ed a ma hema ical model o COVID-19 conside ing mul iple doses o accina ions. They iden ified bo h he disease- ee equilib ium (DFE) and he disease-endemic equilib ium (DEE) and conduc ed s abili y analyses o de e mine he pa ame e space in which he disease would ei he ade ou o pe sis in he popula ion. Addi ionally, hey explo ed bi u ca ion analysis and assessed he sensi i i y o he COVID-19 ma hema ical model. Gao e al. [31] in es iga ed he gene a ion and s abili y o nonlinea wa es in incomp essible bounda y laye s, ocusing on he effec s o dual s eady wa es. They employed a pe u bed wa e oscilla o sys em o analyze he s abili y o hese nonlinea wa es, e ealing ha he loss o s abili y in low-o de oscilla o s p ima ily go e ns wa e ansi ion and ene gy dynamics. In hei s udy, Mungal e al. [32]conduc a case s udy o explo e he op imal gene a ion sys em pa ame e s o a financially cons ained elec ic u ili y si ua ed in he Ca ibbean. Thei objec i e is o de e mine he specific pa ame e s ha he u ili y should p io i ize o in es men . To achie e his goal, hey unde ake a sensi i i y analysis ocusing on he cos - ela ed pa ame e s wi hin he con ex o he Uni Commi men (UC) p oblem. Jamal e al. [33]u ilized he ex ended di ec algeb aic echnique o in es iga e he No iko -Veselo equa ion o iden i ying soli on solu ions. The au ho s also co e ed bi u ca ion, sensi i i y, mul is abili y, and chaos analysis o he No iko -Veselo equa ion. Samina e al. [34]conduc ed a s udy on he (2+1)-dimensional ellip ic nonlinea Sch ödinge equa ion, ocusing on soli on solu ions, bi u ca ion, chaos, and mul is abili y analysis. Jamal e al. [35]explo ed soli on solu ions, bi u ca ion, chaos examina ion echniques, and sensi i i y analysis wi hin he ne e impulse model. Rafiq e al. [36] applied Lie symme ies analysis o he (3+1)-dimensional Kadom se -Pe iash ili equa ion, in es iga ing soli on solu ions, bi u ca ion analysis, chaos iden ifica ion echniques, and s abili y analysis o he conside ed equa ion. Va ious me hodologies such as phase po ai s, ime se ies in es iga ions, Poinca ’e g aphs, powe spec a, bi u ca ion diag ams, and Lyapuno exponen s a e commonly u ilized in he examina ion o dynamical sys ems and chao ic beha iou [37,38]. These me hodologies find ex ensi e applica ions ac oss di e se academic disciplines, offe ing aluable insigh s in o he s abili y, beha iou al ends, and dynamics o complex sys ems. To achie e i s objec i es, his s udy will p ima ily ocus on ou dis inc pe spec i es. Ini ially, employing he one-pa ame e Lie symme y app oach, we aim o iden i y all po en ial se s o symme ies o ans o ma ions. This me hod aids in ei he educing he complexi y o he sys em unde in es iga ion o ans o ming i in o an o dina y diffe en ial equa ion. Secondly, we will u ilize a modified auxilia y equa ion me hod o asce ain he soli on solu ion. Thi dly, a pa ame e -based dynamical analysis inco po a ing bi u ca ion and chaos heo ies mus be conduc ed o he sys em unde examina ion. Phase diag ams will explain and illus a e he bi u ca ion analysis o he unpe u bed dynamical sys em. Addi ionally, he chao ic examina ion o he pe u bed dynamical s uc u e will be defined and shown using a ange o me hods o de ec ing chao ic beha iou , such as bi u ca ion diag ams, ime se ies, Poinca é maps, phase po ai s, and mul is abili y analysis. We will also alk abou he analysed model’s sensi i i y analysis. Ou unde s anding leads us o belie e ha his s udy adds some hing esh ha hasn’ been co e ed in he co pus o ecen li e a u e. The pape ollows he sugges ed o ma . Using he Lie symme y echnique, Sec ion 2aims o iden i y all possible symme y se s o ans o - ma ions o he model unde in es iga ion. Sec ion 3p esen s a ho ough analysis o he modified auxilia y equa ion me hod o ob aining soli on esul s. The phase pic u es o he bi u ca ion a he dynamical s uc u e’s balancing poin s a e shown and examined in Sec ion 4. A numbe o me h- Resul s in Enginee ing 22 (2024) 102194 3 A. Jhangee , T. Jamal, A.M. Tala ha e al. ods a e used in Sec ion 5 o de e mine he chao ic beha iou o he dis u bed dynamical sys em. In Sec ion 6, he sensi i i y analysis is co e ed. Ul ima ely, he model unde examina ion is concluded, and ou esul s a e alked abou . 2. Applica ion o Lie g oup analysis De e mining he ans o ma ions, o symme ies, which main ain he o iginal diffe en ial equa ion is he ocus o Lie symme y examina ion. These symme ies a e equen ly ob ained om Lie ope a o s, which a e infini esimal ope a o s ha cons i u e a Lie algeb a. The symme ies ha p ese e he diffe en ial equa ion’s s uc u e can be ound by p ac icing hese ope a o s o i . Conside he subsequen one-pa ame e Lie g oup o ans o ma ions o de i e symme ies o equa ion (1): 𝑥 =𝑥+𝜈𝑍1(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)+𝑂(𝜈2), 𝑦 =𝑦+𝜈𝑍2(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)+𝑂(𝜈2), 𝑧 =𝑧+𝜈𝑍3(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)+𝑂(𝜈2),  𝑡=𝑡+𝜈𝑍4(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)+𝑂(𝜈2), 𝑢 =𝑢+𝜈Φ(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)+𝑂(𝜈2). (2) Whe e 𝑍1(𝑥, 𝑦, 𝑧, 𝑡, 𝑢), 𝑍2(𝑥, 𝑦, 𝑧, 𝑡, 𝑢), 𝑍3(𝑥, 𝑦, 𝑧, 𝑡, 𝑢), 𝑍4(𝑥, 𝑦, 𝑧, 𝑡, 𝑢), Φ(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)a e infini esimals. The p e ious equa ion (2)’s infini esimal gene - a o is 𝐾=𝑍1(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)𝜕 𝜕𝑥 +𝑍2(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)𝜕 𝜕𝑦 +𝑍3(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)𝜕 𝜕𝑧 +𝑍4(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)𝜕 𝜕𝑡 +Φ(𝑥, 𝑦, 𝑧, 𝑡, 𝑢)𝜕 𝜕𝑢,(3) and he hi d p olonga ion o mula ha co esponds o i is p o ided by 𝑝𝑟(3)𝐾=𝐾+Φ 𝑡𝜕 𝜕𝑢𝑡 +Φ 𝑥𝜕 𝜕𝑢𝑥 +Φ 𝑥𝑥𝑥 𝜕 𝜕𝑢𝑥𝑥𝑥 +Φ 𝑦𝑦𝑥 𝜕 𝜕𝑢𝑦𝑦𝑥 +Φ 𝑧𝑧𝑥 𝜕 𝜕𝑢𝑧𝑧𝑥 .(4) Equa ion (1)has o mee he equi emen s a ed by he in a iance s anda ds. 𝑝𝑟(3)𝐾(𝑢𝑡+𝑎𝑢2𝑢𝑥+𝑏𝑢𝑥𝑥𝑥 +𝑑(𝑢𝑦𝑦 +𝑢𝑧𝑧)𝑥)∣Eq. (1)=0=0.(5) The subsequen Lie poin symme ies can be ob ained by sol ing equa ion (1)wi h he hi d p olonga ion 𝑝𝑟(3): 𝐾1=𝜕 𝜕𝑦,𝐾 2=𝜕 𝜕𝑡,𝐾 3=𝜕 𝜕𝑥,𝐾 4=𝜕 𝜕𝑧,𝐾 5=𝑧𝜕 𝜕𝑦 −𝑦𝜕 𝜕𝑧, 𝐾6=𝑢𝜕 𝜕𝑢 −3𝑡𝜕 𝜕𝑡 −𝑥𝜕 𝜕𝑥 −𝑦𝜕 𝜕𝑦 −𝑧𝜕 𝜕𝑧. (6) We a e now going o minimise he dimension by ei he one o mo e o equa ion (1)by using he symme y adjus men s p o ided in equa ion (6). •Reduc ion h ough ansla ion symme y 𝐾1=𝜕 𝜕𝑦 . The cha ac e is ic equa ion esul ing om 𝐾1 akes he ollowing o m: 𝑑𝑡 0=𝑑𝑥 0=𝑑𝑦 1=𝑑𝑧 0=𝑑𝑢 0.(7) I p oduces he subsequen ou come: 𝑥=𝜌, 𝑧 =𝜁, 𝑡=𝜏, 𝑢 =𝑔(𝜌, 𝜁, 𝜏).(8) The simplified o m o equa ion (1) can be de i ed by applying he simila i y pa ame e s ou lined in Eq. (8) o Eq. (1). 𝑔𝜏+𝑎𝑔2𝑔𝜌+𝑏𝑔𝜌𝜌𝜌 +𝑑𝑔𝜁𝜁𝜌 =0.(9) This is an addi ional simplified e sion o he examined equa ion in h ee dimensions. •Reduc ion h ough ansla ion symme y 𝐾2=𝜕 𝜕𝑡 . he cha ac e is ic equa ion is ep esen ed as 𝐾2. 𝑑𝑡 1=𝑑𝑥 0=𝑑𝑦 0=𝑑𝑧 0=𝑑𝑢 0.(10) I p oduces he subsequen ou come: 𝑥=𝜌, 𝑦 =𝜁, 𝑧=𝜏, 𝑢 =𝑔(𝜌, 𝜁, 𝜏).(11) The simplified o m o equa ion (1) can be de i ed by applying he simila i y pa ame e s ou lined in Eq. (11) o Eq. (1). 𝑎𝑔2𝑔𝜌+𝑏𝑔𝜌𝜌𝜌 +𝑑𝑔𝜁𝜁𝜌 +𝑑𝑔𝜏𝜏𝜌 =0.(12) This is no hing bu a sho ened dimensional e m o a h ee-dimensional obse a ion equa ion. •Reduc ion h ough ansla ion symme y 𝐾3=𝜕 𝜕𝑥 . Resul s in Enginee ing 22 (2024) 102194 4 A. Jhangee , T. Jamal, A.M. Tala ha e al. The esul o he cha ac e is ic equa ion 𝐾3is o he o m: 𝑑𝑡 0=𝑑𝑥 1=𝑑𝑦 0=𝑑𝑧 0=𝑑𝑢 0.(13) P oduces he ollowing esul : 𝑦=𝜌, 𝑧 =𝜁, 𝑡=𝜏, 𝑢 =𝑔(𝜌, 𝜁, 𝜏).(14) The simplified Eq. (1) can be ob ained by subs i u ing he dimensionless pa ame e s p esen ed in Eq. (14) o Eq. (1). 𝑔𝜏=0.(15) This is ano he o m o he simplified exp ession ha was ea lie ob ained. •Reduc ion h ough ansla ion symme y 𝐾4=𝜕 𝜕𝑧 . The esul ing cha ac e is ic equa ion gene a ed by 𝐾4has he ollowing o m: 𝑑𝑡 0=𝑑𝑥 0=𝑑𝑦 0=𝑑𝑧 1=𝑑𝑢 0.(16) I p oduces he subsequen ou come: 𝑥=𝜌, 𝑦 =𝜁, 𝑡=𝜏, 𝑢 =𝑔(𝜌, 𝜁, 𝜏).(17) He e, we can ge he simplified o m o equa ion (1)by means o applying he simila i y pa ame e s which a e gi en in Eq. (17) o Eq. (1). 𝑔𝜏+𝑎𝑔2𝑔𝜌+𝑏𝑔𝜌𝜌𝜌 +𝑑𝑔𝜁𝜁𝜌 =0.(18) This means ha he gi en equa ion is w i en in he h ee-dimensional amewo k. •Reduc ion h ough ansla ion symme y 𝐾2+𝑐𝐾1=𝜕 𝜕𝑡 +𝑐𝜕 𝜕𝑦 . The esul ing cha ac e is ic equa ion gene a ed by 𝐾2+𝑐𝐾1has he ollowing o m: 𝑑𝑡 1=𝑑𝑥 0=𝑑𝑦 𝑐=𝑑𝑧 0=𝑑𝑢 0.(19) I p oduces he subsequen ou come: 𝑥=𝜌, 𝑧 =𝜁, 𝜏 =𝑦−𝑐𝑡, 𝑢 =𝑔(𝜌, 𝜁, 𝜏).(20) We de i e he simplified e sion o equa ion (1)by employing he simila i y pa ame e s men ioned in Eq. (20) o Eq. (1). −𝑐𝑔𝜏+𝑎𝑔2𝑔𝜌+𝑏𝑔𝜌𝜌𝜌 +𝑑𝑔𝜁𝜁𝜌 +𝑑𝑔𝜏𝜏𝜌 =0.(21) This ep esen s a simple example o he equa ion defined in a h ee-dimensional amewo k. •Reduc ion h ough ansla ion symme y 𝐾2+𝑐𝐾3=𝜕 𝜕𝑡 +𝑐𝜕 𝜕𝑥 . The cha ac e is ic equa ion o 𝐾2+𝑐𝐾3is as ollows: 𝑑𝑡 1=𝑑𝑥 𝑐=𝑑𝑦 0=𝑑𝑧 0=𝑑𝑢 0.(22) I p oduces he subsequen ou come: 𝑦=𝜌, 𝑧 =𝜁, 𝜏 =𝑥−𝑐𝑡, 𝑢 =𝑔(𝜌, 𝜁 , 𝜏).(23) The esul ing simplified o m o equa ion (1) can be ob ained by applying he simila i y pa ame e s gi en in Eq. (23) o Eq. (1). −𝑐𝑔𝜏+𝑎𝑔2𝑔𝜏+𝑏𝑔𝜏𝜏𝜏 +𝑑𝑔𝜁𝜁𝜏 +𝑑𝑔𝜌𝜌𝜏 =0.(24) This is an addi ional simplified e sion o he examined equa ion in h ee dimensions. •Reduc ion h ough ansla ion symme y 𝐾2+𝑐𝐾4=𝜕 𝜕𝑡 +𝑐𝜕 𝜕𝑧 . The o m ha he cha ac e is ic equa ion o 𝐾2+𝑐𝐾𝑧p oduces is as ollows: 𝑑𝑡 1=𝑑𝑥 0=𝑑𝑦 0=𝑑𝑧 𝑐=𝑑𝑢 0.(25) I p oduces he subsequen ou come: 𝑥=𝜁, 𝑦=𝜌, 𝜏 =𝑧−𝑐𝑡, 𝑢 =𝑔(𝜌, 𝜁 , 𝜏).(26) The esul ing simplified o m o equa ion (1) can be ob ained by p ac icing he simila i y pa ame e s gi en in Eq. (26) o Eq. (1). −𝑐𝑔𝜏+𝑎𝑔2𝑔𝜁+𝑏𝑔𝜁𝜁𝜁 +𝑑𝑔𝜌𝜌𝜁 +𝑑𝑔𝜏𝜏𝜁 =0.(27) This is an addi ional simplified e sion o he examined equa ion in h ee dimensions. •T a elling wa e p o o ype h ough abelian algeb a 𝐿 =𝐾1+𝐾2+𝐾3+𝐾4is e iden ly an abelian subalgeb a. He e, we’ll de e mine he a elling wa e solu ion o he conside ed model ha co esponds o he o m’s linea combina ion o ansla ion symme ies 𝑍=( 𝜕 𝜕𝑥 +𝜕 𝜕𝑦 +𝜕 𝜕𝑧)− 1 𝑐 𝜕 𝜕𝑡. Resul s in Enginee ing 22 (2024) 102194 5 A. Jhangee , T. Jamal, A.M. Tala ha e al. Employing he linea combina ion 𝑍o symme ies desc ibed a o esaid, we desc ibe he ans o ma ion ou lined below: 𝑍(𝑥, 𝑦, 𝑧, 𝑡)=𝑈(𝜉),𝜉=𝑥+𝑦+𝑧−𝑐𝑡. (28) U ilizing equa ion (28)in equa ion (1), we ge −𝑐𝑈′+𝑎𝑈2𝑈′+𝑏𝑈′′′ +2𝑑𝑈′′′ =0.(29) 3. Soli on solu ion analysis o he gKdV-ZK model using modified auxilia y equa ion me hod The p ac iced echnique is beneficial o collec he exac ou comes known as soli on ou comes. The homogeneous balance concep is used o figu e ou he deg ee o he ou come unde conside a ion. We ob ain he balance numbe 𝑛 =1by balancing he highes nonlinea and dispe si e ph ase in Eq. (29). Acco ding o he modified auxilia y equa ion me hod, we can p esume he succeeding pa e n o solu ion: 𝑈(𝜉)=𝑎0+ 𝑛 ∑ 𝑖=1 [𝑎𝑖(𝑘𝑓)𝑖+𝑏𝑖(𝑘𝑓)−𝑖)]=𝑎0+𝑎1𝑘𝑓+𝑏1𝑘−𝑓,(30) whe e 𝑓(𝜉)mee s he ollowing auxilia y equa ion and cons an s a e 𝑎𝑖𝑠and 𝑏𝑖𝑠 ha need o be calcula ed. 𝑓′(𝜉)= 𝛽+𝛼𝑘−𝑓+𝜎𝑘𝑓 ln 𝑘.(31) Whe e 𝜎, 𝛽, 𝛼and 𝑘a e unspecified cons an s ha ing 𝑘 >0, 𝑘 ≠1. The esul s o Eq. (31)a e as ollows: ∙I 𝛽2−4𝜎𝛼 < 0and 𝜎≠0, 𝑘𝑓(𝜉)= −𝛽+√4𝜎𝛼−𝛽2 an(√4𝜎𝛼−𝛽2𝜉 2) 2𝜎o 𝑘𝑓(𝜉)=− 𝛽+√4𝜎𝛼−𝛽2co (√4𝜎𝛼−𝛽2𝜉 2) 2𝜎. ∙I 𝛽2−4𝜎𝛼 > 0and 𝜎≠0, 𝑘𝑓(𝜉)=− 𝛽+√𝛽2−4𝜎𝛼 anh(√𝛽2−4𝜎𝛼𝜉 2) 2𝜎o 𝑘𝑓(𝜉)=− 𝛽+√𝛽2−4𝜎𝛼 co h (√𝛽2−4𝜎𝛼𝜉 2) 2𝜎. ∙I 𝛽2−4𝜎𝛼 =0and 𝜎≠0, 𝑘𝑓(𝜉)=−2+𝛽𝜉 2𝜎𝜉 . Using Eq. (30) oge he wi h Eq. (31)in Eq. (29)and he se o algeb aic equa ions appea s as ollows when all he ac o s o he a ious powe s o 𝑘𝑓a e equalled o ze o, as (𝑘𝑓)−4 ∶𝑎𝑎3 1𝜎+6𝑎1𝑏𝜎3+ 12𝑎1𝑑𝜎3=0, (𝑘𝑓)−3 ∶2𝑎𝑎0𝑎2 1𝜎+𝑎𝑎3 1𝛽+ 12𝑎1𝑏𝛽𝜎2+ 24𝑎1𝛽𝑑𝜎2=0, (𝑘𝑓)−2 ∶𝑎𝑎2 0𝑎1𝜎+2𝑎𝑎0𝑎2 1𝛽+𝑎𝑎3 1𝛼+𝑎𝑎2 1𝑏1𝜎+8𝑎1𝛼𝑏𝜎2+ 16𝑎1𝛼𝑑𝜎2+7𝑎1𝑏𝛽2𝜎+ 14𝑎1𝛽2𝑑𝜎 −𝑎1𝑐𝜎 =0, (𝑘𝑓)−1 ∶𝑎𝑎2 0𝑎1𝛽+2𝑎𝑎0𝑎2 1𝛼+𝑎𝑎2 1𝑏1𝛽+8𝑎1𝛼𝑏𝛽𝜎 + 16𝑎1𝛼𝛽𝑑𝜎 +𝑎1𝑏𝛽3+2𝑎1𝛽3𝑑−𝑎1𝛽𝑐 =0, (𝑘𝑓)0∶𝑎𝑎2 0𝑎1𝛼−𝑎𝑎2 0𝑏1𝜎+𝑎𝑎2 1𝛼𝑏1−𝑎1𝑏2 1𝜎+2𝑎1𝛼2𝑏𝜎 +4𝑎1𝛼2𝑑𝜎 +𝑎1𝛼𝑏𝛽2+2𝑎1𝛼𝛽2𝑑−2𝛼𝑏𝑏1𝜎2− 4𝛼𝑏1𝑑𝜎2−𝑏𝑏1𝛽2𝜎−2𝑏1𝛽2𝑑𝜎 −𝑎1𝛼𝑐 +𝑏1𝑐𝜎 =0, (𝑘𝑓)1∶−𝑎𝑎2 0𝑏1𝛽−2𝑎𝑎0𝑏2 1𝜎−𝑎𝑎1𝑏2 1𝛽−8𝛼𝑏𝑏1𝛽𝜎 − 16𝛼𝑏1𝛽𝑑𝜎 −𝑏𝑏1𝛽3−2𝑏1𝛽3𝑑+𝑏1𝛽𝑐 =0, (𝑘𝑓)2∶−𝑎𝑎2 0𝛼𝑏1−2𝑎𝑎0𝑏2 1𝛽−𝑎𝑎1𝛼𝑏2 1−𝑎𝑏3 1𝜎−8𝛼2𝑏𝑏1𝜎− 16𝛼2𝑏1𝑑𝜎 −7𝛼𝑏𝑏1𝛽2− 14𝛼𝑏1𝛽2𝑑+𝛼𝑏1𝑐=0, (𝑘𝑓)3∶−2𝑎𝑎0𝛼𝑏2 1−𝑎𝑏3 1𝛽− 12𝛼2𝑏𝑏1𝛽− 24𝛼2𝑏1𝛽𝑑 =0, (𝑘𝑓)4∶−𝑎𝛼𝑏3 1−6𝛼3𝑏𝑏1− 12𝛼3𝑏1𝑑=0. The abo e sys em o algeb aic equa ions has he ollowing solu ions: 𝐅𝐚𝐦𝐢𝐥𝐲𝟏 ∶[𝑎0=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ,𝑎 1=0,𝑏 1=± √−6𝑏+ 12𝑑 𝑎𝛼,𝑐 =2𝛼𝑏𝜎 +4𝑑𝛼𝜎 −1 2𝑏𝛽2−𝑑𝛽2], 𝐅𝐚𝐦𝐢𝐥𝐲𝟐 ∶[𝑎0=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ,𝑎 1=± √−6𝑏+ 12𝑑 𝑎𝛼,𝑏1=0,𝑐 =2𝛼𝑏𝜎 +4𝑑𝛼𝜎 −1 2𝑏𝛽2−𝑑𝛽2], 𝐅𝐚𝐦𝐢𝐥𝐲𝟑 ∶[𝑎0=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ,𝑎 1=± √−6𝑏+ 12𝑑 𝑎𝜎,𝑏1=± 6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ,𝑐=−4𝛼𝑏𝜎 −8𝑑𝛼𝜎 −1 2𝑏𝛽2−𝑑𝛽2]. (32) Family 1: Fo 𝛽2−4𝜎𝛼 < 0and 𝜎≠0bes ow, 𝑈1,1=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[−𝛽+√4𝜎𝛼 −𝛽2 an(√4𝜎𝛼−𝛽2𝜉 2) 2𝜎]−1 ,(33) o Resul s in Enginee ing 22 (2024) 102194 6 A. Jhangee , T. Jamal, A.M. Tala ha e al. 𝑈1,2=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√4𝜎𝛼 −𝛽2co (√4𝜎𝛼−𝛽2𝜉 2) 2𝜎]−1 .(34) Fo 𝛽2−4𝜎𝛼 > 0and 𝜎≠0bes ow, 𝑈1,3=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√𝛽2−4𝜎𝛼 anh(√𝛽2−4𝜎𝛼𝜉 2) 2𝜎]−1 ,(35) o 𝑈1,4=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√𝛽2−4𝜎𝛼 co h (√𝛽2−4𝜎𝛼𝜉 2) 2𝜎]−1 .(36) Fo 𝛽2−4𝜎𝛼 =0and 𝜎≠0p o ide, 𝑈1,5=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[−2+𝛽𝜉 2𝜎𝜉 ]−1 .(37) Family 2: Fo 𝛽2−4𝜎𝛼 < 0and 𝜎≠0bes ow, 𝑈2,1=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[−𝛽+√4𝜎𝛼 −𝛽2 an(√4𝜎𝛼−𝛽2𝜉 2) 2𝜎],(38) o 𝑈2,2=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√4𝜎𝛼 −𝛽2co (√4𝜎𝛼−𝛽2𝜉 2) 2𝜎].(39) Fo 𝛽2−4𝜎𝛼 > 0and 𝜎≠0bes ow, 𝑈2,3=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√𝛽2−4𝜎𝛼 anh(√𝛽2−4𝜎𝛼𝜉 2) 2𝜎],(40) o 𝑈2,4=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[− 𝛽+√𝛽2−4𝜎𝛼 co h (√𝛽2−4𝜎𝛼𝜉 2) 2𝜎].(41) Fo 𝛽2−4𝜎𝛼 =0and 𝜎≠0p o ide, 𝑈2,5=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝛼[−2+𝛽𝜉 2𝜎𝜉 ].(42) Family 3: Fo 𝛽2−4𝜎𝛼 < 0and 𝜎≠0bes ow, 𝑈3,1=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+ 12𝑑 𝑎𝜎[−𝛽+√4𝜎𝛼 −𝛽2 an(√4𝜎𝛼−𝛽2𝜉 2) 2𝜎] ±6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎[−𝛽+√4𝜎𝛼 −𝛽2 an(√4𝜎𝛼−𝛽2𝜉 2) 2𝜎]−1 , (43) o Resul s in Enginee ing 22 (2024) 102194 7 A. Jhangee , T. Jamal, A.M. Tala ha e al. Fig. 1. 2D and 3D pic u es o Eq. (35) 𝑈3,2=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+12𝑑 𝑎𝜎[− 𝛽+√4𝜎𝛼 −𝛽2co (√4𝜎𝛼−𝛽2𝜉 2) 2𝜎] ±6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎[− 𝛽+√4𝜎𝛼 −𝛽2co (√4𝜎𝛼−𝛽2𝜉 2) 2𝜎]−1 . (44) Fo 𝛽2−4𝜎𝛼 > 0and 𝜎≠0bes ow, 𝑈3,3=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+12𝑑 𝑎𝜎[− 𝛽+√𝛽2−4𝜎𝛼 anh(√𝛽2−4𝜎𝛼𝜉 2) 2𝜎] ±6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎[− 𝛽+√𝛽2−4𝜎𝛼 anh(√𝛽2−4𝜎𝛼𝜉 2) 2𝜎]−1 , (45) o 𝑈3,4=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+12𝑑 𝑎𝜎[− 𝛽+√𝛽2−4𝜎𝛼 co h (√𝛽2−4𝜎𝛼𝜉 2) 2𝜎] ±6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎[− 𝛽+√𝛽2−4𝜎𝛼 co h (√𝛽2−4𝜎𝛼𝜉 2) 2𝜎]−1 . (46) Fo 𝛽2−4𝜎𝛼 =0and 𝜎≠0p o ide, 𝑈3,5=± 3𝛽(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎 ±√−6𝑏+12𝑑 𝑎𝜎[−2+𝛽𝜉 2𝜎𝜉 ]±6𝛼(2𝑑+𝑏) 𝑎√−6𝑏+12𝑑 𝑎[−2+𝛽𝜉 2𝜎𝜉 ]−1 .(47) Assigning app op ia e alues o he a bi a y cons an s enables he p edic ion o physical phenomena in he nonlinea model. By employing he modified auxilia y equa ion app oach, a ious solu ions including hype bolic, a ional unc ions, and igonome ic solu ions a e gene a ed. Fu - he mo e, s able a elling wa es wi h dis inc shapes, known as soli ons, a e ob ained, which a e he specific findings o in e es . Fig. 1illus a es he an i-kink soli on solu ion o Eq. (35) co esponding o he pa ame e alues 𝑎 =−1, 𝑏 =2.5, 𝑑=0.5, 𝛼=1 4, 𝜎=1, 𝛽=5, 𝑦 =−3, 𝑧 =−1.7. The figu e showcases he a ia ions in he empo al and eloci y componen s, specifically o 𝑡 =0, 𝑡 =1, 𝑡 =2, 𝑐=0.2, 𝑐=0.6and 𝑐=1, wi hin he in e al (-10,10). The Fig. 2delinea es he b igh da k combo soli on solu ion o Eq. (41) co esponding o he pa ame e alues 𝑎 =−1, 𝑏 =2.5, 𝑑=6, 𝛼=1 4, 𝜎=1, 𝛽=5, 𝑦 =4, 𝑧 =−0.07. The figu e showcases he a ia ions in he empo al and eloci y componen s, specifically o 𝑡 =0, 𝑡 =1, 𝑡 =2, 𝑐=0.2, 𝑐=0.6and 𝑐=1wi hin he in e al (-10,10). The Fig. 3po ays he kink soli on solu ion o Eq. (40) co esponding o he pa ame e alues 𝑎 =−1, 𝑏 =2.5, 𝑑=0.5, 𝛼=1 4, 𝜎=1, 𝛽=5, 𝑦 =4, 𝑧 =−0.07. The figu e showcase he a ia ions in he empo al and eloci y componen s, specially o 𝑡 =0, 𝑡 =1, 𝑡 =2, 𝑐=0.2, 𝑐=0.6and 𝑐=1 wi hin he in e al (-10,10). Resul s in Enginee ing 22 (2024) 102194 8 A. Jhangee , T. Jamal, A.M. Tala ha e al. Fig. 2. 2D and 3D pic u es o Eq. (41) Fig. 3. 2D and 3D pic u es o Eq. (40) 4. Bi u ca ion analysis In his sec ion, he dynamical s uc u e app oach will be used o analyse he model. The e is a fi s in eg al in Eq. (29)o he ype: (𝑏+2𝑑)𝑈′′ +1 3𝑎𝑈3−𝑐𝑈 =0.(48) Eq. (48)will be e alua ed using bi u ca ion heo y. Applying he Galilean ans o ma ion Eq. (48) in o p ac ice so ha i may be shown as a dynamical plane sys em as: {𝑑𝑈 𝑑𝜉 =𝑉, 𝑑𝑉 𝑑𝜉 =𝜓1𝑈−𝜓2𝑈3,(49) whe e 𝜓1=𝑐 𝑏+2𝑑, 𝜓2=𝑎 3(𝑏+2𝑑). Wi hin sys em (49) he e a e ac ually h ee s able poin s, which a e lis ed as ollows: 𝑈1=(0, 0), 𝑈2=(−𝜓1 𝜓2 , 0), 𝑈3=(𝜓1 𝜓2 , 0). The jacobian ma ix o he linea ized s uc u e o a angemen (49)is hen 𝐽(𝑈,𝑉)=||||| 01 𝜓1−3𝜓2𝑈20|||||.(50) We a e all awa e ha he equilib ium poin 𝐸𝑖ac s as a cen e when 𝐽>0, a saddle poin when 𝐽<0, and a cusp poin when 𝐽=0 o e alua ing he phase po ai a equilib ium poin s. To ca ego ise diffe en ou es in a dynamical sys em’s (49)phase diag ams, use he no a ions below. ∙𝑁𝐻𝑇(𝑈, 𝑉) ep esen s nonlinea homoclinic ajec o y, ∙𝑁𝑃𝑇(𝑈, 𝑉) ep esen s nonlinea pe iodic ajec o y, ∙𝑆𝑁𝑃𝑇(𝑈, 𝑉) ep esen s supe nonlinea pe iodic ajec o y. This segmen includes a g aphic illus a ion o he de i a ion, whe e 𝑉depic s dis inc laye s enclosed by a ajec o y and 𝑈indica es s able poin s. Each phase ajec o y is a closed, non sel in e sec ing cu e on he phase plane. Dynamical sys em phase diag am is a subse o hese laye ed phase di ec ions. Resul s in Enginee ing 22 (2024) 102194 9 A. Jhangee , T. Jamal, A.M. Tala ha e al. Fig. 4. Phase Po ai o 𝜓1<0,𝜓2>0 Fig. 5. Phase Po ai o 𝜓1>0,𝜓2<0 4.1. Case 1 The phase po ai o he dis inc pa ame e s op ions below is depic ed in Fig. 4: ∙𝜓1<0, 𝜓2>0, he e a e h ee equilib ium poin s in he sys em (49) desc ibed abo e: 𝑈1, 𝑈2, 𝑈3. Fo his 𝐽𝑖(𝑈1) >0, 𝐽𝑖(𝑈2) >0and 𝐽𝑖(𝑈3) >0 espec i ely. He e a e h ee s able poin s, 𝑈1is inco po a ed in he (49), p oducing 𝑈1 he cen e. The phase pic u e o he scena io is shown in Fig. 4, which demons a es ha a amily o 𝑁𝑃𝑇(1, 0) con ains 𝑈1. 4.2. Case 2 The phase po ai o he dis inc pa ame e s op ions below is depic ed in Fig. 5: ∙𝜓1>0, 𝜓2<0, he e a e h ee equilib ium poin s in he sys em (49) desc ibed abo e: 𝑈1, 𝑈2, 𝑈3wi h 𝑈1a saddle poin . Fo his 𝐽𝑖(𝑈1) <0, 𝐽𝑖(𝑈2) <0and 𝐽𝑖(𝑈3) <0 espec i ely. The phase pic u e o he scena io is shown in Fig. 5, which demons a es ha 𝑈1is a saddle poin . 4.3. Case 3 The phase po ai o he dis inc pa ame e s op ions below is depic ed in Fig. 6: ∙𝜓1>0, 𝜓2>0, he e a e h ee equilib ium poin s in he sys em (49) desc ibed abo e: 𝑈1, 𝑈2, 𝑈3. 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