Unveiling solitons and dynamic patterns for a (3+1)-dimensional model describing nonlinear wave motion

https://www.aimspress.com/journal/Math AIMS Mathematics, 9(8): 20390–20412. DOI: 10.3934/math.2024992 Received: 17 March 2024 Revised: 29 April 2024 Accepted: 07 May 2024 Published: 24 June 2024 Research article Unveiling solitons and dynamic patterns for a (3+1)-dimensional model describing nonlinear wave motion Muhammad Bilal Riaz1,2,*, Syeda Sarwat Kazmi1, Adil Jhangeer1,3and Jan Martinovic1 1IT4 Innovations, VSB–Technical University of Ostrava, Ostrava, Czech Republic 2Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon 3Department of Mathematics, Namal University, Talagang Road, Mianwali 42250, Pakistan *Correspondence: Email: [email protected]. Abstract: In this study, the underlying traits of the new wave equation in extended (3+1) dimensions, utilized in the field of plasma physics and fluids to comprehend nonlinear wave scenarios in various physical systems, were explored. Furthermore, this investigation enhanced comprehension of the characteristics of nonlinear waves present in seas and oceans. The analytical solutions of models under consideration were retrieved using the sub-equation approach and Sardar sub-equation approach. A diverse range of solitons, including bright, dark, combined dark-bright, and periodic singular solitons, was made available through the proposed methods. These solutions were illustrated through visual depictions utilizing 2D, 3D, and density plots with carefully chosen parameters. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation analysis was conducted via phase portraits at critical points, revealing the system’s transition dynamics. Introducing an external periodic force induced chaotic phenomena in the dynamical system, which were visualized through time plots, twodimensional plots, three-dimensional plots, and the presentation of Lyapunov exponents. Furthermore, the sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. The obtained findings indicated the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems. Keywords: new integrable wave equation; soliton solutions; sub-equation method; Sardar sub-equation method; bifurcation; chaotic behavior; sensitivity Mathematics Subject Classification: 34H10, 35C08 20391 1. Introduction In light of its growing significance, researchers have placed a great deal of emphasis on studying nonlinear partial differential equations (NPDEs). Multiple scientific areas, including hydrodynamics, fluids, engineering, and other domains, have made use of these nonlinear equations [1–3]. Many solitary waves associated with NPDEs have been identified in the search for exact solutions, especially in disciplines such as nonlinear optics, quantum physics, plasma, and many others [4–6]. Scholars have centered on solving these NPDEs using different analytical approaches. Among the notable approaches are the sub-equation approach [7], Painlev´ e test [8], bilinear approach [9], auxiliary equation method [10], modified auxiliary equation method [11,12], and Lie symmetry approach [13]. The solitons which usually underpin the telecommunication sector are now of special interests. Solitons also matter significantly as they are pivotal for the advancement of computer systems’ computing power while they present wide variety of applications. Such applications cover image processing, data analysis, neurology and fluids, among other fields [14, 15]. The dynamics of the solitons have been studied in great deal using a wide range of nonlinear equations including the Fokas-Lenells equation [16], Manakov model [17], Sakovich model [18], Born-Infeld equation [19], Schr¨ odinger equation [20], complex short pulse equation [21], and the Wadati-Konno-Ichikawa equation [22,23]. The analysis of integrable equations in (3+1) dimensions has been getting greater exposure recently. These equations are vital for deciphering the physics behind a number of industrial and scientific phenomena. This rising demand has led to the development of multiple nonlinear extended equations, among which are the modified Kadomtsev-Petviashvili (KP) equation and the Korteweg-de Vries (KdV) equation. Akinyemi [24] examined the equation in (2+1) dimensions: AΩxt +aΩxx +b(Ω2)xx +cΩxxxx +dΩyy =0,(1.1) and its extended version, comprising two additional linear terms. AΩxt +aΩxx +b(Ω2)xx +cΩxxxx +dΩyy +eΩty +hΩtt =0,(1.2) here, the constants A,a,b,c,d,e, and hin the given context are unrestricted real values. It is worth highlighting that when A=e=d=0, Eq (1.2) simplifies to the Boussinesq equation. Ωtt +aΩxx +b(Ω2)xx +cΩxxxx =0.(1.3) Equations (1.1) and (1.2) have been demonstrated to exhibit Painlev´ e integrability, and their multiple solitons have also been obtained [24]. In the ongoing study, our intention is to address an extended version of Eq (1.2) [25]. Ωxt +aΩxx +b(Ω2)xx +cΩxxxx +dΩyy +eΩty +hΩtt +kΩxy +mΩxz =0,(1.4) where Ω = Ω(x,y,z,t). It is evident that Eq (1.4) is constructed by introducing two additional linear terms, specifically kΩxy, and mΩxz to Eq (1.2). Furthermore, the coefficients a,b,c,d,e,h,kand m are random real values. Wazwaz et al. [25] investigated the integrability criteria for the discussed AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20392 model (1.4) through the use of the Painlev´ e test. The study concluded by analyzing a set of lump solutions for the suggested equation. The suggested model is designed to aid numerous researchers engaged in plasma physics and fluid mechanics by providing insights into the characteristics of nonlinear waves occurring in various physical systems. Furthermore, this investigation will enhance comprehension of the characteristics of nonlinear waves present in seas and oceans. In this study, we undertake a comprehensive exploration of a nonlinear model (1.4), employing two distinct but complementary methodologies: the sub-equation (SE) approach [7], and the Sardar sub-equation (SSE) approach [26]. The SE approach yields results expressed in trigonometric and hyperbolic functions. To deepen our understanding of the equation, we introduce the SSE approach alongside the SE approach. The SSE approach, an extension of the SE approach, emerges as an effective analytical tool for extracting precise solutions from nonlinear models. By employing the SSE technique, bright, dark, dark-bright (combo), and periodic solitons are discovered and visually portrayed using density plots, two dimensional and three dimensional plots. It is notable that Eq (1.4) has never been addressed using the techniques outlined in this paper. The results of this study expand theoretical comprehension in the domain of NPDEs. A rising tide of attention has been devoted in recent years to the analysis of differential equation (DE) dynamics via the facets of chaos and bifurcation. Dynamical systems find wide-ranging applications in several domains, including economics, engineering, and biology [27, 28]. Bifurcation analysis studies how an orbit evolves with respect to distinct parameters. Moreover, there has been a noticeable emphasis on studying NPDEs when an outside periodic disturbance is involved. This focus has led due to the realization that a fully integrable nonlinear model falls short in clarifying quasiperiodic and chaotic features. On the contrary, these irregular patterns may be elicited by applying an outward periodic disturbance to a nonlinear system. For instance, in their work, Riaz et al. [28] investigated the dynamics of bifurcation, chaos, and solitons for the oskolkov equation. Similarly, Rafiq et al. [29] studied the dynamic nature of shallow equations, and extracted the multi-wave solitons. Furthermore, the study conducted by Hosseini et al. [30] concentrated on examining the sensitive and dynamic features of the Schr¨ odinger equation. In this work, we studied the chaotic phenomena of the discussed equation through the presentation of two-dimensional plots, three-dimensional plots, time plots, and Lyapunov exponents. •Phase plots offer a graphic depiction of a planar system’s characteristics. They involve plotting one state variable against another. Examining the structure of the resultant graph can furnish valuable insights into the system’s dynamics, encompassing aspects such as periodicity, or inclination towards chaos. •A time plot refers to a set of points that are systematically accumulated over a specific time span. In this method, the state variables of system undergo scrutiny, and if they exhibit irregular patterns, they are classified as chaotic. Conversely, if these variables manifest periodicity, or quasi-periodic tendencies, they are categorized as non-chaotic. •Lyapunov exponents (LE) measure how a dynamic system reacts to changes in its initial conditions, assessing the extent to which nearby trajectories either diverge or converge as time progresses. Chaotic behavior is indicated by positive Lyapunov exponents, whereas negative exponents signify stability. Through the computation of LE, we can ascertain whether a system demonstrates chaotic feature or maintains a stable state. These principles are widely applied in analyzing intricate systems across diverse fields, offering AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20393 valuable insights into their dynamics, stability, and behavioral patterns. This research concentrates on exploring a newly extended equation within a (3+1)-dimensional framework, examining it from various perspectives. We employ the sub-equation and Sardar sub-equation techniques to derive analytical solutions, investigate bifurcations at equilibrium points, utilize various chaos detection methods to pinpoint chaotic behavior, and explore the sensitivity of the discussed model. Based on our comprehension, this study seems to be an innovative contribution not previously encountered in the available literature. The structure of the manuscript is as follows: In Section 2, the description of SE method and SSE method has been given. The mathematical analysis and the application of SE and SSE approaches for finding soliton solutions of the model is presented in Section 3. Results obtained from the analysis are discussed in Section 4. A comprehensive investigation of the dynamic characteristics of the proposed equation, utilizing phase portraits of bifurcation, is conducted in Section 5. Chaotic phenomena is explored in Section 6. In Section 7, sensitivity of the suggested equation is presented. Finally, the conclusion is provided in the last section. 2. Description of proposed methods In this section, two analytical techniques are examined: the SSE approach and the SE approach. The detail description of the proposed techniques is given in this segment. Step 1. Let us consider the NPDE as follows: P(Ω,Ωt,Ωx,Ωy,Ωz,Ωxx,Ωtt, ...)=0.(2.1) Then, using the transformation Ω(x,y,z,t)= Ψ(γ1x+γ2y+γ3z+µt), Eq (2.1) is changed into an ordinary DE as shown: E=(Ψ,Ψ0,Ψ00, ...)=0.(2.2) Step 2. The initial solution for these approaches is outlined below: Ψ(ζ)= j X i=0 σiΛi(ζ),(2.3) here, σi,(i=0,1,2,3, ..., j) are random constants to be resolved. 2.1. Description of the SE approach In this method, the function Λi(ζ) fulfils the following auxiliary equation, Λ0(ζ)=β+δΛ2(ζ), β, δ ∈R,(2.4) here, βand δare random constants to be assessed afterward. Equation (2.4) has the following solutions: Case 1: If χ=β δ<0, then, Λ1(ζ)=−√−χtanh(δ√−χζ), Λ2(ζ)=−√−χcoth(δ√−χζ), Λ3(ζ)=−√−χtanh(2δ√−χζ)±ι√−χsech(2δ√−χζ). (2.5) AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20394 Case 2: If χ=β δ>0, then, Λ4(ζ)=√χtan(δ√χζ), Λ5(ζ)=√χcot(δ√χζ), Λ6(ζ)=√χtan(2δ√χζ)±ι√χsec(2δ√χζ). (2.6) Case 3: If χ=β δ=0, then, Λ7(ζ)=−δ ζ+ζ0 , ζ0∈R.(2.7) 2.2. Description of the SSE approach In this method, the function Λi(ζ) fulfils the following auxiliary equation, Λ0(ζ)=pδ+αΛ(ζ)2+λΛ(ζ)4,(2.8) here, δ,αand λare constants and Eq (2.8) presents solution as: Case 1: If α > 0 and δ=0, then        Λ± 1(ζ)=±p−mnα λsechmn(√αζ),(λ < 0), Λ± 2(ζ)=±pmnα λcschmn(√αζ),(λ > 0).(2.9) Here, sechmn(ζ)=2 meζ+ne−ζ, csch(ζ)=2 meζ−ne−ζ. Case 2: If α < 0, λ > 0, and δ=0, then        Λ± 3(ζ)=±p−mnα λsecmn(√−αζ), Λ± 4(ζ)=±p−mnα λcscmn(√−αζ).(2.10) Here, secmn(ζ)=2 meιζ +ne−ιζ , csc(ζ)=2 meιζ −ne−ιζ . Case 3: If α < 0, λ > 0 and δ=α2 4λ, then                                    Λ± 5(ζ)=±q−α 2λtanhmn(q−α 2ζ), Λ± 6(ζ)=±q−α 2λcothmn(q−α 2ζ), Λ± 7(ζ)=±q−α 2λtanhmn(√−2αζ)±ι√mn sechmn(√−2αζ), Λ± 8(ζ)=±q−α 2λcothmn(√−2αζ)±√mn cschmn(√−2αζ), Λ± 9(ζ)=±q−α 8λtanhmn(q−α 8ζ)+cothmn(q−α 8ζ). (2.11) Here, tanhmn(ζ)=meζ−ne−ζ meζ+ne−ζ, coth(ζ)=meζ+ne−ζ meζ−ne−ζ. Case 4: If α > 0, λ > 0 and δ=α2 4λ, then                            Λ± 10(ζ)=±pα 2λtanmn(pα 2ζ), Λ± 11(ζ)=±pα 2λcotmn(pα 2ζ), Λ± 12(ζ)=±pα 2λtanmn(√2αζ)±√mn secmn(√2αζ), Λ± 13(ζ)=±pα 2λcotmn(√2αζ)±√mn cscmn(√2αζ), Λ± 14(ζ)=±pα 8λtanmn(pα 8ζ)+cotmn(pα 8ζ). (2.12) AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20395 Here, tanmn(ζ)=−ιmeιζ −ne−ιζ meιζ +ne−ιζ , cot(ζ)=ιmeιζ +ne−ιζ meιζ −ne−ιζ with parameters mand n. Upon m=n=1, they become known trigonometric and hyperbolic functions. Step 3. The positive integer jis determined by homogeneous balance approach. Step 4. A system of equations for Λ0 isare generated by substituting Eq (2.4) from SE approach and Eq (2.8) from SSE approach into Eq (2.2) and reducing the coefficients of Λito zero. To determine the solution of Eq (2.2), we then solve the resulting system using tools like Mathematica. 3. Mathematical analysis The extended wave equation in (3+1) dimensions can be expressed as: Ωxt +aΩxx +b(Ω2)xx +cΩxxxx +dΩyy +eΩty +hΩtt +kΩxy +mΩxz =0.(3.1) By assuming the traveling wave transformation as: Ω(x,y,z,t)= Ψ(ζ), ζ =γ1x+γ2y+γ3z+µt.(3.2) Here, Ψ(ζ) and µrepresent the characteristics of the traveling wave, specifically referring to its shape and velocity. Additionally, γ1, γ2, γ3serve as random parameters. Upon substituting the expression from Eq (3.2) into Eq (3.1), we obtain the resulting equation. (cγ4 1)Ψ(4) +(γ1µ+aγ2 1+dγ2 2+eµγ2+hµ2+kγ1γ2+mγ1γ3)Ψ00 +2bγ2 1ΨΨ00 +2bγ2 1(Ψ0)2=0.(3.3) By integrating Eq (3.3) twice with respect to ζ, we acquire; (cγ4 1)Ψ00 +(γ1µ+aγ2 1+dγ2 2+eµγ2+hµ2+kγ1γ2+mγ1γ3)Ψ + bγ2 1Ψ2=0.(3.4) This section is focused on the application of SSE approach and SE approach for extraction of analytical solutions. By setting Ψ00 equal to Ψ2in Eq (3.4), as j+2=2j, results j=2. 3.1. Application of the SE approach This section is focused on the application of SE approach for extraction of analytical solutions. The initial solution in this case becomes: Ψ(ζ)=σ0+σ1Λ + σ2Λ2.(3.5) Using Eqs (3.5) and (2.4) into Eq (3.4) and solving the resulting system for σ0,σ1,σ2, and a, yields the following solution: σ0=−2cβδγ2 1 b, σ1=0, σ2=−6cδ2γ2 1 b, a=−dγ2 2−mγ1γ3−kγ1γ2−4cβδγ4 1−µγ1−eµγ2−hµ2 γ2 1 .(3.6) By putting above values in Eq (3.5) the solutions of Eq (1.4) are as follows: AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20396 Case 1: If χ=β δ<0, then, Ω1(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b−√−χtanh(δ√−χζ)2 , Ω2(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b−√−χcoth(δ√−χζ)2 , Ω3(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b−√−χtanh(2δ√−χζ)±ι√−χsech(2δ√−χζ)2 . (3.7) Case 2: If χ=β δ>0, then, Ω4(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b√χtan(δ√χζ)2 , Ω5(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b√χcot(δ√χζ)2 , Ω6(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b√χtan(2δ√χζ)±ι√χsec(2δ√χζ)2 . (3.8) Case 3: If χ=β δ=0, then, Ω7(x,y,z,t)=−2cβδγ2 1 b−6cδ2γ2 1 b−δ ζ+ζ02 , ζ0∈R.(3.9) In all above cases ζ=γ1x+γ2y+γ3z+µt. 3.2. Application of SSE approach In this part, we employ the SSE method to solve Eq (3.4). The initial solution for j=2 becomes: Ψ(ζ)=σ0+σ1Λ + σ2Λ2,(3.10) where σ0,σ1, and σ2are constants to be extracted. Using Eqs (3.10) and (2.8) into Eq (3.4) and solving the resulting system for σ0,σ1,σ2, and a, yields the following solution: σ0= 2−α+√α2−3δ λγ12c b, σ1=0, σ2=−6cλ γ12 b, a=−4cγ14√α2−3δ λ +dγ22+eµ γ2+hµ2+kγ1γ2+mγ3γ1+γ1µ γ12.(3.11) By putting above values in Eq (3.10) the solutions of Eq (1.4) are as follows: Case 1: If α > 0 and δ=0, then              Ω± 1(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 b±p−mnα λsechmn(√αζ)2 ,(λ < 0), Ω± 2(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 b±pmnα λcschmn(√αζ)2 ,(λ > 0). (3.12) AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20397 Case 2: If α < 0, λ > 0, and δ=0, then              Ω± 3(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 b±p−mnα λsecmn(√−αζ)2 , Ω± 4(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 b±p−mnα λcscmn(√−αζ)2 . (3.13) Case 3: If α < 0, λ > 0 and δ=α2 4λ, then                                          Ω± 5(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bq−α 2λtanhmn(q−α 2ζ)2 , Ω± 6(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bq−α 2λcothmn(q−α 2ζ)2 , Ω± 7(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bq−α 2λtanhmn(√−2αζ)±ι√mn sechmn(√−2αζ)2 , Ω± 8(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bq−α 2λcothmn(√−2αζ)±√mn cschmn(√−2αζ)2 , Ω± 9(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 b±q−α 8λtanhmn(q−α 8ζ)+cothmn(q−α 8ζ)2 . (3.14) Case 4: If α > 0, λ > 0 and δ=α2 4λ, then                                          Ω± 10(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bpα 2λtanmn(pα 2ζ)2 , Ω± 11(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bpα 2λcotmn(pα 2ζ)2 , Ω± 12(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bpα 2λtanmn(√2αζ)±√mn secmn(√2αζ)2 , Ω± 13(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bpα 2λcotmn(√2αζ)±√mn cscmn(√2αζ)2 , Ω± 14(x,y,z,t)=2−α+√α2−3δ λγ12c b−6cλ γ12 bpα 8λtanmn(pα 8ζ)+cotmn(pα 8ζ)2 . (3.15) In all above cases ζ=γ1x+γ2y+γ3z+µt. 4. Results and discussions In this part, we analyze the attributes of a varied set of acquired solutions. The first portion examines the solutions to the proposed equation utilizing the SE method, whereas the subsequent section investigates solutions to the suggested equation employing the SSE method. The visualization of the results was conducted through three-dimensional, density, and two-dimensional plots. It is important to highlight that within the suggested model, the velocity wave, µ, exhibits two unique values, signifying the existence of dual-wave propagation within the nonlinear system, known as the left-wave and right-wave, both propagating simultaneously. 4.1. Solutions by sub-equation approach Initially, an array of soliton structures is generated by employing distinct parameter values through the use of the SE technique. Furthermore, employing suitable parameter values has led to the AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20398 observation of various distinct structures, including bright solitons, dark solitons, dark-bright(combo) solitons, and singular periodic solitons. In Figure 1, the solution Ω1(x,y,z,t) is visually represented as a bright soliton, where the central region exhibits higher intensity while the surrounding area displays lower intensity. The parameter values for this representation are as follows: γ1=γ2=γ3=b=1, c=−1, β=0.1, δ=−0.2, z=1.5, t=2.1, and µ=1.5. Moving to Figure 2, a 3D, density, and 2D visualization of Ω1(x,y,z,t) is presented with a modified parameter c=1, while all other parameters remain the same as in the previous case, resulting in a recorded dark soliton identified by a localized depression in the surrounding field. Dark solitons, which attract significant attention in optics due to their stable transmission, are highlighted. Shifting to Figure 3, the solution |Ω3(x,y,z,t)| is depicted as a dark-bright (combo) soliton, along with its 2D and density plots. Dark-bright solitons combine both dark and bright features within their structure, featuring a localized region of decreased intensity (dark soliton) embedded within a localized region of increased intensity (bright soliton). The parameter values for this scenario are: γ1=γ2=γ3=b=c=1, β=0.1, δ=−2, y=0.5, t=0.1, and µ=1.5. In Figure 4, the solution |Ω6(x,y,z,t)|is presented with the following parameters: γ1=γ2=γ3=b=c=1, β=2.1, δ=0.2, y=0.5, t=0.1, resulting in singular periodic solitons displaying discontinuity at the lower ends. (a) 3D Plot (b) 2D Plot (c) Density Plot Figure 1. Visual representation of the solution Ω1(x,y,z,t) through 3-dimensional, 2dimensional and density plots. (a) 3D Plot (b) 2D Plot (c) Density Plot Figure 2. Visual representation of the solution Ω1(x,y,z,t) through 3-dimensional, 2dimensional and density plots. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20405 (a) κ0=0.02, θ=0.03 (b) κ0=1.3, θ=π (c) κ0=3.7, θ=2π(d) κ0=8.4, θ=3π Figure 12. Detection of chaotic phenomena in the perturbed system (6.1) via time plots. (a) κ0=0.02, θ=0.03 (b) κ0=1.3, θ=π (c) κ0=3.7, θ=2π(d) κ0=8.4, θ=3π Figure 13. Detection of chaotic phenomena in the perturbed system (6.1) via two dimensional plots. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20406 (a) κ0=0.02, θ=0.03 (b) κ0=1.3, θ=π (c) κ0=3.7, θ=2π(d) κ0=8.4, θ=3π Figure 14. Detection of chaotic phenomena in the perturbed system (6.1) via three dimensional plots. Figure 15. Detection of chaos in the system (6.1) via Lyapunov exponent with initial condition (0.05, 0.05, 0.05). $0=2.5, $1=1, κ0=3.7, θ =2π. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20407 Figure 16. Detection of chaos in the system (6.1) via Lyapunov exponent with initial condition (0.1, 0.1, 0.1). $0=2.5, $1=1, κ0=3.7, θ =2π. Figure 17. Detection of chaotic phenomena in the perturbed system (6.1) via Lyapunov exponent with initial condition (0.3, 0.3, 0.3). $0=2.5, $1=1, κ0=8.4, θ =3π. Figure 18. Detection of chaotic phenomena in the perturbed system (6.1) via Lyapunov exponent with initial condition (0.5, 0.5, 0.5). $0=2.5, $1=1, κ0=8.4, θ =3π. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20408 7. Sensitivity analysis In this section, we explore the response of the suggested equation to variations in initial conditions. To assess the model’s sensitivity, we examine four distinct sets of initial conditions. Figure 19 illustrates four solutions: (Ψ,Γ)=(0.01,0.01) in green, (Ψ,Γ)=(0.03,0.01) in red, (Ψ,Γ)= (0.05,0.01) in black, and (Ψ,Γ)=(0.07,0.01) in blue. Additionally, Figure 20 displays four solutions: (Ψ,Γ)=(0.01,0.01) in green, (Ψ,Γ)=(0.06,0.06) in red, (Ψ,Γ)=(0.2,0.2) in black, and (Ψ,Γ)=(0.3,0.3) in blue. It is apparent that even slight variations in the initial conditions can lead to subtle shifts in the dynamics of the system (5.1). Alternatively, we can assert that the two solution curves never overlap under any circumstances. Consequently, we infer that the proposed system exhibits sensitivity, though it is not excessively so. Figure 19. Sensitivity analysis across various initial values. Figure 20. Sensitivity analysis across various initial values. 8. Conclusions In brief, we delve into the extended integrable wave equation, a frequently employed concept in plasma physics, fluids, and various scientific fields. Our main concern is to conduct a thorough analysis of this equation, covering diverse facets such as the identification of solitons, examination of bifurcation phenomena, chaos analysis, and an investigation into the sensitivity of the proposed equation. At first, analytical solutions for the model under consideration were obtained using two effective and powerful approaches: the sub-equation approach and the Sardar sub-equation approach. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20409 The suggested approaches offer a wide variety of solitons, such as bright, dark, combined dark-bright, and periodic solitary solitons. Bright and dark solitons are characterized by their distinctive intensity profiles; bright solitons exhibit a peak in intensity, whereas dark solitons display a dip. Additionally, dark-bright solitons combine both dark and bright features within their structure, featuring a localized region of decreased intensity (dark soliton) embedded within a localized region of increased intensity (bright soliton). Periodic soliton solutions demonstrate a repetitive structure. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation has been examined at critical points, and the dynamical system, subjected to an outward periodic force, revealed chaotic phenomena. Chaotic behaviors has been illustrated through time plots, two-dimensional plots, threedimensional plots, and the presentation of Lyapunov exponents, as illustrated in Figures 10)–18. The sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. Future investigations into the extended integrable wave equation using alternative approaches remain a potential avenue for exploration. Thus, substantial research endeavors lie ahead to fully comprehend and explore the capabilities of this model. Such investigations hold promise for unveiling new insights and improving our comprehension of the behavior and characteristics of the discussed equation. Such advancements could potentially pave the way for the development of more precise and effective mathematical models and numerical techniques designed for solving this equation. The obtained findings indicate the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems. Author contributions Conceptualization, M.B.R. and A.J.; methodology, M.B.R. and A.J.; software, J.M. and M.B.R.; validation, S.S.K., M.B.R., and A.J.; formal analysis, M.B.R. and A.J.; investigation, M.B.R. and A.J.; data curation, S.S.K.; writing–original draft preparation, S.S.K.; writing–review and editing, M.B.R., J.M. and A.J.; visualization, M.B.R. and A.J.; supervision, A.J.; project administration, J.M. 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. Conflict of interest The authors declare that they have no conflicts of interest. AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20410 References 1. S. Kumar, A. Kumar, B. Mohan, Evolutionary dynamics of solitary wave profiles and abundant analytical solutions to a (3+1)-dimensional burgers system in ocean physics and hydrodynamics, J. Ocean Eng. Sci.,8(2021), 1–14. https://doi.org/10.1016/j.joes.2021.11.002 2. N. Zobeiry, K. D. Humfeld, A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications, Eng. Appl. Artif. Intel., 101 (2021), 104232. https://doi.org/10.1016/j.engappai.2021.104232 3. S. Kumar, S. Rani, N. Mann, Diverse analytical wave solutions and dynamical behaviors of the new (2+1)-dimensional Sakovich equation emerging in fluid dynamics, Eur. Phys. J. Plus,137 (2022), 1226. https://doi.org/10.1140/epjp/s13360-022-03397-w 4. V. Jadaun, N. R. Singh, S. Singh, R. Shankar, Impact of solitons on the progression of initial lesion in aortic dissection, Int. J. Biomath.,15 (2022), 2150096. https://doi.org/10.1142/S1793524521500960 5. V. Jadaun, A. Srivastav, A special phenomenon of wave interactions: an application of nonlinear evolution equation in (3+1)-dimension, Commun. Nonlinear Sci. Numer. Simul.,130 (2024), 107733. https://doi.org/10.1016/j.cnsns.2023.107733 6. N. Raza, S. S. Kazmi, Qualitative analysis and stationary optical patterns of nonlinear Schr¨ odinger equation including nonlinear chromatic dispersion, Opt. Quant. Electron.,55 (2023), 718. https://doi.org/10.1007/s11082-023-04978-4 7. S. Duran, B. Karabulut, Nematicons in liquid crystals with Kerr Law by sub-equation method, Alex. Eng. J.,61 (2022), 1695–1700. https://doi.org/10.1016/j.aej.2021.06.077 8. A. M. Wazwaz, Painlev´ e integrability and lump solutions for two extended (3+1)-and (2+1)-dimensional Kadomtsev-Petviashvili equations, Nonlinear Dyn.,111 (2023), 3623–3632. https://doi.org/10.1007/s11071-022-08074-2 9. R. R. Yuan, Y. Shi, S. L. Zhao, J. X. Zhao, The combined KdV-mKdV equation: bilinear approach and rational solutions with free multi-parameters, Results Phys.,55 (2023), 107188. https://doi.org/10.1016/j.rinp.2023.107188 10. S. S. Kazmi, A. Jhangeer, N. Raza, H. I. Alrebdi, A. H. Abdel-Aty, H. Eleuch, The analysis of bifurcation, quasi-periodic and solitons patterns to the new form of the generalized q-deformed Sinh-Gordon equation, Symmetry,15 (2023), 1324. https://doi.org/10.3390/sym15071324 11. G. Akram, I. Zainab, M. Sadaf, A. Bucur, Solitons, one line rogue wave and breather wave solutions of a new extended KP-equation, Results Phys.,55 (2023), 107147. https://doi.org/10.1016/j.rinp.2023.107147 12. M. A. Ullah, K. Rehan, Z. Perveen, M. Sadaf, G. Akram, Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena, Nonlinear Eng.,13 (2024), 20220318. https://doi.org/10.1515/nleng-2022-0318 13. V. Jadaun, Soliton solutions of a (3+1)-dimensional nonlinear evolution equation for modeling the dynamics of ocean waves, Phys. Scripta,96 (2021), 095204. https://doi.org/10.1088/14024896/ac0031 AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20411 14. O. Gonz´ alez-Gaxiola, A. Biswas, L. Moraru, A. A. Alghamdi, Solitons in neurosciences by the Laplace-Adomian decomposition scheme, Mathematics,11 (2023), 1080. https://doi.org/10.3390/math11051080 15. B. Li, J. Zhao, W. Liu, Analysis of interaction between two solitons based on computerized symbolic computation, Optik,206 (2020), 164210. https://doi.org/10.1016/j.ijleo.2020.164210 16. M. Sadaf, S. Arshed, G. Akram, E. Husaain, Dynamical behavior of nonlinear cubic-quartic FokasLenells equation with third and fourth order dispersion in optical pulse propagation, Opt. Quant. Electron.,55 (2023), 1207. https://doi.org/10.1007/s11082-023-05389-1 17. G. Akram, M. Sadaf, S. Arshed, M. Farrukh, Optical soliton solutions of Manakov model arising in the description of wave propagation through optical fibers, Opt. Quant. Electron.,56 (2024), 906. https://doi.org/10.1007/s11082-024-06735-7 18. M. Vivas-Cortez, N. Raza, S. S. Kazmi, Y. Chahlaoui, G. A. Basendwah, A novel investigation of dynamical behavior to describe nonlinear wave motion in (3+1)-dimensions, Results Phys., 55 (2023), 107131. https://doi.org/10.1016/j.rinp.2023.107131 19. S. Kumar, V. Jadaun, Symmetry analysis and some new exact solutions of Born-Infeld equation, Int. J. Geom. Methods Mod. Phys.,15 (2018), 1850183. https://doi.org/10.1142/S0219887818501839 20. Y. Li, S. F. Tian, J. J. Yang, Riemann-Hilbert problem and interactions of solitons in the n-component nonlinear Schr¨ odinger equations, Stud. Appl. Math.,148 (2022), 577–605. https://doi.org/10.1111/sapm.12450 21. Z. Q. Li, S. F. Tian, J. J. Yang, E. Fan, Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions, J. Differ. Equations,329 (2022), 31–88. https://doi.org/10.1016/j.jde.2022.05.003 22. Z. Q. Li, S. F. Tian, J. J. Yang, Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data, Ann. Henri Poincar´e,23 (2022), 2611–2655. https://doi.org/10.1007/s00023-021-01143-z 23. Z. Q. Li, S. F. Tian, J. J. Yang, On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions, Adv. Math.,409 (2022), 108639. https://doi.org/10.1016/j.aim.2022.108639 24. L. Akinyemi, Shallow ocean soliton and localized waves in extended (2+1)- dimensional nonlinear evolution equations, Phys. Lett. A,463 (2023), 128668. https://doi.org/10.1016/j.physleta.2023.128668 25. A. M. Wazwaz, W. Alhejaili, S. A. El-Tantawy, Analytical study on two new (3+1)-dimensional Painlev´ e integrable equations: Kink, lump, and multiple soliton solutions in fluid mediums, Phys. Fluids,35 (2023), 093119. https://doi.org/10.1063/5.0169763 26. N. Ullah, M. I. Asjad, A. Hussanan, A. Akg¨ ul, W. R. Alharbi, H. Algarni, et al., Novel waves structures for two nonlinear partial differential equations arising in the nonlinear optics via Sardarsubequation method, Ale. Eng. J.,71 (2023), 105–113. https://doi.org/10.1016/j.aej.2023.03.023 27. D. Lathrop, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Phys. Today,68 (2015), 54–55. https://doi.org/10.1063/PT.3.2751 AIMS Mathematics Volume 9, Issue 8, 20390–20412. 20412 28. M. B. Riaz, A. Jhangeer, J. Martinovic, S. S. Kazmi, Dynamics and soliton propagation in a modified Oskolkov equation: phase plot insights, Symmetry,15 (2023), 2171. https://doi.org/10.3390/sym15122171 29. M. H. Rafiq, N. Raza, A. Jhangeer, Dynamic study of bifurcation, chaotic behavior and multisoliton profiles for the system of shallow water wave equations with their stability, Chaos Soliton. Fract.,171 (2023), 113436. https://doi.org/10.1016/j.chaos.2023.113436 30. K. Hosseini, E. Hinc¸al, M. Ilie, Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schr¨ odinger equation, Nonlinear Dyn.,111 (2023), 17455– 17462. https://doi.org/10.1007/s11071-023-08759-2 31. A. M. Talafha, A. Jhangeer, S. S. Kazmi, Dynamical analysis of (4+1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation by employing Lie symmetry approach, Ain Shams Eng. J.,14 (2023), 102537. https://doi.org/10.1016/j.asej.2023.102537 32. L. Bai, J. Qi, Y. Sun, Further physical study about solution structures for nonlinear q-deformed Sinh-Gordon equation along with bifurcation and chaotic behaviors, Nonlinear Dyn.,111 (2023), 20165–20199. https://doi.org/10.1007/s11071-023-08882-0 33. L. Yang, M. ur Rahman, M. A. Khan, Complex dynamics, sensitivity analysis and soliton solutions in the (2+1)-dimensional nonlinear Zoomeron model, Results Phys.,56 (2024), 107261. https://doi.org/10.1016/j.rinp.2023.107261 ©2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0) AIMS Mathematics Volume 9, Issue 8, 20390–20412.