Periodic, Cross-Kink, and InteractionbetweenStripe andPeriodic WaveSolutions forGeneralizedHietarintaEquation:Prospects for Applications in Environmental Engineering

Cangzhou Preschool Teachers College, Cangzhou Hebei 062150, China Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov str., Lankaran, Azerbaijan Department of Civil and Architectural Engineering, University of Wyoming, Laramie 82071, USA Department of Mathematics, D. G. Vaishnav College, Chennai, India Ganja State University, Haydar Aliyev ave., 429, Ganja, Azerbaijan


Introduction
Research in the field of nonlinear wave theory has been become very interesting due to its applications in sciences and engineering. Many physical phenomena are represented as models in the structure of nonlinear PDEs, mostly in the form of nonlinear integrable equations. ese models clearly indicate the parameters that affect the phenomenon that are not seen directly by observing the phenomenon. Various models have been made in the field of science and engineering that are representing the different phenomenon. For example, mostly naturally occurring phenomenons are modeled as modified tan(φ/2)-expansion technique [1], the homotopy perturbation scheme [2], the csch-function method [3], the Lie symmetry analysis [4], the Bäcklund transformation method [5], the sine-Gordon expansion approach [6], a new nonlinear mathematical programming model for dynamic cell formation [7], the (G′/G, 1/G), the modified (G′/ G 2 ) and (1/G′)-expansion schemes [8], and imperialist competitive algorithm [9]. Except of these methods, there are other powerful methods such as the multiple exp-function method [10][11][12][13], a new fuzzy classification algorithm [14], Hirota's bilinear method [15][16][17][18][19][20][21], a deterministic mathematical mixed integer linear programming model [22], the coupled modified Korteweg-de Vries equation with nonzero boundary conditions at infinity [23], the high-order rogue wave of generalized non-linear Schrödinger equation with nonzero boundary [24], the supersymmetric constrained B type and C type KP hierarchies of Manin-Radul and Jacobian types [25], the (3 + 1)-dimensional extended Jimbo-Miwa equations [26], understanding by design model as useful tool for a meaningful and permanent learning [27], and the first integral method for constructing the exact solutions of the time-fractional Wu-Zhang system [28].
In Ref. [29], the authors suggested the fuzzy clustering to discover the optimal number of clusters as an innovation clustering algorithm in marketing to determine the best group of customers, similar items, and products. In a valuable research, the Bayer-Hanck cointegration test, wavelet coherence, Fourier Toda-Yamamoto, and Breitung-Candelon frequency-domain spectral causality tests were investigated the causal relationships among carbon emissions, economic growth, and life expectancy in [30]. Adinda et al. [31] studied students' metacognitive awareness failures about solving absolute value problems (AVPs) in mathematics education, and they found that there was a significant failure, and three students were sampled from who had experienced different metacognitive awareness failures in solving AVPs. In [32], the residual power series method to solve the (3 + 1)-dimensional nonlinear conformable Schrödinger equation with cubic-quintic-septic nonlinearities along with three test applications was considered subject to different initial conditions. Two classes of lump and line rogue wave solutions for a new (2 + 1)-dimensional extension of the Hietarinta equation were obtained by means of the Hirota bilinear scheme by Manukure and Zhou [32]. In [33], the authors showed the existence of the three-periodic wave solutions numerically for the Hietarinta equation by using the direct method. Both Dirichlet and Neumann data on some part of the domain boundary for a family of quasilinear inverse problems to the Laplace equation coupled with a sequence of nonlinear scalar equations were recovered [34]. A novel integral transform involving the product of the Whittaker function and two Bessel functions of the first kind was employed to Bessel-Circular-Gaussian beam to generate a new laser beam called Exton-Gaussian beams [35]. e complete discrimination system method was used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives by Han and Li [36]. e periodic, cross-kink wave solutions were obtained by the authors of [37] by the help of Hirota bilinear operator, and also, the semi-inverse variational principle was utilized for the (2 + 1)-dimensional generalized Hirota-Satsuma-Ito equation. In [38], the effects of Mobile Ad Wearout on irritation, intrusiveness, engagement, and loyalty via social media outlets were studied. Author of [39] studied the mathematical models for global solar radiation intensity estimation at Shakardara area which is to estimate atmospheric transparency percentage. Fauzi and Respati [40] analyzed and studied the differences in students' critical thinking skills utilizing the guided discovery learning model and the problem-based learning model including both theoretical and practical knowledge and skills, and also, they used quantitative methods through an experimental approach. e present research focuses on the Hirota bilinear scheme to getting the analytical solutions of nonlinear (2 + 1)-dimensional wave equation. In this considered scheme, the solutions are written as a combination of trigonometric and hyperbolic waves and also a combination of trigonometric and exponential waves so that the solutions can adapt easily made by symbolic estimations. e fundamental work of this paper is to extract new analytical findings of (2 + 1)-D generalized Hietarinta model. For the purpose, determining the solutions of the shown model by powerful technique has been made. Many kinds of schemes have been used to determine the new kinds of solitons of this model, such as, two good papers in references [41,42]. According to used algorithm in reference [41] the bilinear shape can be driven as follows in which u � u(x, y, t) is a unfamiliar solution and h s (s � 1, 2, 3) are all free quantities. According to expansion and generalization of the Hietarinta-type model, [43] was studied with the below bilinear model form: In addition, by using the following relations the following nonlinear equation will be arisen as in which v x � u, and h 1 , h 2 , and h 3 are arbitrary quantities. Besides, a new (2 + 1)-D extension of equation (4) was proposed in [44]. On the basis of Hirota bilinear method, a few nonlinear models have been investigated as the valuable researches, for example, the coupled nonlinear Schrödinger equations [45]; the modified coupled Hirota equation by help of Riemann-Hilbert approach [46]; an extended (2 + 1)dimensional Calogero-Bogoyavlenskii-Schiff-like equation by using the generalized bilinear operators [47]; a generalized (3 + 1) shallow water-like equation through the Hirota bilinear method and the Cole-Hopf transformation [48]; a new (3 + 1)-dimensional weakly coupled B-type Kadomtsev-Petviashvili equation by constructing the symmetric positive semidefinite matrix technique [49]. Wave solutions have been used for different purposes as modeling of contaminant distribution or biodegradation in environmental engineering [50][51][52]. Specifically, Janssen et al. modeled the biodegradation of contaminants in heterogeneous aquifers using a semianalytical traveling wave solution for the one-dimensional reactive transport [50], and Wang et al. suggested a multimedia fate model to evaluate the fate of an organic contaminant by a one-dimensional network kinematic wave equation [51]. Moreover, wave equations have also been exploited in the analysis of transient flow in large distribution systems like groundwater [52]. In this regard, Jaradat et al. analyzed the health risks from the intrusion of contaminants into the distribution system from pressure transients. In [53], the multiple-kink solutions and 2 Advances in Mathematical Physics singular-kink solutions for (2 + 1)-D coupled Burgers system with time variable coefficients were obtained by Jaradat and coworkers. is paper investigates new the periodic-solitary, periodic wave, cross-kink wave, and interaction between stripe and periodic wave solutions for the generalized Hietarinta equation. We seek to explore two types of soliton solutions using two different formulas according to trigonometric, hyperbolic, and rational functions. In addition, we establish singular and dark soliton findings according to trigonometric and hyperbolic, respectively. e fundamental work of this paper is to extract new exact findings of equation (4), and the paper is organized as follows: in Section 2, the analysis of the governing system via bilinear form polynomials is formulated to the generalized (2 + 1)-dimensional nonlinear model. In Sections 3-6, we obtain the periodic-solitary, periodic wave, cross-kink wave, and interaction between stripe and periodic wave solutions, respectively, for the generalized (2 + 1)-dimensional Hietarinta equation. Some conclusions that be gained throughout the paper have been presented in Section 7.

The Bilinear Formalism Euations
rough ref. [21], take λ � λ(x 1 , x 2 , . . . , x n ) be a C ∞ function with multivariables as follows: x j e λ , (5) with the below formalism (BBPs [21]) and we have e multidimensional binary Bell polynomial can be written as We have the following conditions as Proposition 1. Let μ 1 � ln(Ω 1 /Ω 2 ), μ 2 � ln(Ω 1 Ω 2 ), then the relations between binary Bell polynomials and Hirota D-operator reads Σ n 1 x 1 ,...,n j x j μ 1 , with Hirota operator which need to satisfy e generalized Bell polynomials Υ n 1 x 1 ,...,n j x j (ξ) is as Advances in Mathematical Physics e Cole-Hopf relation is as follows: with Also, based on the above writings, the bilinear frame to the aforementioned nonlinear model will be as

Periodic-Solitary Solutions
Here, we utilize to formulate the new exact solutions to the (2 + 1)-dimensional generalized Hietarinta equation. Consider the following function for studying the periodic-solitary solutions as Afterwards, the values α s , β s , δ s , ϵ s (s � 1: 5) will be found. By making use of equation (18) into equation (17) and taking the coefficients, each powers of sin(x, y, t) and sinh(x, y, t) to zero, a system of equations (algebraic) (these are not collected here for minimalist) for α s , β s , δ s , ϵ s (s � 1: 5) is yielded. ese algebraic equations by using the emblematic computation software like, Maple, give the following solutions with using u � 2(lnϕ) x and v � 2(lnϕ).

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Here, α d , δ d , and ε k for d � 1: 2, k � 1: 4, β 2 are the unknown parameters. By considering the necessary assumption: by substituting the received above parameters into equation (18), we obtain an analytical form of rational equation: If τ 2 ⟶ ∞, ϕ 1 will be constant with any time Figure 1 shows the analysis of treatment of periodic and progress of soliton wave as hyperbolic function with graphs of ϕ 1 with the following selected parameters: in equation (21).

Set II Solutions
Here, α d , ϵ k for d � 1: 2, k � 1: 4, δ 1 are the unknown parameters. By considering the necessary assumption, by substituting the above parameters into equation (18), we obtain an analytical form of rational equation: If τ 2 ⟶ ∞, the periodic-solitary wave outputs u ⟶ 2α 2 at every time. Figure 2 shows the analysis of treatment of periodic and progress of soliton wave as hyperbolic function with graphs of ϕ 2 with the following selected parameters: in equation (25).
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Here, δ d and ϵ k for d � 1: 2, k � 1: 4, α 1 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation: If τ 3 ⟶ ∞, the periodic-solitary solution u ⟶ 0 at every time.
If τ 2 ⟶ ∞, the periodic-solitary solution u ⟶ 2α 2 at every time. Figure 3 offers the analysis of treatment of periodic and progress of soliton as hyperbolic function with graphs of ϕ 4 with the following selected parameters: in equation (31).

Collection V Findings
Here, δ d , ϵ j for d � 1, 2, k � 1: 4, and β 2 are the unknown parameters. By considering the necessary assumption and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:

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Here, δ d , ϵ k for d � 1, 2, k � 1: 5 are the unknown parameters. By considering the necessary assumptions and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation

Periodic Wave Solutions
In this paragraph, we find out some advanced exact periodic wave soliton solutions to the (2 + 1)-dimensional generalized Hietarinta equation. Assume the stated function for studying the periodic wave solutions which is as follows: Afterwards, the values α s , β s , δ s , ε s (s � 1: 5) will be found. By making use of equation (39) into equation (17) and taking the coefficients, each powers of e Φ 1 (x,y,t) , e Φ 2 (x,y,t) , and trigonometric function cos (Φ(x, y, t)) to zero yield a system of equations (algebraic) (these are not collected here for minimalist) for α s , β s , δ s , ε s (s � 1: 5). ese algebraic equations by using the emblematic computation software like, Maple, give the solutions as follows with using u � 2(lnϕ) x and v � 2(lnϕ).

Set I Findings
Here, δ d , ϵ k for d � 1, 2, k � 1: 4 are the unknown parameters and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation: If τ 1 ⟶ ∞, the breather outputs u ⟶ 2δ 1 at every time.   Advances in Mathematical Physics

Set II Solutions
Here, α d , ϵ k for d � 1, 2, k � 1: 5, and δ 1 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into the equation (36), we obtain an analytical form of rational equation: If τ 1 ⟶ ∞, the breather outputs u ⟶ 2α 1 at every time. Figure 4 shows the analysis of treatment of periodic and progress of breather-wave solutions as exponential and trigonometric functions with graphs of ϕ 1 with the following selected parameters: in equation (41).

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If τ 1 ⟶ ∞, the periodic outputs u ⟶ 2α 1 at any time. Figure 5 shows the analysis of treatment of periodic and progress of periodic wave solutions as exponential and trigonometric functions with graphs of ϕ 3 with the following selected parameters: in equation (48).

Set IV Solutions
Here, δ d , ϵ k for d � 1, 2, k � 1: 5, α 2 , and β 2 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation: If τ 1 ⟶ ∞, the breather outputs u ⟶ 2α 2 δ 1 /δ 2 at every time. Figure 6 shows the analysis of treatment of periodic and progress of periodic wave solutions as exponential and trigonometric functions with graphs of ϕ 4 with the following selected parameters: in equation (48).

Collection V Outputs
Here, δ d , ϵ k for d � 1, 2, k � 1: 5, and β 2 are the unknown parameters. By considering the necessary assumption,  Advances in Mathematical Physics and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation: (56) 4.6. Set VI Solutions.
, k � 1: 5 and β 2 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:

Set VII Solutions
Here, α d , ϵ k for d � 1, 2, k � 1: 5 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:
Afterwards, the values α s , β s , δ s , ϵ s (s � 1: 3) will be found. By making use of equation (63) into (17) and taking the coefficients, each powers of e Φ(x,y,t) , sin(x, y, t), and sinh(x, y, t) to zero yield a system of equations (algebraic) (these are not collected here for minimalist) for α s , β s , δ s , ϵ s (s � 1: 3). ese algebraic equations by using the emblematic computation software like, Maple, give the following solutions with using u � 2(lnϕ) x and v � 2(lnϕ).

Set I Solutions
Here, α d , ϵ k for d � 1: 2, k � 1: 6, β 2 , δ 1 , and δ 2 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation: If τ 1 ⟶ ∞, the breather outputs u ⟶ 2α 3 at every time.

Set IV Solutions
and β 3 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation: If τ 3 > τ 1 ⟶ ∞, the cross-kink outputs u ⟶ 2α 3 at every time, but if τ 3 < τ 1 ⟶ ∞, the cross-kink outputs u ⟶ 2α 1 at every time.

Set V Solutions.
for d � 1: 3, k � 1: 6, β 2 , and β 3 are the unknown parameters. By considering the necessary assumption, and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation: If τ 3 > τ 1 ⟶ ∞, the cross-kink outputs u ⟶ 2α 3 at every time, but if τ 3 < τ 1 ⟶ ∞, the cross-kink outputs u ⟶ 2α 1 at any time. Figure 7 show the analysis of treatment of cross-kink wave as periodic and hyperbolic function with graphs of ϕ 5 with the following selected parameters: in equation (74).

Set VIII Solutions
Here δ d , ϵ k for d � 1: 3, k � 1: 5, α 3 , β 2 , and β 3 are arbitrary inputs, and the following case is considered as By substituting the above parameters into equation (63), we obtain an analytical form of rational equation: If τ 1 ⟶ ∞, the cross-kink wave outputs u ⟶ 2α 1 at any time.

Conclusion
is article investigated the soliton and periodic solutions of the generalized Hietarinta equation. e Cole-Hopf algorithm has been described by means of binary Bell polynomials.
e governing equation is translated to nonlinear ODE using Hirota transformation. Various types of soliton, breather, and periodic solutions have been constructed in terms of exponential, hyperbolic, trigonometric, and rational functions. e dynamic features of different types of traveling waves are analyzed in detail through numerical simulation. Meanwhile, the profiles of the surface for the deduced solutions have been depicted in 2D and 3D for the obtained solutions. e gained solutions may be applied to explain the model in simple and straight forward way. At the end, it is concluded that, to handle nonlinear partial differential equations, Hirota bilinear technique suggested an effective and well-built mathematical tools. ese solutions are also verified by using Maple software.
Data Availability e datasets supporting the conclusions of this article are included in the article.

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