# Periodic, Cross-Kink, and Interaction between Stripe and Periodic Wave Solutions for Generalized Hietarinta Equation: Prospects for Applications in Environmental Engineering

Authors: Guangping Li; Jalil Manafian; Mirmehdi Seyyedi; R. Sivaraman; Subhiya M. Zeynalli
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In the current work, the modified (2 + 1)-dimensional Hietarinta model is considered by employing Hirota’s bilinear scheme. Likewise, the bilinear formalism is obtained for the considered model. In addition, the periodic-solitary, periodic wave, cross-kink wave, and interaction between stripe and periodic wave solutions of the mentioned equation by particular coefficients are offered. The obtained results may be used in the description of the model in fruitful way. Finally, by using the available situations, the physical demeanor of solutions is discussed in the given method. We demonstrated that these solutions validated the program using Maple and found them correct. Moreover, a lot of graphs in some sections to determine the analysis of obtained findings for the aforementioned equation are given. The achieved solutions are also verified by using the Maple software. These periodic wave solutions suggest that these three methods are useful, easy to use, and effective than other methods.


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## Body

## 1. 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. These 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 modifiedtanφ/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′/ G2) 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 [10–13], a new fuzzy classification algorithm [14], Hirota’s bilinear method [15–21], 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]. The 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]. The 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. The 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.The 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(1)Dx4−DxDt3+h1Dx2+h2DxDt+h3Dt2ϕ.ϕ=0,in which u=ux,y,t is a unfamiliar solution and hss=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:(2)Dx4+DxDt3+h1Dx2+h2DxDt+h3Dt2−DtDyϕ.ϕ=0.In addition, by using the following relations(3)u=2lnϕx,v=2lnϕ,the following nonlinear equation will be arisen as(4)6uxuxx+uxxxx+3ututt+3utxvtt+uxttt+h1uxx+h2utx+h3utt−uty=0,in which vx=u, and h1,h2, and h3 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 [50–52]. 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 singular-kink solutions for (2 + 1)-D coupled Burgers system with time variable coefficients were obtained by Jaradat and coworkers.This 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.The 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.


## 2. The Bilinear Formalism Euations

Through ref. [21], take λ=λx1,x2,…,xn be a C∞ function with multivariables as follows:(5)Υn1x1,…,njxjλ≡Υn1,…,njλd1x1,…,djxj=e−λ∂x1n1⋯∂xjnjeλ,with the below formalism (BBPs [21])(6)λd1x1,…,djxj=∂x1d1⋯∂xjdjλ,λ0xi≡λ,d1=0,…,n1;…;dj=0,…,nj,and we have(7)Υ1λ=λx,Υ2λ=λ2x+λx2,Υ3λ=λ3x+3λxλ2x+λx3,…,λ=λx,t,Υx,tλ=λx,t+λxλt,Υ2x,tλ=λ2x,t+λ2xλt+2λx,tλx+λx2λt,….The multidimensional binary Bell polynomial can be written as(8)Σn1x1,…,njxjμ1,μ2=Υn1,…,njλ|λd1x1,…,djxj=μ1d1x1,…,djxj,d1+d2+⋯+dj,is oddμ2d1x1,…,djxj,d1+d2+⋯+dj,is even.We have the following conditions as(9)Σxμ1=μ1x,Σ2xμ1,μ2=μ2x+μ1x2,Σx,tμ1,μ2=μ2x,t+μ1xμ1t,….Proposition 1.
Letμ1=lnΩ1/Ω2,μ2=lnΩ1Ω2, then the relations between binary Bell polynomials and Hirota D-operator reads(10)Σn1x1,…,njxjμ1,μ2|μ1=lnΩ1/Ω2,μ2=lnΩ1Ω2=Ω1Ω2−1Dx1n1…DxjnjΩ1Ω2,with Hirota operator(11)∏i=1jDξinig.η=∏i=1j∂∂ξi−∂∂ξi′niΩ1ξ1,…,ξjΩ2ξ1′,…,ξj′|ξ1=ξ1′,…,ξj=ξj′.Proposition 2.
ConsideringΞγ=∑iδiPd1ξ1,…,djξj=0 and μ1=lnΩ1/Ω2,μ1=lnΩ1Ω2, we receive(12)∑iδ1iΥn1ξ1,…,njξjμ1,μ2=0,∑iδ1iΥd1ξ1,…,djξjμ1,μ2=0,which need to satisfy(13)ℜγ′,γ=ℜγ′−ℜγ=ℜμ2+μ1−ℜμ2−μ1=0.The generalized Bell polynomialsΥn1x1,…,njxjξ is as(14)Ω1Ω2−1Dξ1n1…DξjnjΩ1Ω2=Σn1ξ1,…,njξjμ1,μ2|μ1=lnΩ1/Ω2,μ2=lnΩ1Ω2=Σn1ξ1,…,njξjμ1,μ1+γ|μ1=lnΩ1/Ω2,γ=lnΩ1Ω2=∑k1n1…∑kjnj∏i=1jnikiPk1ξ1,…,kjξjγΥn1−k1ξ1,…,nj−kjξjμ1.The Cole–Hopf relation is as follows:(15)Υk1ξ1,…,kjξjμ1=lnτ=τn1ξ1,…,njξjτ,Ω1Ω2−1Dx1n1…DxjnjΩ1Ω2|Ω2=expγ/2,Ω1/Ω2=τ=τ−1∑k1n1…∑kjnj∏l=1jnlklPk1ξ1,…,klξlγτn1−k1ξ1,…,nd−klξl,with(16)Υtμ1=τtτ,Υ2xμ1,β=γ2x+τ2xτ,Υ2x,yμ1,μ2=γ2xτyτ+2γx,yτxτ+τ2x,yτ.Also, based on the above writings, the bilinear frame to the aforementioned nonlinear model will be as(17)Dx4+DxDt3+h1Dx2+h2DxDt+h3Dt2−DtDyϕ.ϕ=2ϕxxxxϕ−4ϕxxxϕx+3ϕxx2+ϕxtttϕ−4ϕtttϕx+6ϕttϕtx−4ϕtϕttx+ϕxϕttt+h1ϕxxϕ−ϕx2+h2ϕxtϕ−ϕxϕt+h3ϕttϕ−ϕt2−ϕytϕ−ϕyϕt]=0.


## 3. 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(18)ϕ=ϵ3sinτ1+ϵ4sinhτ2+ϵ5,τs=αsx+βsy+δst+ϵs,s=1,2.Afterwards, the valuesαs,βs,δs,ϵss=1:5 will be found. By making use of equation (18) into equation (17) and taking the coefficients, each powers of sinx,y,t and sinhx,y,t to zero, a system of equations (algebraic) (these are not collected here for minimalist) for αs,βs,δs,ϵss=1:5 is yielded. These algebraic equations by using the emblematic computation software like, Maple, give the following solutions with using u=2lnϕx and v=2lnϕ.


### 3.1. Set I Solutions

(19)β1=3α12+α222α12ε32−α22ε42+3δ12+δ22α12ε32−α22ε42α1δ1+α2δ2+ε32+ε42α1δ2−α2δ1−α1δ2h3+α2δ1h3+α1β2α1δ2−α2δ1α2ε32+ε42,h1=δ12+δ223α1δ13ε32+4α1δ1δ22ε32+α1δ1δ22ε42−α2δ12δ2ε32−4α2δ12δ2ε42−3α2δ32ε42+Φ1ε32+ε42α1δ2−α2δ12,Φ1=δ123ε32α14+6ε32α12α22−ε32α24−4ε42α24+δ224ε32α14+α14ε42−6α12α22ε42−3ε42α24−4α1α2δ1δ2ε32+ε42α12−α22,h2=Φ2α2ε32+ε42α1δ2−α2δ12,Φ2=α12+α22α1ε323α13δ2−6α12α2δ1−α1α22δ2−2α32δ1−α22ε42α12δ2−4α1α2δ1−3α22δ2+ε32+ε42α1δ2−α2δ12−δ2h3+β2−α12α26δ14ε32+3δ12δ22ε32+δ24ε32+4δ24ε42+3α13δ1δ2ε32δ12+δ22−α1α22δ1δ210δ12ε32+δ12ε42+2δ22ε32−7δ22ε42+α234δ14ε32+4δ14ε42+3δ12δ22ε42+3δ24ε42,ε5=0.Here,αd,δd,andεk for d=1:2,k=1:4, β2 are the unknown parameters. By considering the necessary assumption:(20)α2ε32+ε42α1δ2−α2δ12≠0,by substituting the received above parameters into equation (18), we obtain an analytical form of rational equation:(21)u=2lnϕ1x=2ϵ3costδ1+xα1+yβ1+ϵ1α1+ϵ4coshtδ2+xα2+yβ2+ϵ2α2ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhtδ2+xα2+yβ2+ϵ2,ϕ1=ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhtδ2+xα2+yβ2+ϵ2.Ifτ2⟶∞, ϕ1 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 ϕ1 with the following selected parameters:(22)δ1=1,δ2=0.5,α1=0.1,α2=0.5,β2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,t=1,in equation (21).Figure 1
Periodic-solitary solution (24) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1andx=−1,0,1.
(a)(b)


### 3.2. Set II Solutions

(23)β1=α12+α223ε32α14−ε32α12α22−α12α22ε42+3ε42α22+α22δ1ε32+ε42α1h2+δ1h3+α1δ133α12ε32−4α22ε32−α22ε42α22δ1ε32+ε42,β2=2α1α12+α223α12ε32+α22ε32−2α22ε42+2δ133α12ε32−2α22ε32−2α22ε42+α22δ1h2ε32+ε42α2ε32+ε42δ1,h1=3ε32α14+6ε32α12α22+3α1δ13ε32−ε32α24−4ε42α24α22ε42+ε42,δ2=ε5=0.Here,αd,ϵk for d=1:2,k=1:4, δ1 are the unknown parameters. By considering the necessary assumption,(24)α22ε32+ε42δ1≠0,by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(25)u2=2lnϕ2x=2ϵ3costδ1+xα1+yβ1+ϵ1α1+ϵ4coshxα2+yβ2+ϵ2α2ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhxα2+yβ2+ϵ2+ϵ5.Ifτ2⟶∞, the periodic-solitary wave outputs u⟶2α2 at every time. Figure 2 shows the analysis of treatment of periodic and progress of soliton wave as hyperbolic function with graphs of ϕ2 with the following selected parameters:(26)δ1=1,δ2=0.5,α1=0.1,α2=0.5,β2=1,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,t=1,in equation (25).Figure 2
Periodic-solitary solution (26) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


### 3.3. Set III Solutions

(27)α2=ε5=0,β1=α13α13δ1ε32+3δ14ε32+3δ12δ22ε32+4δ24ε32+4δ24ε42+δ22ε32+ε42α1h2+δ1h3δ22ε32+ε42,β2=−α1ε32α13+δ13+δ1δ22−δ22h3ε32+ε42δ2ε32+ε42,h1=α133δ12ε32+4δ22ε32+δ22ε42+δ1δ12+δ223δ12ε32+4δ22ε32+δ22ε42α1δ22ε32+ε42.Here,δdandϵk for d=1:2,k=1:4, α1 are the unknown parameters. By considering the necessary assumption,(28)α1δ22ε32+ε42≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(29)u3=2lnϕ3x=2ε3costδ1+xα1+yβ1+ε1α1ε3sintδ1+xα1+yβ1+ε1+ε4sinhtδ2+xα2+yβ2+ε2.Ifτ3⟶∞, the periodic-solitary solution u⟶0 at every time.


### 3.4. Set IV Solutions

(30)α1=α2ε4ε3,β1=ε42ε32α24+2ε42α24+2α2δ23ε32+2α2δ23ε42+β2δ2ε42ε33δ2,δ1=δ2ε4ε3,h2=−ε32α24−3ε42α24+α2ε23ε32−3α2δ23ε42+α22h1ε42+δ22h3ε32−β2δ2δ32δ32δ2α2,ε5=0.Here,εk for k=1:4, and α2andδ2 are the unknown parameters. By considering the necessary assumption,(31)ε32δ2α2≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(32)u4=2lnϕ4x=2costδ2ε4/ε3+xα2ε4/ε3+yβ1+ε1α2ε4+ε4coshtδ2+xα2+yβ2+ε2α2ε3sintδ2ε4/ε3+xα2ε4/ε3+yβ1+ε1+ε4sinhtδ2+xα2+yβ2+ε2.Ifτ2⟶∞, the periodic-solitary solution u⟶2α2 at every time. Figure 3 offers the analysis of treatment of periodic and progress of soliton as hyperbolic function with graphs of ϕ4 with the following selected parameters:(33)δ1=1,δ2=2,α1=0.1,α2=0.25,β2=1,h2=1,h3=1.5,ϵ1=1,ϵ2=2,ϵ3=4,ϵ4=2,t=1,in equation (31).Figure 3
Periodic-solitary solution (33) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


### 3.5. Collection V Findings

(34)αi=−δi,i=1,2,β1=β2δ1δ2,h2=δ22h1+δ22h3−β2δ2δ22,ε5=0.Here,δd,ϵj for d=1,2,k=1:4, and β2 are the unknown parameters. By considering the necessary assumption(35)δ2≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(36)u5=2lnϕ5x=2−ε3costδ1−xδ1+yβ2δ1/δ2+ε1δ1−ε4coshtδ2−xδ2+yβ2+ε2δ2ε3sintδ1−xδ1+yβ2δ1/δ2+ε1+ε4sinhtδ2−xδ2+yβ2+ε2.


### 3.6. Set VI Solutions

(37)αi=−δi,i=1,2,β1=δ1h1−h2+h3,β2=δ2h1−h2+h3.Here,δd,ϵk 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(38)u6=2lnϕ6x=2−ε3cosζ1δ1−ε4coshtδ2−xδ2+yδ2h1−h2+h3+ε2δ2ε3sinζ1+ε4sinhtδ2−xδ2+yδ2h1−h2+h3+ε2+ε5,ζ1=tδ1−xδ1+yδ1h1−h2+h3+ε1.


## 3.1. Set I Solutions

(19)β1=3α12+α222α12ε32−α22ε42+3δ12+δ22α12ε32−α22ε42α1δ1+α2δ2+ε32+ε42α1δ2−α2δ1−α1δ2h3+α2δ1h3+α1β2α1δ2−α2δ1α2ε32+ε42,h1=δ12+δ223α1δ13ε32+4α1δ1δ22ε32+α1δ1δ22ε42−α2δ12δ2ε32−4α2δ12δ2ε42−3α2δ32ε42+Φ1ε32+ε42α1δ2−α2δ12,Φ1=δ123ε32α14+6ε32α12α22−ε32α24−4ε42α24+δ224ε32α14+α14ε42−6α12α22ε42−3ε42α24−4α1α2δ1δ2ε32+ε42α12−α22,h2=Φ2α2ε32+ε42α1δ2−α2δ12,Φ2=α12+α22α1ε323α13δ2−6α12α2δ1−α1α22δ2−2α32δ1−α22ε42α12δ2−4α1α2δ1−3α22δ2+ε32+ε42α1δ2−α2δ12−δ2h3+β2−α12α26δ14ε32+3δ12δ22ε32+δ24ε32+4δ24ε42+3α13δ1δ2ε32δ12+δ22−α1α22δ1δ210δ12ε32+δ12ε42+2δ22ε32−7δ22ε42+α234δ14ε32+4δ14ε42+3δ12δ22ε42+3δ24ε42,ε5=0.Here,αd,δd,andεk for d=1:2,k=1:4, β2 are the unknown parameters. By considering the necessary assumption:(20)α2ε32+ε42α1δ2−α2δ12≠0,by substituting the received above parameters into equation (18), we obtain an analytical form of rational equation:(21)u=2lnϕ1x=2ϵ3costδ1+xα1+yβ1+ϵ1α1+ϵ4coshtδ2+xα2+yβ2+ϵ2α2ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhtδ2+xα2+yβ2+ϵ2,ϕ1=ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhtδ2+xα2+yβ2+ϵ2.Ifτ2⟶∞, ϕ1 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 ϕ1 with the following selected parameters:(22)δ1=1,δ2=0.5,α1=0.1,α2=0.5,β2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,t=1,in equation (21).Figure 1
Periodic-solitary solution (24) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1andx=−1,0,1.
(a)(b)


## 3.2. Set II Solutions

(23)β1=α12+α223ε32α14−ε32α12α22−α12α22ε42+3ε42α22+α22δ1ε32+ε42α1h2+δ1h3+α1δ133α12ε32−4α22ε32−α22ε42α22δ1ε32+ε42,β2=2α1α12+α223α12ε32+α22ε32−2α22ε42+2δ133α12ε32−2α22ε32−2α22ε42+α22δ1h2ε32+ε42α2ε32+ε42δ1,h1=3ε32α14+6ε32α12α22+3α1δ13ε32−ε32α24−4ε42α24α22ε42+ε42,δ2=ε5=0.Here,αd,ϵk for d=1:2,k=1:4, δ1 are the unknown parameters. By considering the necessary assumption,(24)α22ε32+ε42δ1≠0,by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(25)u2=2lnϕ2x=2ϵ3costδ1+xα1+yβ1+ϵ1α1+ϵ4coshxα2+yβ2+ϵ2α2ϵ3sintδ1+xα1+yβ1+ϵ1+ϵ4sinhxα2+yβ2+ϵ2+ϵ5.Ifτ2⟶∞, the periodic-solitary wave outputs u⟶2α2 at every time. Figure 2 shows the analysis of treatment of periodic and progress of soliton wave as hyperbolic function with graphs of ϕ2 with the following selected parameters:(26)δ1=1,δ2=0.5,α1=0.1,α2=0.5,β2=1,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,t=1,in equation (25).Figure 2
Periodic-solitary solution (26) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


## 3.3. Set III Solutions

(27)α2=ε5=0,β1=α13α13δ1ε32+3δ14ε32+3δ12δ22ε32+4δ24ε32+4δ24ε42+δ22ε32+ε42α1h2+δ1h3δ22ε32+ε42,β2=−α1ε32α13+δ13+δ1δ22−δ22h3ε32+ε42δ2ε32+ε42,h1=α133δ12ε32+4δ22ε32+δ22ε42+δ1δ12+δ223δ12ε32+4δ22ε32+δ22ε42α1δ22ε32+ε42.Here,δdandϵk for d=1:2,k=1:4, α1 are the unknown parameters. By considering the necessary assumption,(28)α1δ22ε32+ε42≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(29)u3=2lnϕ3x=2ε3costδ1+xα1+yβ1+ε1α1ε3sintδ1+xα1+yβ1+ε1+ε4sinhtδ2+xα2+yβ2+ε2.Ifτ3⟶∞, the periodic-solitary solution u⟶0 at every time.


## 3.4. Set IV Solutions

(30)α1=α2ε4ε3,β1=ε42ε32α24+2ε42α24+2α2δ23ε32+2α2δ23ε42+β2δ2ε42ε33δ2,δ1=δ2ε4ε3,h2=−ε32α24−3ε42α24+α2ε23ε32−3α2δ23ε42+α22h1ε42+δ22h3ε32−β2δ2δ32δ32δ2α2,ε5=0.Here,εk for k=1:4, and α2andδ2 are the unknown parameters. By considering the necessary assumption,(31)ε32δ2α2≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(32)u4=2lnϕ4x=2costδ2ε4/ε3+xα2ε4/ε3+yβ1+ε1α2ε4+ε4coshtδ2+xα2+yβ2+ε2α2ε3sintδ2ε4/ε3+xα2ε4/ε3+yβ1+ε1+ε4sinhtδ2+xα2+yβ2+ε2.Ifτ2⟶∞, the periodic-solitary solution u⟶2α2 at every time. Figure 3 offers the analysis of treatment of periodic and progress of soliton as hyperbolic function with graphs of ϕ4 with the following selected parameters:(33)δ1=1,δ2=2,α1=0.1,α2=0.25,β2=1,h2=1,h3=1.5,ϵ1=1,ϵ2=2,ϵ3=4,ϵ4=2,t=1,in equation (31).Figure 3
Periodic-solitary solution (33) such that (a) 3-D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


## 3.5. Collection V Findings

(34)αi=−δi,i=1,2,β1=β2δ1δ2,h2=δ22h1+δ22h3−β2δ2δ22,ε5=0.Here,δd,ϵj for d=1,2,k=1:4, and β2 are the unknown parameters. By considering the necessary assumption(35)δ2≠0,and by substituting the above parameters into equation (18), we obtain an analytical form of rational equation:(36)u5=2lnϕ5x=2−ε3costδ1−xδ1+yβ2δ1/δ2+ε1δ1−ε4coshtδ2−xδ2+yβ2+ε2δ2ε3sintδ1−xδ1+yβ2δ1/δ2+ε1+ε4sinhtδ2−xδ2+yβ2+ε2.


## 3.6. Set VI Solutions

(37)αi=−δi,i=1,2,β1=δ1h1−h2+h3,β2=δ2h1−h2+h3.Here,δd,ϵk 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(38)u6=2lnϕ6x=2−ε3cosζ1δ1−ε4coshtδ2−xδ2+yδ2h1−h2+h3+ε2δ2ε3sinζ1+ε4sinhtδ2−xδ2+yδ2h1−h2+h3+ε2+ε5,ζ1=tδ1−xδ1+yδ1h1−h2+h3+ε1.


## 4. 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:(39)ϕ=ε3eτ1+ε4e−τ1+ε5cosτ2,τs=αsx+βsy+δst+εs,s=1,2.Afterwards, the valuesαs,βs,δs,εss=1:5 will be found. By making use of equation (39) into equation (17) and taking the coefficients, each powers of eΦ1x,y,t,eΦ2x,y,t, and trigonometric function cosΦx,y,t to zero yield a system of equations (algebraic) (these are not collected here for minimalist) for αs,βs,δs,εss=1:5. These algebraic equations by using the emblematic computation software like, Maple, give the solutions as follows with using u=2lnϕx and v=2lnϕ.


### 4.1. Set I Findings

(40)αl=−δl,βl=δlh1−h2+h3,l=1,2,ϵ3=0.Here,δd,ϵk 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:(41)u=2lnϕ1x=2ε4δ1eζ1+ε5sintδ2−xδ2+yδ2h1−h2+h3+ε2δ2ε4eζ1+ε5costδ2−xδ2+yδ2h1−h2+h3+ε2,ζ1=−tδ1+xδ1−yδ1h1−h2+h3−ε1.Ifτ1⟶∞, the breather outputs u⟶2δ1 at every time.


### 4.2. Set II Solutions

(42)β1=−4α1ε3ε43α15+2α13α22+3α12δ13−α1α24−4α22δ13+α1α22ε52α13−2α1α22+δ13−3α26ε52−α22δ14ε3ε4−ε52α1h2+δ1h3α22δ14ε3ε4−ε52,β2=−8ε3ε43α15+4α13α22+3α12δ13+α1α24−2α22δ13+4α22ε52α13+α1α22+δ13−α22δ1h24ε3ε4−ε52α24ε3ε4−ε52δ1,δ2=0,h1=−43α14ε3ε4+6α12α22ε3ε4+3α1δ13ε3ε4−α24ε3ε4+α24ε52α224ε3ε4−ε52.Here,αd,ϵk for d=1,2,k=1:5, and δ1 are the unknown parameters. By considering the necessary assumption,(43)α24ε3ε4−ε52δ1≠0,and by substituting the above parameters into the equation (36), we obtain an analytical form of rational equation:(44)u2=2lnϕ2x=2ε3α1etδ1+xα1+yβ1+ε1−ε4α1e−tδ1−xα1−yβ1−ε1−ε5sinxα2+yβ2+ε2α2ε3etδ1+xα1+yβ1+ε1+ε4e−tδ1−xα1−yβ1−ε1+ε5cosxα2+yβ2+ε2.Ifτ1⟶∞, the breather outputs u⟶2α1 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 ϕ1 with the following selected parameters:(45)δ1=1,α1=0.1,α2=0.5,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,ε5=3,t=1,in equation (41).Figure 4
Periodic-wave solution (41) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−1,0,1.
(a)(b)


### 4.3. Collection III Outputs

(46)α2=0,β1=−4α1ε3ε43α13δ1+3δ14+3δ12δ22+4δ24−4α1δ24ε52−δ224ε3ε4−ε52α1h2+δ1h3δ224ε3ε4−ε52,β2=4ε3ε43α14+3α1δ13+3α1δ1δ22+δ22h3−δ22h3ε52δ24ε3ε4−ε52,h1=−4ε3ε43δ12+4δ22α13+δ13+δ1δ22−δ22ε52α13+δ13+δ1δ22α1δ224ε3ε4−ε52.Here,δd,ϵk for d=1,2,k=1:5, α1,andβ2 are the unknown parameters. By considering the necessary assumption,(47)α1δ224ε3ε4−ε52≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(48)u3=2lnϕ3x=2ε3α1etδ1+xα1+yβ1+ε1−ε4α1e−tδ1−xα1−yβ1−ε1ε3etδ1+xα1+yβ1+ε1+ε4e−tδ1−xα1−yβ1−ε1+ε5costδ2+xα2+yβ2+ε2.Ifτ1⟶∞, the periodic outputs u⟶2α1 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 ϕ3 with the following selected parameters:(49)δ1=1,δ−2=2,α1=0.1,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,ε5=3,t=1,in equation (48).Figure 5
Periodic-wave solution (48) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


### 4.4. Set IV Solutions

(50)α1=α2δ1δ2,β1=−δ12α24δ12+2α24δ22+2α2δ12δ23+2α2δ25−β2δ23δ24,h2=−3α24δ12−α24δ22+3α2δ12δ23−α2δ25+α22δ22h1+δ24h3−β2δ23α2δ23,ε3=−14δ22δ52δ12ε4.Here,δd,ϵk for d=1,2,k=1:5, α2,andβ2 are the unknown parameters. By considering the necessary assumption,(51)α2δ2≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(52)u4=2lnϕ4x=2−1/4δ22ε52α1/δ12ε4etδ1+xα2δ1/δ2+yβ1+ε1−ε4α1e−tδ1−xα2δ1/δ2−yβ1−ε1−ε5sintδ2+xα2+yβ2+ε2α2−1/4δ22ε52/δ12ε4etδ1+xα2δ1/δ2+yβ1+ε1+ε4e−tδ1−xα2δ1/δ2−yβ1−ε1+ε5costδ2+xα2+yβ2+ε2..Ifτ1⟶∞, the breather outputs u⟶2α2δ1/δ2 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 ϕ4 with the following selected parameters:(53)δ1=1,δ2=0.5,α2=0.5,β2=0.2,h1=1,h3=2,ε1=1,ε2=0.1,ε4=2,ε5=3,t=0.5,in equation (48).Figure 6
Periodic-wave solution (48) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


### 4.5. Collection V Outputs

(54)αl=−δl,l=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2.Here,δd,ϵk for d=1,2,k=1:5, and β2 are the unknown parameters. By considering the necessary assumption,(55)δ2≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(56)u5=2lnϕ5x=2−ε3δ1etδ1−xδ1+yβ2δ1/δ2+ε1+ε4δ1e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε5sintδ2−xδ2+yβ2+ε2δ2ε3etδ1−xδ1+yβ2δ1/δ2+ε1+ε4e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε5costδ2−xδ2+yβ2+ε2.


### 4.6. Set VI Solutions

(57)α1=θ,α2=0,β1=−4θδ23−θδ2h2−2δ1δ2h3+β2δ1δ2,h1=−−δ12δ2h3−δ23h3+β2δ12+β2δ22θ2δ2,ε3=14ε52ε4.Here,θ=3−δ14−δ1δ22,δd,εk for d=1,2,k=1:5 and β2 are the unknown parameters. By considering the necessary assumption,(58)δ2θ≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(59)u6=2lnϕ6x=21/4ε52θetδ1+xα1+yβ1+ε1/ε4−ε4α1e−tδ1−xα1−yβ1−ε11/4ε52etδ1+xα1+yβ1+ε1/ε4+ε4e−tδ1−xα1−yβ1−ε1+ε5costδ2+yβ2+ε2.


### 4.7. Set VII Solutions

(60)β1=−8α12α22+θ2h3+α1θh2+α12h1−α22h1θ,β2=α2−12α12α22−4α24+α1θh2+2α12h1α1θ,δ1=θ,δ2=0,θ=3−α14−2α12α22−α24α12α1,ε3=14ε52ε4.Here,αd,ϵk for d=1,2,k=1:5 are the unknown parameters. By considering the necessary assumption,(61)α1θ≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(62)u7=2lnϕ7x=21/4ε52α1etδ1+xα1+yβ1+ε1/ε4−ϵ4α1e−tθ−xα1−yβ1−ϵ1−ϵ5sinxα2+yβ2+ϵ2α21/4ε52etδ1+xα1+yβ1+ε1/ε4+ϵ4e−tθ−xα1−yβ1−ϵ1+ϵ5cosxα2+yβ2+ϵ2.


## 4.1. Set I Findings

(40)αl=−δl,βl=δlh1−h2+h3,l=1,2,ϵ3=0.Here,δd,ϵk 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:(41)u=2lnϕ1x=2ε4δ1eζ1+ε5sintδ2−xδ2+yδ2h1−h2+h3+ε2δ2ε4eζ1+ε5costδ2−xδ2+yδ2h1−h2+h3+ε2,ζ1=−tδ1+xδ1−yδ1h1−h2+h3−ε1.Ifτ1⟶∞, the breather outputs u⟶2δ1 at every time.


## 4.2. Set II Solutions

(42)β1=−4α1ε3ε43α15+2α13α22+3α12δ13−α1α24−4α22δ13+α1α22ε52α13−2α1α22+δ13−3α26ε52−α22δ14ε3ε4−ε52α1h2+δ1h3α22δ14ε3ε4−ε52,β2=−8ε3ε43α15+4α13α22+3α12δ13+α1α24−2α22δ13+4α22ε52α13+α1α22+δ13−α22δ1h24ε3ε4−ε52α24ε3ε4−ε52δ1,δ2=0,h1=−43α14ε3ε4+6α12α22ε3ε4+3α1δ13ε3ε4−α24ε3ε4+α24ε52α224ε3ε4−ε52.Here,αd,ϵk for d=1,2,k=1:5, and δ1 are the unknown parameters. By considering the necessary assumption,(43)α24ε3ε4−ε52δ1≠0,and by substituting the above parameters into the equation (36), we obtain an analytical form of rational equation:(44)u2=2lnϕ2x=2ε3α1etδ1+xα1+yβ1+ε1−ε4α1e−tδ1−xα1−yβ1−ε1−ε5sinxα2+yβ2+ε2α2ε3etδ1+xα1+yβ1+ε1+ε4e−tδ1−xα1−yβ1−ε1+ε5cosxα2+yβ2+ε2.Ifτ1⟶∞, the breather outputs u⟶2α1 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 ϕ1 with the following selected parameters:(45)δ1=1,α1=0.1,α2=0.5,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,ε5=3,t=1,in equation (41).Figure 4
Periodic-wave solution (41) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−1,0,1.
(a)(b)


## 4.3. Collection III Outputs

(46)α2=0,β1=−4α1ε3ε43α13δ1+3δ14+3δ12δ22+4δ24−4α1δ24ε52−δ224ε3ε4−ε52α1h2+δ1h3δ224ε3ε4−ε52,β2=4ε3ε43α14+3α1δ13+3α1δ1δ22+δ22h3−δ22h3ε52δ24ε3ε4−ε52,h1=−4ε3ε43δ12+4δ22α13+δ13+δ1δ22−δ22ε52α13+δ13+δ1δ22α1δ224ε3ε4−ε52.Here,δd,ϵk for d=1,2,k=1:5, α1,andβ2 are the unknown parameters. By considering the necessary assumption,(47)α1δ224ε3ε4−ε52≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(48)u3=2lnϕ3x=2ε3α1etδ1+xα1+yβ1+ε1−ε4α1e−tδ1−xα1−yβ1−ε1ε3etδ1+xα1+yβ1+ε1+ε4e−tδ1−xα1−yβ1−ε1+ε5costδ2+xα2+yβ2+ε2.Ifτ1⟶∞, the periodic outputs u⟶2α1 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 ϕ3 with the following selected parameters:(49)δ1=1,δ−2=2,α1=0.1,h2=1,h3=2,ε1=1,ε2=2,ε3=4,ε4=2,ε5=3,t=1,in equation (48).Figure 5
Periodic-wave solution (48) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


## 4.4. Set IV Solutions

(50)α1=α2δ1δ2,β1=−δ12α24δ12+2α24δ22+2α2δ12δ23+2α2δ25−β2δ23δ24,h2=−3α24δ12−α24δ22+3α2δ12δ23−α2δ25+α22δ22h1+δ24h3−β2δ23α2δ23,ε3=−14δ22δ52δ12ε4.Here,δd,ϵk for d=1,2,k=1:5, α2,andβ2 are the unknown parameters. By considering the necessary assumption,(51)α2δ2≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(52)u4=2lnϕ4x=2−1/4δ22ε52α1/δ12ε4etδ1+xα2δ1/δ2+yβ1+ε1−ε4α1e−tδ1−xα2δ1/δ2−yβ1−ε1−ε5sintδ2+xα2+yβ2+ε2α2−1/4δ22ε52/δ12ε4etδ1+xα2δ1/δ2+yβ1+ε1+ε4e−tδ1−xα2δ1/δ2−yβ1−ε1+ε5costδ2+xα2+yβ2+ε2..Ifτ1⟶∞, the breather outputs u⟶2α2δ1/δ2 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 ϕ4 with the following selected parameters:(53)δ1=1,δ2=0.5,α2=0.5,β2=0.2,h1=1,h3=2,ε1=1,ε2=0.1,ε4=2,ε5=3,t=0.5,in equation (48).Figure 6
Periodic-wave solution (48) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−2,0,2.
(a)(b)


## 4.5. Collection V Outputs

(54)αl=−δl,l=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2.Here,δd,ϵk for d=1,2,k=1:5, and β2 are the unknown parameters. By considering the necessary assumption,(55)δ2≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(56)u5=2lnϕ5x=2−ε3δ1etδ1−xδ1+yβ2δ1/δ2+ε1+ε4δ1e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε5sintδ2−xδ2+yβ2+ε2δ2ε3etδ1−xδ1+yβ2δ1/δ2+ε1+ε4e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε5costδ2−xδ2+yβ2+ε2.


## 4.6. Set VI Solutions

(57)α1=θ,α2=0,β1=−4θδ23−θδ2h2−2δ1δ2h3+β2δ1δ2,h1=−−δ12δ2h3−δ23h3+β2δ12+β2δ22θ2δ2,ε3=14ε52ε4.Here,θ=3−δ14−δ1δ22,δd,εk for d=1,2,k=1:5 and β2 are the unknown parameters. By considering the necessary assumption,(58)δ2θ≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(59)u6=2lnϕ6x=21/4ε52θetδ1+xα1+yβ1+ε1/ε4−ε4α1e−tδ1−xα1−yβ1−ε11/4ε52etδ1+xα1+yβ1+ε1/ε4+ε4e−tδ1−xα1−yβ1−ε1+ε5costδ2+yβ2+ε2.


## 4.7. Set VII Solutions

(60)β1=−8α12α22+θ2h3+α1θh2+α12h1−α22h1θ,β2=α2−12α12α22−4α24+α1θh2+2α12h1α1θ,δ1=θ,δ2=0,θ=3−α14−2α12α22−α24α12α1,ε3=14ε52ε4.Here,αd,ϵk for d=1,2,k=1:5 are the unknown parameters. By considering the necessary assumption,(61)α1θ≠0,and by substituting the above parameters into equation (39), we obtain an analytical form of rational equation:(62)u7=2lnϕ7x=21/4ε52α1etδ1+xα1+yβ1+ε1/ε4−ϵ4α1e−tθ−xα1−yβ1−ϵ1−ϵ5sinxα2+yβ2+ϵ2α21/4ε52etδ1+xα1+yβ1+ε1/ε4+ϵ4e−tθ−xα1−yβ1−ϵ1+ϵ5cosxα2+yβ2+ϵ2.


## 5. Cross-Kink Wave Solutions

In this segment, we utilize to formulate the new exact solutions to the (2 + 1)-dimensional generalized Hietarinta equation. Consider the following function for studying the cross-kink wave solutions as(63)ϕ=e−τ1+ε4eτ1+ε5sinτ2+ε6sinhτ3,τs=αsx+βsy+δst+εs,s=1:3.Afterwards, the valuesαs,βs,δs,ϵss=1:3 will be found. By making use of equation (63) into (17) and taking the coefficients, each powers of eΦx,y,t, sinx,y,t, and sinhx,y,t to zero yield a system of equations (algebraic) (these are not collected here for minimalist) for αs,βs,δs,ϵss=1:3. These algebraic equations by using the emblematic computation software like, Maple, give the following solutions with using u=2lnϕx and v=2lnϕ.


### 5.1. Set I Solutions

(64)β1=α32ε62α14+2α12α32+α1δ13−3α34+4α1ε43α15−2α13α32+3α12δ13−α1α34+4α32δ13+α32δ1ε62+4ε4α1h2+δ1h3α32δ1ε62+4ε4,β3=4α32ε62α13−α1α32+δ13+8ε43α15−4α13α32+3α12δ13+α1α34+2α32δ13+α32δ1h2ε62+4ε4α3ε62+4ε4δ1,δ3=0,h1=4−α34ε62+ε43α14−6α12α32+3α1δ13−α34α32ε62+4ε4,ε5=0.Here,αd,ϵk for d=1:2,k=1:6, β2,δ1,andδ2 are the unknown parameters. By considering the necessary assumption,(65)α32δ1ε62+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(66)u1=2lnϕ1x=2−α1e−ζ1+ε4α1eζ1+ε6coshxα3+yβ3+ε3α3e−ζ1+ε4eζ1+ε6sinhxα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ1⟶∞, the breather outputs u⟶2α3 at every time.


### 5.2. Set II Solutions

(67)α3=ε5=0,β1=4α1δ34ε62+4α1ε43α13δ1+3δ14−3δ12δ32+4δ34+δ32ε62+4ε4α1h2+δ1h3δ32ε62+4ε4,β3=−12α1ε4α13+δ13−δ1δ32−δ32h3ε62+4ε4δ3ε62+4ε4,h1=−δ32ε62α13+δ13−δ1δ32+4ε43δ12−4δ32α13+δ13−δ1δ32α1δ32ε62+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α1,α2,andβ2 are the unknown parameters. By considering the necessary assumption,(68)α1δ32ε62+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(69)u2=2lnϕ2x=2−α1e−ζ1+ε4α1eζ1e−ζ1+ε4eζ1+ε6sinhtδ3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶0 at every t, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


### 5.3. Set III Solutions

(70)β1=8α34+8α3δ33+2α32h1+2α3δ3h2+2δ32h3−β3δ3δ3,ε4=−14ε62,ε5=0.Hereαd,δd,ϵk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(71)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(72)u3=2lnϕ3x=2−α1e−tδ1−xα1−yβ1−ε1−1/4ε62α1etδ1+xα1+yβ1+ε1+ε6coshtδ3+xα3+yβ3+ε3α3e−tδ1−xα1−yβ1−ε1−1/4ε62etδ1+xα1+yβ1+ε1+ε6sinhtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any t.


### 5.4. Set IV Solutions

(73)α1=θ,α3=0,β1=4θδ33+θδ3h2+2δ1δ3h3−β3δ1δ3,h1=−−δ12δ3h3+δ33h3+β3δ12−β3δ32θ2δ3,ε4=−14ε62,ε5=0.Here,θ=3−δ13+δ1δ32,αd,δd,εk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(74)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(75)u4=2lnϕ4x=2−α1e−ζ1−1/4ε62α1eζ1+ε6coshtδ3+xα3+yβ3+ε3α3e−ζ1−1/4ε62eζ1+ε6sinhtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


### 5.5. Set V Solutions

(76)α1=θ,α3=0,β1=4θδ33+θδ3h2+2δ1δ3h3−β3δ1δ3,h1=−−δ12δ3h3+δ33h3+β3δ12−β3δ32θ2δ3,ε4=−14ε62,ε5=0.Here,θ=3−δ13+δ1δ32,αd,δd,εk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(77)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(78)u5=2lnϕ5x=2−α1e−ζ1−1/4ε62α1eζ1+ε6coshtδ3+xα3+yβ3+ε3α3e−ζ1−1/4ε62eζ1+ε6sinhtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any time. Figure 7 show the analysis of treatment of cross-kink wave as periodic and hyperbolic function with graphs of ϕ5 with the following selected parameters:(79)δ1=1,δ3=0.5,α1=1,α2=2,α3=3,β3=0.2,h2=1,h3=2,ε1=1,ε2=0.1,ε3=2,ε4=3,ε5=2,t=0.5,in equation (74).Figure 7
Cross-kink solution (78) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−0.2,0,0.2.
(a)(b)


### 5.6. Set VI Solutions

(80)α2=ε6=0,β1=−−4α1δ24ε52+4α1ε43α13δ1+3δ14+3δ12δ22+4δ24−δ22−ε52+4ε4α1h2+δ1h3δ22−ε53+4ε4,β2=12α1ε4α13+δ13+δ1δ22−δ22h3ε52+4δ22h3ε4δ2−ε53+4ε4,h1=−−δ22ε52α13+δ13+δ1δ22+4ε43δ12+4δ22α13+δ13+δ1δ22α1δ22−ε52+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α1,α3,andβ3 are the unknown parameters. By considering the necessary assumption,(81)α1δ22−ε52+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(82)u6=2lnf6x=2−α1e−tδ1−xα1−yβ1−ε1+ε4α1etδ1+xα1+yβ1+ε1e−tδ1−xα1−yβ1−ε1+ε4etδ1+xα1+yβ1+ε1+ε5sintδ2+yβ2+ε2.Ifτ2>τ1⟶∞, the cross-kink outputs u⟶0 at every time, but if τ2<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


### 5.7. Set VII Solutions

(83)α1=α2δ1δ2,β1=−δ12α24δ12+2α24δ22+2α2δ12δ23+2α2δ25−β2δ23δ24,h2=−3α24δ12−α24δ22+3α2δ12δ23−α2δ25+α22δ22h1+δ24h3−β2δ23α2δ23,ε4=−14δ22ε52δ12.Hereδd,ϵk for d=1:3,k=1:6, α2,α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(84)α2δ2≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(85)u7=2lnf7x=2−α1e−tδ1−xα2δ1/δ2−yβ1−ε1−1/4δ2ε52α2/δ1etδ1+xα2δ1/δ2+yβ1+ε1+ε5costδ2+xα2+yβ2+ε2α2e−tδ1−xα2δ1/δ2−yβ1−ε1−1/4δ22ε52/δ12etδ1+xα2δ1/δ2+yβ1+ε1+ε5sintδ2+xα2+yβ2+ε2.Ifτ1⟶∞, the cross-kink wave outputs u⟶2α2δ1/δ2 at any time.


### 5.8. Set VIII Solutions

(86)αl=−δl,l=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2,ε6=0,θ=3−α14−2α12α22−α24α12α1.Hereδd,ϵk for d=1:3,k=1:5, α3,β2,andβ3 are arbitrary inputs, and the following case is considered as(87)δ2≠0.By substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(88)u8=2lnϕ8x=2−α1e−tθ−xα1−yβ1−ε1+1/4ε52α1etθ+xα1+yβ1+ε1+ε5cosxα2+yβ2+ε2α2e−tθ−xα1−yβ1−ε1+1/4ε52etθ+xα1+yβ1+ε1+ε5sinxα2+yβ2+ε2.Ifτ1⟶∞, the cross-kink wave outputs u⟶2α1 at any time.


## 5.1. Set I Solutions

(64)β1=α32ε62α14+2α12α32+α1δ13−3α34+4α1ε43α15−2α13α32+3α12δ13−α1α34+4α32δ13+α32δ1ε62+4ε4α1h2+δ1h3α32δ1ε62+4ε4,β3=4α32ε62α13−α1α32+δ13+8ε43α15−4α13α32+3α12δ13+α1α34+2α32δ13+α32δ1h2ε62+4ε4α3ε62+4ε4δ1,δ3=0,h1=4−α34ε62+ε43α14−6α12α32+3α1δ13−α34α32ε62+4ε4,ε5=0.Here,αd,ϵk for d=1:2,k=1:6, β2,δ1,andδ2 are the unknown parameters. By considering the necessary assumption,(65)α32δ1ε62+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(66)u1=2lnϕ1x=2−α1e−ζ1+ε4α1eζ1+ε6coshxα3+yβ3+ε3α3e−ζ1+ε4eζ1+ε6sinhxα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ1⟶∞, the breather outputs u⟶2α3 at every time.


## 5.2. Set II Solutions

(67)α3=ε5=0,β1=4α1δ34ε62+4α1ε43α13δ1+3δ14−3δ12δ32+4δ34+δ32ε62+4ε4α1h2+δ1h3δ32ε62+4ε4,β3=−12α1ε4α13+δ13−δ1δ32−δ32h3ε62+4ε4δ3ε62+4ε4,h1=−δ32ε62α13+δ13−δ1δ32+4ε43δ12−4δ32α13+δ13−δ1δ32α1δ32ε62+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α1,α2,andβ2 are the unknown parameters. By considering the necessary assumption,(68)α1δ32ε62+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(69)u2=2lnϕ2x=2−α1e−ζ1+ε4α1eζ1e−ζ1+ε4eζ1+ε6sinhtδ3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶0 at every t, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


## 5.3. Set III Solutions

(70)β1=8α34+8α3δ33+2α32h1+2α3δ3h2+2δ32h3−β3δ3δ3,ε4=−14ε62,ε5=0.Hereαd,δd,ϵk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(71)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(72)u3=2lnϕ3x=2−α1e−tδ1−xα1−yβ1−ε1−1/4ε62α1etδ1+xα1+yβ1+ε1+ε6coshtδ3+xα3+yβ3+ε3α3e−tδ1−xα1−yβ1−ε1−1/4ε62etδ1+xα1+yβ1+ε1+ε6sinhtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any t.


## 5.4. Set IV Solutions

(73)α1=θ,α3=0,β1=4θδ33+θδ3h2+2δ1δ3h3−β3δ1δ3,h1=−−δ12δ3h3+δ33h3+β3δ12−β3δ32θ2δ3,ε4=−14ε62,ε5=0.Here,θ=3−δ13+δ1δ32,αd,δd,εk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(74)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(75)u4=2lnϕ4x=2−α1e−ζ1−1/4ε62α1eζ1+ε6coshtδ3+xα3+yβ3+ε3α3e−ζ1−1/4ε62eζ1+ε6sinhtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


## 5.5. Set V Solutions

(76)α1=θ,α3=0,β1=4θδ33+θδ3h2+2δ1δ3h3−β3δ1δ3,h1=−−δ12δ3h3+δ33h3+β3δ12−β3δ32θ2δ3,ε4=−14ε62,ε5=0.Here,θ=3−δ13+δ1δ32,αd,δd,εk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(77)δ3≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(78)u5=2lnϕ5x=2−α1e−ζ1−1/4ε62α1eζ1+ε6coshtδ3+xα3+yβ3+ε3α3e−ζ1−1/4ε62eζ1+ε6sinhtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any time. Figure 7 show the analysis of treatment of cross-kink wave as periodic and hyperbolic function with graphs of ϕ5 with the following selected parameters:(79)δ1=1,δ3=0.5,α1=1,α2=2,α3=3,β3=0.2,h2=1,h3=2,ε1=1,ε2=0.1,ε3=2,ε4=3,ε5=2,t=0.5,in equation (74).Figure 7
Cross-kink solution (78) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−0.2,0,0.2.
(a)(b)


## 5.6. Set VI Solutions

(80)α2=ε6=0,β1=−−4α1δ24ε52+4α1ε43α13δ1+3δ14+3δ12δ22+4δ24−δ22−ε52+4ε4α1h2+δ1h3δ22−ε53+4ε4,β2=12α1ε4α13+δ13+δ1δ22−δ22h3ε52+4δ22h3ε4δ2−ε53+4ε4,h1=−−δ22ε52α13+δ13+δ1δ22+4ε43δ12+4δ22α13+δ13+δ1δ22α1δ22−ε52+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α1,α3,andβ3 are the unknown parameters. By considering the necessary assumption,(81)α1δ22−ε52+4ε4≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(82)u6=2lnf6x=2−α1e−tδ1−xα1−yβ1−ε1+ε4α1etδ1+xα1+yβ1+ε1e−tδ1−xα1−yβ1−ε1+ε4etδ1+xα1+yβ1+ε1+ε5sintδ2+yβ2+ε2.Ifτ2>τ1⟶∞, the cross-kink outputs u⟶0 at every time, but if τ2<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


## 5.7. Set VII Solutions

(83)α1=α2δ1δ2,β1=−δ12α24δ12+2α24δ22+2α2δ12δ23+2α2δ25−β2δ23δ24,h2=−3α24δ12−α24δ22+3α2δ12δ23−α2δ25+α22δ22h1+δ24h3−β2δ23α2δ23,ε4=−14δ22ε52δ12.Hereδd,ϵk for d=1:3,k=1:6, α2,α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(84)α2δ2≠0,and by substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(85)u7=2lnf7x=2−α1e−tδ1−xα2δ1/δ2−yβ1−ε1−1/4δ2ε52α2/δ1etδ1+xα2δ1/δ2+yβ1+ε1+ε5costδ2+xα2+yβ2+ε2α2e−tδ1−xα2δ1/δ2−yβ1−ε1−1/4δ22ε52/δ12etδ1+xα2δ1/δ2+yβ1+ε1+ε5sintδ2+xα2+yβ2+ε2.Ifτ1⟶∞, the cross-kink wave outputs u⟶2α2δ1/δ2 at any time.


## 5.8. Set VIII Solutions

(86)αl=−δl,l=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2,ε6=0,θ=3−α14−2α12α22−α24α12α1.Hereδd,ϵk for d=1:3,k=1:5, α3,β2,andβ3 are arbitrary inputs, and the following case is considered as(87)δ2≠0.By substituting the above parameters into equation (63), we obtain an analytical form of rational equation:(88)u8=2lnϕ8x=2−α1e−tθ−xα1−yβ1−ε1+1/4ε52α1etθ+xα1+yβ1+ε1+ε5cosxα2+yβ2+ε2α2e−tθ−xα1−yβ1−ε1+1/4ε52etθ+xα1+yβ1+ε1+ε5sinxα2+yβ2+ε2.Ifτ1⟶∞, the cross-kink wave outputs u⟶2α1 at any time.


## 6. Interaction between Stripe and Periodic Wave Solutions

In this paragraph, we find out some advanced exact interaction between stripe and periodic wave solutions to the (2 + 1)-dimensional generalized Hietarinta equation. Assume the stated function for studying the interaction of solutions as(89)ϕ=e−τ1+ε4eτ1+ε5cosτ2+ε6coshτ3,τs=αsx+βsy+δst+εs,s=1:3.Afterwards, the valuesαs,βs,δs,ϵss=1:3 will be found. By making use of equation (18) into (17) and taking the coefficients, each powers of cosx,y,t and coshx,y,t and exponential function to zero yield a system of equations (algebraic) (these are not collected here for minimalist) for αs,βs,δs,ϵss=1:3. These algebraic equations by using the emblematic computation software like, Maple, give the following solutions with using u=2lnϕx and v=2lnϕ.


### 6.1. Set I Solutions

(90)β1=−α32ε62α13+2α12α32+α1δ13−3α34+4α1ε43α15−2α13α32+3α12δ13−α1α34+4α32δ13+α32δ1−ε62+4ε4α1h2+δ1h3α32δ1−ε62+4ε4,β3=−4α32ε62α13−α1α32+δ13+8ε43α15−4α13α32+3α12δ13+α1α34+2α32δ13+α32δ1h2−ε62+4ε4α3−ε62+4ε4δ1,δ3=0,h1=4α34ε62+3α14ε4−6α12α32ε4+3α1δ13ε4−α34ε4α32−ε62+4ε4,ε5=0.Here,αd,ϵk for d=1:2,k=1:6, β2,δ1,andδ2 are the unknown parameters. By considering the necessary assumption,(91)α32δ1ε62+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(92)u1=2lnϕ1x=2−α1e−ζ1+ε4α1eζ1+ε6sinhxα3+yβ3+ε3α3e−ζ1+ε4eζ1+ε6coshxα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ1⟶∞, the breather outputs u⟶2α3 at very time.


### 6.2. Set II Solutions

(93)α3=ε5=0,β1=−4α1δ34ε62+4α1ε43α13δ1+3δ14−3δ12δ32+4δ34+δ32−ε62+4ε4α1h2+δ1h3−ε62+4ε4δ32,β3=−12α14ε4+12α1δ13ε4−12α1δ1δ32ε4+δ32h3ε62−4δ32h3ε4δ3−ε62+4ε4,h1=δ32ε62α13+δ13−δ1δ32+4ε43δ12−4δ32α14+δ13−δ1δ32α1δ32−ε62+4ε4.Hereδd,ϵk for d=1:3,k=1:6, α1,α2,β2 are the unknown parameters. By considering the necessary assumption,(94)α1δ32ε62+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(95)u2=2lnϕ2x=2−α1e−ζ1+ε4α1eζ1e−ζ1+ε4eζ1+ε6coshtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the interaction between stripe and periodic wave outputs u⟶0 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


### 6.3. Set III Solutions

(96)β1=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,β3=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,ε5=0.Here,αd,δd,ϵk for d=1:3,k=1:6, β2, andβ3 are the unknown parameters. By considering the necessary assumption,(97)δ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(98)u3=2lnϕ3x=2−α1e−ζ1+ε4α1eζ1+ε6sinhtδ3+xα3+yβ3+ε3α3e−tδ1−xα1−yβ1−ε1+ε4eζ1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any time.


### 6.4. Set IV Solutions

(99)α1=−α3,β1=2δ3h3−β3,h1=−4α32,ε4=14ε62,ε5=0.Here,δd,ϵk for d=1:3,k=1:6, α2,α3,β2,andβ3 are the unknown parameters, and by considering the necessary assumption and also by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(100)u4=2lnϕ4x=2−α3e−tδ1+xα3−y2δ3h3−β3−ε1−1/4α3ε62etδ1−xα3+y2δ3h3−β3+ε1+ε6sinhtδ3+xα3+yβ3+ε3α3etδ1−xα3+y2δ3h3−β3+ε1+1/4ε62etδ1−xα3+y2δ3h3−β3+ε1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the interaction of outputs u⟶2α3 at any t, but if τ3<τ1⟶∞, the the interaction of outputs u⟶2α1 at any time.


### 6.5. Set V Solutions

(101)α1=−α3,β1=−4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,β3=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,δ1=−δ3,ε5=0.Here,αd,δd,ϵk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(102)δ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation equation:(103)u5=2lnϕ5x=2−α1etδ3+xα3−yβ1−ε1+ε4α1e−tδ3−xα3+yβ1+ε1+ε6sinhtδ3+xα3+yβ3+ε3α3etδ3+xα3−yβ1−ε1+ε4e−tδ3−xα3+yβ1+ε1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the interaction of solution u⟶2α3 at any t, but if τ3<τ1⟶∞, the interaction of solution u⟶2α1 at any time.


### 6.6. Set VI Solutions

(104)α2=ε6=0,β1=−−4α1δ24ε52+4α1ε43α13δ1+3δ14+3δ12δ22+4δ24−δ22−ε52+4ε4α1h2+δ1h3δ22−ε52+4ε4,β2=12α1ε4α13+δ13+δ1δ22+δ22h3−ε52+4ε4δ2−ε52+4ε4,h1=−−δ22ε52α13+δ13+δ1δ22+4ε4312+4δ22α13+δ13+δ1δ22α1δ22−ε52+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α2,α3,andβ3 are the unknown parameters. By considering the necessary assumption,(105)α1δ22−ε52+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(106)u6=2lnϕ6x=2−α1e−tδ1−xα1−yβ1−ε1+ε4α1etδ1+xα1+yβ1+ε1e−tδ1−xα1−yβ1−ε1+ε4etδ1+xα1+yβ1+ε1+ε5costδ2+yβ2+ε2.Ifτ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 8 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ6 with the following selected parameters:(107)δ1=0.3,δ2=2,δ3=1,α1=0.1,α3=0.5,β3=1,h2=2,h3=3,ε1=1,ε2=2,ε3=4,ε4=2,ε5=1,t=0.1,in equation (102).Figure 8
Interaction between stripe and periodic wave solution (107) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−3,0,3.
(a)(b)


### 6.7. Set VII Solutions

(108)α1=θ,α2=ε6=0,β1=−4θδ23−θδ2h2−2δ1δ2h3+β2δ1δ2,h1=−−δ12δ2h3−δ23h3+β2δ12+β2δ22θ2δ2,ε4=14ε52.Here,θ=3−δ13−δ1δ22,δd,ϵk for d=1:3,k=1:3, α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(109)δ2≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(110)u7=2lnϕ7x=2−α1e−tδ1−x3−δ13−δ1δ22−yβ1−ε1+1/4ε52α1etδ1+x3−δ13−δ1δ22+yβ1+ε1e−tδ1−x3−δ13−δ1δ22−yβ1−ε1+1/4ε52etδ1+x3−δ13−δ1δ22+yβ1+ε1+ε5costδ2+yβ2+ε2.Ifτ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 9 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ7 with the following selected parameters:(111)δ1=−1.2,δ2=1,δ3=1,α3=0.5,β3=1,h2=2,h3=3,ε1=1,ε2=0.2,ε3=4,ε5=1,t=0.01,in equation (106).Figure 9
Interaction between stripe and periodic wave solution (111) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,y=−1,0,1.
(a)(b)


### 6.8. Set VIII Solutions

(112)αi=−δi,i=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2,ε6=0.Here,δd,ϵk for d=1:3,k=1:5, α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(113)δ2≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(114)u8=2lnϕ8x=2δ1e−tδ1+xδ1−yβ2δ1/δ2−ε1−ε4δ1etδ1−xδ1+yβ2δ1/δ2+ε1+ε5sintδ2−xδ2+yβ2+ε2δ2e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε4etδ1−xδ1+yβ2δ1/δ2+ε1+ε5costδ2−xδ2+yβ2+ε2.


### 6.9. Set IX Solutions

(115)α2=θ,β1=−4α36+4α33δ33+θ24α34−4α3δ33−α3δ3h2−δ32h3θ2δ3,β2=−4α35+4α32δ33+θ24α33−4δ33−δ3h2θδ3,β3=−4α36+4α33δ33+θ24α34−4α3δ33−α3δ3h2−δ32h3θ2δ3,δ2=0,h1=−4α32α3θ2+α33+δ33θ2,α1=α3,δ1=δ3.Here,ϵk for k=1:6, α3,andδ3 are free values. Also, θ solves the θ4+2θ2α32+α34+α3δ33=0. By considering the necessary assumption,(116)θδ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(117)u9=2−α3e−tδ3−xα3−yβ1−ε1+ε4α3etδ3+xα3+yβ1+ε1−ε5sinxθ+yβ2+ε2θ+ε6sinhtδ3+xα3+yβ3+ε3α3e−tδ3−xα3−yβ1−ε1+ε4etδ3+xα3+yβ1+ε1+ε5cosxθ+yβ2+ε2+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the the interaction between stripe and periodic wave solution u⟶2α3 at any t, but if τ3<τ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 10 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ9 with the following selected parameters:(118)δ1=−1.2,δ2=1,δ3=3,α3=−0.5,β2=1,β3=1,h2=2,h3=3,ε1=1,ε2=0.1,ε3=3,ε4=4,ε5=5,ε6=4,t=1,in equation (117).Figure 10
Interaction between stripe and periodic wave solution (117) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−3,0,3.
(a)(b)


## 6.1. Set I Solutions

(90)β1=−α32ε62α13+2α12α32+α1δ13−3α34+4α1ε43α15−2α13α32+3α12δ13−α1α34+4α32δ13+α32δ1−ε62+4ε4α1h2+δ1h3α32δ1−ε62+4ε4,β3=−4α32ε62α13−α1α32+δ13+8ε43α15−4α13α32+3α12δ13+α1α34+2α32δ13+α32δ1h2−ε62+4ε4α3−ε62+4ε4δ1,δ3=0,h1=4α34ε62+3α14ε4−6α12α32ε4+3α1δ13ε4−α34ε4α32−ε62+4ε4,ε5=0.Here,αd,ϵk for d=1:2,k=1:6, β2,δ1,andδ2 are the unknown parameters. By considering the necessary assumption,(91)α32δ1ε62+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(92)u1=2lnϕ1x=2−α1e−ζ1+ε4α1eζ1+ε6sinhxα3+yβ3+ε3α3e−ζ1+ε4eζ1+ε6coshxα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ1⟶∞, the breather outputs u⟶2α3 at very time.


## 6.2. Set II Solutions

(93)α3=ε5=0,β1=−4α1δ34ε62+4α1ε43α13δ1+3δ14−3δ12δ32+4δ34+δ32−ε62+4ε4α1h2+δ1h3−ε62+4ε4δ32,β3=−12α14ε4+12α1δ13ε4−12α1δ1δ32ε4+δ32h3ε62−4δ32h3ε4δ3−ε62+4ε4,h1=δ32ε62α13+δ13−δ1δ32+4ε43δ12−4δ32α14+δ13−δ1δ32α1δ32−ε62+4ε4.Hereδd,ϵk for d=1:3,k=1:6, α1,α2,β2 are the unknown parameters. By considering the necessary assumption,(94)α1δ32ε62+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(95)u2=2lnϕ2x=2−α1e−ζ1+ε4α1eζ1e−ζ1+ε4eζ1+ε6coshtδ3+xα3+yβ3+ε3,ζ1=tδ1+xα1+yβ1+ε1.Ifτ3>τ1⟶∞, the interaction between stripe and periodic wave outputs u⟶0 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at every time.


## 6.3. Set III Solutions

(96)β1=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,β3=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,ε5=0.Here,αd,δd,ϵk for d=1:3,k=1:6, β2, andβ3 are the unknown parameters. By considering the necessary assumption,(97)δ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(98)u3=2lnϕ3x=2−α1e−ζ1+ε4α1eζ1+ε6sinhtδ3+xα3+yβ3+ε3α3e−tδ1−xα1−yβ1−ε1+ε4eζ1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the cross-kink outputs u⟶2α3 at every time, but if τ3<τ1⟶∞, the cross-kink outputs u⟶2α1 at any time.


## 6.4. Set IV Solutions

(99)α1=−α3,β1=2δ3h3−β3,h1=−4α32,ε4=14ε62,ε5=0.Here,δd,ϵk for d=1:3,k=1:6, α2,α3,β2,andβ3 are the unknown parameters, and by considering the necessary assumption and also by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(100)u4=2lnϕ4x=2−α3e−tδ1+xα3−y2δ3h3−β3−ε1−1/4α3ε62etδ1−xα3+y2δ3h3−β3+ε1+ε6sinhtδ3+xα3+yβ3+ε3α3etδ1−xα3+y2δ3h3−β3+ε1+1/4ε62etδ1−xα3+y2δ3h3−β3+ε1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the interaction of outputs u⟶2α3 at any t, but if τ3<τ1⟶∞, the the interaction of outputs u⟶2α1 at any time.


## 6.5. Set V Solutions

(101)α1=−α3,β1=−4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,β3=4α34+4α3δ33+α32h1+α3δ3h2+δ32h3δ3,δ1=−δ3,ε5=0.Here,αd,δd,ϵk for d=1:3,k=1:6, β2,andβ3 are the unknown parameters. By considering the necessary assumption,(102)δ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation equation:(103)u5=2lnϕ5x=2−α1etδ3+xα3−yβ1−ε1+ε4α1e−tδ3−xα3+yβ1+ε1+ε6sinhtδ3+xα3+yβ3+ε3α3etδ3+xα3−yβ1−ε1+ε4e−tδ3−xα3+yβ1+ε1+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the interaction of solution u⟶2α3 at any t, but if τ3<τ1⟶∞, the interaction of solution u⟶2α1 at any time.


## 6.6. Set VI Solutions

(104)α2=ε6=0,β1=−−4α1δ24ε52+4α1ε43α13δ1+3δ14+3δ12δ22+4δ24−δ22−ε52+4ε4α1h2+δ1h3δ22−ε52+4ε4,β2=12α1ε4α13+δ13+δ1δ22+δ22h3−ε52+4ε4δ2−ε52+4ε4,h1=−−δ22ε52α13+δ13+δ1δ22+4ε4312+4δ22α13+δ13+δ1δ22α1δ22−ε52+4ε4.Here,δd,ϵk for d=1:3,k=1:6, α2,α3,andβ3 are the unknown parameters. By considering the necessary assumption,(105)α1δ22−ε52+4ε4≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(106)u6=2lnϕ6x=2−α1e−tδ1−xα1−yβ1−ε1+ε4α1etδ1+xα1+yβ1+ε1e−tδ1−xα1−yβ1−ε1+ε4etδ1+xα1+yβ1+ε1+ε5costδ2+yβ2+ε2.Ifτ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 8 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ6 with the following selected parameters:(107)δ1=0.3,δ2=2,δ3=1,α1=0.1,α3=0.5,β3=1,h2=2,h3=3,ε1=1,ε2=2,ε3=4,ε4=2,ε5=1,t=0.1,in equation (102).Figure 8
Interaction between stripe and periodic wave solution (107) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−3,0,3.
(a)(b)


## 6.7. Set VII Solutions

(108)α1=θ,α2=ε6=0,β1=−4θδ23−θδ2h2−2δ1δ2h3+β2δ1δ2,h1=−−δ12δ2h3−δ23h3+β2δ12+β2δ22θ2δ2,ε4=14ε52.Here,θ=3−δ13−δ1δ22,δd,ϵk for d=1:3,k=1:3, α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(109)δ2≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(110)u7=2lnϕ7x=2−α1e−tδ1−x3−δ13−δ1δ22−yβ1−ε1+1/4ε52α1etδ1+x3−δ13−δ1δ22+yβ1+ε1e−tδ1−x3−δ13−δ1δ22−yβ1−ε1+1/4ε52etδ1+x3−δ13−δ1δ22+yβ1+ε1+ε5costδ2+yβ2+ε2.Ifτ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 9 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ7 with the following selected parameters:(111)δ1=−1.2,δ2=1,δ3=1,α3=0.5,β3=1,h2=2,h3=3,ε1=1,ε2=0.2,ε3=4,ε5=1,t=0.01,in equation (106).Figure 9
Interaction between stripe and periodic wave solution (111) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,y=−1,0,1.
(a)(b)


## 6.8. Set VIII Solutions

(112)αi=−δi,i=1,2,β1=β2δ1δ2,h2=−−δ2h1−δ2h3+β2δ2,ε6=0.Here,δd,ϵk for d=1:3,k=1:5, α3,β2,andβ3 are the unknown parameters. By considering the necessary assumption,(113)δ2≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(114)u8=2lnϕ8x=2δ1e−tδ1+xδ1−yβ2δ1/δ2−ε1−ε4δ1etδ1−xδ1+yβ2δ1/δ2+ε1+ε5sintδ2−xδ2+yβ2+ε2δ2e−tδ1+xδ1−yβ2δ1/δ2−ε1+ε4etδ1−xδ1+yβ2δ1/δ2+ε1+ε5costδ2−xδ2+yβ2+ε2.


## 6.9. Set IX Solutions

(115)α2=θ,β1=−4α36+4α33δ33+θ24α34−4α3δ33−α3δ3h2−δ32h3θ2δ3,β2=−4α35+4α32δ33+θ24α33−4δ33−δ3h2θδ3,β3=−4α36+4α33δ33+θ24α34−4α3δ33−α3δ3h2−δ32h3θ2δ3,δ2=0,h1=−4α32α3θ2+α33+δ33θ2,α1=α3,δ1=δ3.Here,ϵk for k=1:6, α3,andδ3 are free values. Also, θ solves the θ4+2θ2α32+α34+α3δ33=0. By considering the necessary assumption,(116)θδ3≠0,and by substituting the above parameters into equation (89), we obtain an analytical form of rational equation:(117)u9=2−α3e−tδ3−xα3−yβ1−ε1+ε4α3etδ3+xα3+yβ1+ε1−ε5sinxθ+yβ2+ε2θ+ε6sinhtδ3+xα3+yβ3+ε3α3e−tδ3−xα3−yβ1−ε1+ε4etδ3+xα3+yβ1+ε1+ε5cosxθ+yβ2+ε2+ε6coshtδ3+xα3+yβ3+ε3.Ifτ3>τ1⟶∞, the the interaction between stripe and periodic wave solution u⟶2α3 at any t, but if τ3<τ1⟶∞, the interaction between stripe and periodic wave solution u⟶2α1 at any t. Figure 10 shows the analysis of treatment of interaction of solutions as periodic and hyperbolic functions with graphs of ϕ9 with the following selected parameters:(118)δ1=−1.2,δ2=1,δ3=3,α3=−0.5,β2=1,β3=1,h2=2,h3=3,ε1=1,ε2=0.1,ε3=3,ε4=4,ε5=5,ε6=4,t=1,in equation (117).Figure 10
Interaction between stripe and periodic wave solution (117) such that (a) 3D design of ux,y,t at t=1 and (b) 2D plot of ux,y,t at t=1,x=−3,0,3.
(a)(b)


## 7. Conclusion

This article investigated the soliton and periodic solutions of the generalized Hietarinta equation. The Cole–Hopf algorithm has been described by means of binary Bell polynomials. The 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. The 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. The 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. These solutions are also verified by using Maple software.



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*Source: 6445482-2022-03-27.xml*