Bifurcation Analysis of Travelling Waves and Multi-rogue Wave Solutions for aNonlinearPseudo-ParabolicModel ofVisco-Elastic Kelvin-Voigt Fluid

Department of Mathematics, Pabna University of Science and Technology, Pabna 6600, Bangladesh Department of Applied Mathematics, Gono Bishwabidyalay, Savar, Dhaka, Bangladesh Department of Basic Sciences FSD Campus, UET, Lahore, Pakistan Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, Kuala Lumpur 50603, Malaysia School of Mathematics and Statistics, Qujing Normal University, Qujing 650011, China Siirt University, Art and Science Faculty, Department of Mathematics, Siirt 56100, Turkey Near East University, Mathematics Research Center, Department of Mathematics, Near East Boulevard, PC 99138, Nicosia/Mersin 10, Turkey

Among the above procedures, the planner dynamical scheme is the most important, stable, and reliable approach as it provides various types of phase orbits. Depending on the parametric conditions, each type of orbit (periodic, heteroclinic, and homoclinic) yields periodic, kink, and bell wave solutions, respectively, under selective initial/boundary conditions. is type of approach is still least explored on numerous nonlinear models; hence, it breeds a vast challenge for proximate ascertainment of the models' dynamical behaviors.
e (1 + 1)-dimensional Oskolkov model for Kevin-Voigt fluids is one of the nonlinear models [23][24][25] with huge applications in contemplating incompressible visco-elastic Kelvin-Voigt fluid flows. is category of the pseudo-parabolic model, which involves time derivative in the highest differential term, can also be exemplified for wider purviews such as fluid flows of fissured rock, second order shear of fluids, thermodynamics, consolidation of dust, and transmission of long wave propagations with small amplitudes. Hence, the Oskolkov model is anticipatorily examined via diverse schemes by various researchers across the globe [23][24][25]. Several fundamental and featured procedures have also been established to procure the exact solutions of the Oskolkov model. e modified simple equation method [12] was used to derive more exact and explicit solutions of the Oskolkov model. e tanh-coth method [23] was applied to accomplish exact solutions of the nonlinear pseudo-parabolic model. Asymptotic stable and unstable solutions to the Oskolkov model were analyzed graphically too [24]. e nonlinear shock wave, bell wave, and various dynamical motions in the presence of an exotic periodic force were discussed briefly for the generalized Oskolkov model [25]. Although the enormous investigation and profound dynamical solutions to the model have been scrutinized by many researchers, there are still some unresolved problems like bounded travelling wave solutions with their corresponding phase orbit, solutions with boundary conditions, and rogue wave solutions. All these updates spark inquisitions on how the results are exhibited and how they vary with the changing parameters. To retort such accounts, we need to derive solutions involving some of the prominent parameters with suitable initial/boundary conditions. rough this article, we shed light on the progress of nonlinear travelling waves such as bright and dark bell waves, periodic waves, and singular bright and dark cusp waves as corresponding to each orbit of the phase portraits on the dynamical Oskolkov model by utilizing the planner dynamical strategy [20,21]. Besides, rogue waves, multisoliton, interaction of solitons, and breather wave solutions, which are no less fascinating, have also received gradually increased attention in the field of nonlinear dynamical systems [26][27][28]. e long wave limit of the breather localized wave was derived by Yue et al. [29]. Abundant research on these topics has been performed and different types of nonlinear localized waves have been revisited for strategic recipes of various nonlinear physical models materialized in mathematical physics and engineering applications [30,31]. e Haar wavelet technique has been used to analyze the dynamical phenomena of the fractional-order nonlinear model [32,33]. Nowadays, investigations on the fractional model are studied by analytical techniques [34,35] and numerically [36,37], exposing huge efforts. Notwithstanding, we also aim to construct the solutions to multirogue type waves, shock waves, interactions of bright-dark and darkbright bell solitons, interactions of shock-bell waves, and singular multisoliton to the (1 + 1)-dimensional Oskolkov model by taking benefit of the lavish unified method [38,39]. e creation of this article is designed as follows: in Section 2, we deliberate the bifurcation and phase portrait analyses of the Oskolkov model. In Section 3, we exhibit parametric expressions of explicit solutions of the Oskolkov model. e exact wave solutions and their graphical illustrations of the model via the unified method are presented in Section 4. Finally, the perspectives of the current study on this kind of dynamical wave phenomenon are discussed in the last section.

Bifurcations and Phase Portraits of the
Oskolkov Model e regular form of the (1 + 1) dimensional Oskolkov model (1) is where U(x, t) � U(ξ) and ξ � kx − ωt in which k is a constant whereas ω is the wave speed. Now, we reinstate (1) into the form of (ODE) as follows: where the primes stand for the derivative with respect to ξ Equation (2) in a concise form can be reformulated as where P � α/ωβ, Q � 1/κ 2 β, R � 1/2κωβ , and R ≠ 0. is (3) can further be written into a prototype system of differential form

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Next, we attempt to detect the bifurcation of the phase orbits of the dynamical system in (4) with various conditions on the parameters α, β, κ, ω. From the observation, it is clear that a smooth homoclinic orbit of the system in (4) arises from the smooth solitary waves in equation (1). When the solution of U(κ x − ω t) � U(ξ), (−∞ < ξ < ∞) of the system guarantees lim ξ⟶ −∞ U(ξ) � a and lim ξ⟶ ∞ U(ξ) � b, U(ξ) is said to be homoclinic and heteroclinic orbits for a � b and a ≠ b respectively. Normally, a homoclinic orbit of the (4) correlates with a solitary wave solution of the (1) and a heteroclinic orbit of the (4) correlates with a king (antiking) wave solution of the (1). Alike, a periodic orbit of the system in (4) keeps up a correspondence with a periodic travelling wave solution of (1). erefore, we can explore to locate all possible phase portraits of the dynamical system in (4) with the help of different constraints on the parameters α, β, κ, ω.
For critical points in an equilibrium situation, we have to consider U ′ � 0 and V ′ � 0 then that prototype (4) will provide two equilibrium points Ν 0 (0, 0) and Ν 1 (Q/R, 0), if Q ≠ 0 . Furthermore, the dynamical prototype (4) yields one equilibrium point Ν(0, 0) at Q � 0 . Now, let us assume, A(U Ν , V Ν ) to be the coefficients' matrix of the linearized prototype (4) at equilibrium points Ν i (i � 0, 1) and let I � det A(U Ν , V Ν ) . erefore, we have With the help of the bifurcation theorem [21,22] and the above inspection, we will get the following remarks: Group-1: for Q > 0 the nature at the origin Ν 0 (0, 0) is a saddle point and Ν 1 (Q/R, 0) is a center point, whereas the corresponding bifurcations of phase portraits of the prototype (4)

Explicit Expressions of Solution for the Model (1)
In this segment, we derive various explicit parametric illustrations of travelling wave solutions for the model (1). For effortlessness, the energy level of the Hamiltonian is specified as Z 0 � H(0, 0) � 0 and

Examining Group-1 Based on Figure 1 Observation (Part 2)
where the model (1) suggests itself as a valley type smooth solitary wave solution specified by Figure 1(a). Using the relation H(U, V) � Z 0 � 0 in equation (5), yields Combining the first equation of the prototype (4) and (7), we attain the parametric expression of the valley-type smooth solitary wave solution as follows: where (1) suggests itself as a valley-type smooth solitary wave solution portrayed in Figure 1(b). Using the relation H(U, V) � Z 0 � 0 in equation (5), yields Combining the first equation of the prototype (5) and (9), we attain the parametric expression of the valley-type smooth solitary wave solution as follows: where (10) is shown in Figure 4 (1) represents a set of smooth periodic wave solutions described by H(U, V) � Z, Z ∈ (Z 1 , 0) as can be viewed in Figures 1(a) and 1(b), respectively. For the state Q > 0, R > 0, the phase portrait of the prototype (4) is shown in Figure 1(a). In this phase, the formulation of the closed domain can be written as: Combining the rst equation of the prototype equations (4) and (11), we attain the parametric formulation of the periodic solution as follows: e shape of the periodic outcomes of (12) is displayed in Figure 4(c) with ω 1, κ −1, Q 1, R 0.5.

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e shape of the periodic outcomes of (13) is presented in Figure 4

Examining
en, by means of the first equation of the prototype (4) and (14), we attain the parametric expression of the valley-type smooth solitary wave solution that has a similar nature as depicted in Figure 4(a) as follows: where ξ � κ x − ω t, P � α/ωβ, Q � 1/κ 2 β, R � 1/2 κωβ (II) For the state Q < 0, R < 0, it is related to the homoclinic orbit at the stable point Ν 1 (Q/R, 0) identified by H(U, V) � Z 1 , where the model (1) represents a valley-type smooth solitary wave solution as displayed in Figure 2(b). With the help of the relation H(U, V) � Z 1 , yields en, by means of the first equation of the prototype (4) and (16), we attain the parametric expression of the valley-type smooth solitary wave solution that has similar nature as depicted in Figure 4(b) as follows: where ξ � κ x − ω t, P � α/ωβ, Q � 1/κ 2 β, R � 1/2 κωβ (III) For the state Q < 0, R > 0 or Q < 0, R < 0, the prototype (4) has a group of smooth periodic wave solutions identified by H(U, V) � Z, Z ∈ (0, Z 1 ) , which can be observed in Figures 2(a) and 2(b), respectively. For this aspect, the formulation of the periodic solution is identical to the solutions in (12) and (13). Figure 3 Observation (Part 2).

Examining Group-3 Based on
(I) For the state Q � 0, R > 0, there is an unrestricted orbit with the alike Hamiltonian as the origin Ν 0 (0 , 0) (Figure 3(a)). e open cusp orbit can be specified by en, by means of the first equation of the prototype (4) and (18), we acquire the periodic cusp wave solution as follows: where ξ � κ x − ω t, P � α/ωβ, Q � 1/κ 2 β, R � 1/2 κωβ e shape of the singular bright cusp wave of (19) is depicted in Figure 5 (II) For the state Q � 0, R < 0, there is also an unrestricted orbit with the alike Hamiltonian as the origin Ν 0 (0, 0) (Figure 3(b)). Due to this condition, we also catch out the similar periodic cusp wave solution in the form as follows: where ξ � κx − ω t, P � α/ωβ, Q � 1/κ 2 β,R � 1/2κωβ e profile of the singular dark cusp wave solution in (20) is depicted in Figure 5

Solutions of the Oskolkov Model via the Unified Method and Graphical Illustration
Let us apply a steadfast treatment of the unified method [32,33] to the (1 + 1) dimensional Oskolkov model (1) in this portion. Graphical presentations are also outlined here.

Solutions of the Oskolkov Model.
Using the travelling wave variable, we reinstate (1) into the ODE (2). Next, we enumerate the balance number of expressions in (2) between the linear term, U ″ and the nonlinear term, U 2 by letting N � 2. rough this effort, a trial solution can be formed as follows: Here a i , b i ; i � 0, 1, 2 are unfamiliar constants to be formed subsequently and we consider the derivative of S(ξ) satisfying an ODE namely the Riccati differential equation 6 Mathematical Problems in Engineering e solutions of the given Riccati equation are given as follows: Case-01: hyperbolic function (when λ < 0) as follows: Case-02: trigonometric function (when λ > 0) as follows: where l ≠ 0 and d, S(ξ) are real arbitrary constants. Case-03: rational function solutions (when λ 0.)  (19) and (b) dark cusp wave of equation (20).

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Differentiating (21) as many times as necessary while satisfying (22), these expressions are substituted back into (1). Next, we attain a polynomial of S(ξ) J where (J � ± 0, ± 1, ± 2, ..., ± N), and then by letting the coefficients of S(ξ) J equal to zero yields as follows: By solving the above system of equations, many sets of solutions can be attained as follows: Set-01: Set-02:

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Set-03: Set-04: Set-05: Set-06: Now, by plugging these expressions into (21) for unfamiliar constants, different types of solutions for (1) can be acquired.

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For the Set-01, if λ < 0, then we obtain the hyperbolic function solutions
If λ 0, we can search out rational function type solutions, but in this case, if we consider λ 0 then the executed values of k, ω, a 2 , b 1 and b 2 will be unde ned. So, the solution is rejected. Similarly, the other sets (Set-02 to Set-06) each provide us with eight solutions in the same way.

Numerical Illustrations of the Obtained Results and
Discussion. In this section, we will deliberate the nature and behavior of all the solutions physically with 2-D and 3-D graphs, as well as density plots of the (1 + 1) dimensional Oskolkov model by instigating the most recent method, namely, the uni ed method. e solution U 1,1 conveys the multipeaked bright solitonic nature for the parametric values λ −0.5, α 1, β 0.2, d 1, E 0.5, l 2 . Oppositely, the solution U 1,2 conveys the multipeaked dark solitonic nature for the parametric values λ −0.5, α 1, β 0.2, d 1, E 0.5, l 2 as illustrated in Figure 6. e real part of U 1,5 presents the kink type shock wave whereas its imaginary part presents the combo brightdark bell wave for the parameters λ 2, α −1, β 1, d 1, E 3, l 2 as illustrated in Figure 7. On the other hand, the real part of U 1,6 conveys the combo dark-bright bell wave with a shock wave, whereas its imaginary part expresses the bright bell wave with a shock for the parameter λ 2, α β 1, d 3, E −3, l 2, as illustrated in Figure 8. e real part of the solution U 1,7 presents the dark bell as a double shock wave but its imaginary part presents the double bright-dark bell wave with a shock for the parameters λ 2, α β 1, d 3, E −3, l 2, as demonstrated in Figure 9. e real part of U 1,8 conveys the bright bell wave with double shocks but its imaginary part conveys the bright-dark bell wave with a shock for the parametric values λ 2, α −1, β 1, d 3, E −3, l 2, as demonstrated in Figure 10. Both the real and imaginary parts of U 5,5 represents the multi-rogue type waves with singularities for λ 2, α 1, β 0.2, d 1, E −1, l −2, as demonstrated in Figure 11. Yet, every solution obtained for the model by using this technique can be reduced to the multirogue (bright-dark) type wave with the changing signs of the     parameters β and E. Such a dynamic is displayed in Figure 12 of the solution U 2,1 for the parametric values λ −1, α −1, β 0.2, d 1, E 0, l 0.5. Besides, we display its density and 2D plots of corresponding real and imaginary parts altogether.

Conclusions
rough this work, the key e ort is to procure, test, and explore the innovative travelling wave solutions and the physical properties of the nonlinear Oskolkov model for  Kevin-Voigt fluids by utilizing two reliable mathematical techniques. Based on the bifurcation theory of the dynamical scheme, we explored the bifurcation and derived a phase portrait of the model using diverse parametric conditions. We have derived smooth bright bell and dark bell waves, periodic solutions as well as singular dark cusp and bright cusp waves. ese travelling waves are achieved by corresponding to homoclinic, periodic, and open orbits with the cusp of each orbit of the phase portraits, respectively. e unified method as a substantial way to derive the new travelling wave solutions has been utilized through which we have successfully achieved the solutions for multipeaked bright and dark solitons, shock waves, bright bell waves with a shock, bright bell waves with a double shock, combo bright-dark and dark-bright bell waves with a shock, dark bell into a double shock wave, and bright-dark multirogue type waves. e bifurcation analysis of this model is presented first in this article, even though most of the obtained solutions derived by direct integration from the Hamiltonian of the model according to each energy is the first step here. In particular, the bright bell waves with a shock, bright bell waves with double shock, combo bright-dark, combo dark-bright bell waves with a shock, dark bell into a double shock wave, and bright-dark multirogue type waves were not reported in the previous literature. is technique proposes solutions with free parameters that could be significant to deliberate more intricate nonlinear corporal phenomena. e obtained solutions in this paper reaffirmed that the method is very effective and easily applicable to formulate more exact travelling wave solutions than other methods of the nonlinear evolution models occurring in many mathematical physics and engineering applications.

Data Availability
e data used to support the findings of this study are included in the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.