# Dynamical Behaviors of Lumpoff and Rogue Wave Solutions for Nonlocal Gardner Equation

Authors: Jalil Manafian; Seyedeh Fatemeh Shahabi Takami; Baharak Eslami; Somaye Malmir; Anju Sood; Amarpreet Kaur Sabherwal
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we got a novel kind of rogue waves with the predictability of (2 + 1)-dimensional nonlocal Gardner equation with the aid of Maple according to the Hirota bilinear model. We first construct a general quadratic function to derive the general lump solution for the mentioned equation. At the same time, the lumpoff solutions are demonstrated with more free autocephalous parameters, in which the lump solution is localized in all directions in space. Moreover, when the lump solution is cut by twin-solitons, special rogue waves are also introduced. Based on the data available in the literature, the resulting soliton solutions are innovative, developed, distinctive, and significant and can be applied to more complex phenomena, and they are immensely active for nonlinear models of classical and fractional-order type. To examine the dynamic behavior of the waves, contour 3D and 2D plots of several obtained findings are sketched by assigning specific values to the parameters. Furthermore, we obtain new sufficient solutions containing cross-kink, periodic-kink, multi-waves, and solitary wave solutions. It is worth noting that the emerging time and place of the rogue waves depend on the moving path of the lump solution.


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## Body

## 1. Introduction

Nonlinear physical structures have been linked with nonlinear equations concerning several disciplines, like thermodynamics, mechanics, fluid dynamics, wave propagation, plasma physics, fluid flow, nonlinear networks, optical fibers, and soil consolidations, to develop vital phenomenon and implementations ([1–4])[5, 6]. The role of nonlinearity in waves is quite significant, mostly throughout non-linear sciences; research development towards exact solutions of partial differential nonlinear equations has always been a major endeavor for the past couple of years. All the while, several powerful, efficient, and reliable methods for attempting to seek exact analytical solutions to traveling waves have been established: for example, the N-soliton solution [7], a fractional-order multiple-model type-3 fuzzy control [8], a calculation methodology for geometrical characteristics [9], a combination of group method of data handling and computational fluid dynamics [10], the truss optimization with metaheuristic algorithms [11], the extended generalized Darboux method [12], the Lax pair technique [13], intermolecular interactions to the modern acid-base theory [14], the deep learning for Feynman’s path integral [15], the Darboux–Bäcklund technique [16], the Hirota bilinear technique [17], the multiple exp-function method [18], optimal structure design [19], the experimental study on circular steel [20, 21], an influence of seismic orientation on the statistical distribution [22], effects of actual loading waveforms on the fatigue behaviors [23], and so on [24–27]. Hirota bilinear method is an efficient instrument to construct exact solutions of NLEEs, and there exist a lot of completely integrable equations that are investigated in this procedure, for example, the generalized bilinear equations [28], the Kadomtsev–Petviashvili (KP)–Benjamin–Bona–Mahony equation [29], the optimal Galerkin-homotopy asymptotic method [30], the (2 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation [31], the KP equation [32], the B-type KP equation [33], the optical interactions within magnetic layered structures [34], the (2 + 1)-dimensional breaking soliton equation [35], and the bidirectional Sawada–Kotera equation [36].As we all know, the (1 + 1)-dimensional Gardner equation is an important equation and is a mixed version of the well-known Korteweg-De Vries (KdV) equation and the modified KdV (mKdV) equation [37, 38], and its properties are widely used in the field of mathematical physics, fluid dynamics, hydrodynamics, quantum mechanics, and nonlinear optics which is presented as follows:(1)ut+6βuux+uxxx−32α2u2ux=0,the Gardner equation is completely integrable. The diverse kinds of exact solutions of Eq. (1) were studied by several techniques where interested researchers have worked on it, for example, using the inverse spectral transform method and constructing explicitly the exact solutions by the ∂-dressing method [39], utilizing the Casorati and Grammian determinant solutions to the (2 + 1)-dimensional Gardner equation [40], the tan h method with kink solutions [41], and the nonlinearization technique of Lax pairs to find the integrable decompositions for the (2 + 1)-dimensional Gardner [42]. Konopelchenko and Dubrovsky suggested the integro-differential form for the (2 + 1)-dimensional (D) Gardner equation as below:(2)ut+6βuux+uxxx−32α2u2ux+3σ2∂x−1uyy−3ασux∂x−1uy=0,where σ2=±1, α and β are nonzero arbitrary values, and u=ux,y,t. Also, the operator ∂x−1.=∫0x.. All cases for parameters α,β, and σ have been investigated in the valuable works such as nonlocal symmetry and exact solutions [43], N-soliton solution and soliton resonances [44], and lie symmetries and invariant solutions [45] of (2 + 1)-dimensional Gardner equation.Different kinds of probes were studied by proficient scholars in which many of them can be indicated, for instance, the generalized higher-order NLSE [46], the double dispersive equation in Murnaghan’s rod [47], the fractional (2 + 1)-D Boussinesq dynamical model [48], the conformable time-fractional Wu-Zhang system [49], and the time-fractional model of Lassa hemorrhagic fever spreading [50]. In the valuable works, the important periodic and breather solutions to the KP-BBM equation [51] and generalized Bogoyavlensky–Konopelchenko equation [52] with the help of the Hirota technique were obtained. Authors of Reference [53] investigated the exact soliton solutions of a nonlinear Schrödinger equation including Kudryashov’s sextic power-law nonlinearity by two efficient analytical methods. Cinar et al.[54] studied the soliton solutions for the perturbed Fokas–Lenells equation which has a vital role in optics by using Sardar subequation method. In Reference [55], the analytical solutions of simplified modified Camassa–Holm equations with various derivative operators, namely, conformable and M-truncated derivatives were obtained. Also, the analytical solutions of (2 + 1) dimensional Heisenberg ferromagnetic spin equation which describes the nonlinear dynamics of the ferromagnetic materials by using the extended rational sine-cosine and sin h-cos h methods were reached [56]. The dark, singular, combined dark-singular soliton, singular periodic wave, and rational function solutions for the (2 + 1)-dimensional Biswas–Milovic equation for the description of pulse propagation in optical fiber were constructed [57]. Some authors worked on the nonlinear PDEs including the nonlinear directional couplers in nonlinear optics [58]; an improved perturbed Schrödinger equation with Kerr law nonlinearity in nonlinear optics [59]; the fractional analysis of fusion and fission process in plasma physics [60]; traveling, periodic, quasiperiodic, and chaotic structures of perturbed Fokas–Lenells model [61]; and dynamics of longitudinal bud equation among a magneto-electro-elastic round rod [62]. In Reference [63], the (G’/G)-expansion and the Ricatti transformation method were used for solving the (2 + 1)-dimensional Boussinesq equation. The analytical solutions to integer and fractional-order to KdV and Fornberg–Whitham equations were obtained by using the Laplace transforms, Sumudu transforms, and Elzaki transforms [64]. Scholars of Reference [65] studied Yang transform homotopy perturbation method to nonlinear fractional order KdV and Burger equation with exponential-decay kernel and provided formulae for the Yang transform of Caputo–Fabrizio fractional-order derivatives.The lump, lumpoff, rogue wave, lump-periodic, periodic wave, cross-kink, multiwave, and solitary wave solutions of (2) will be investigated and analyzed. According to the Hirota bilinear model (2), the general periodic wave solution is concluded in Section 2 and also the new periodic solutions which can be arisen with more than five classes. Finally, conclusions are given is Section 10.


## 2. General Lump Solutions

According to Reference [66], the Hirota bilinear form of the Gardner equation is as follows:(3)2σαDy−2αDx2−4βα2Dxg.f=0,Dx3+Dt+3DxDy+6σβαDyg.f=0,under the condition of transformation:(4)u=−2αlngfx.Here, theD-operator is defined as follows:(5)DxaDybDtcF1⋅F2=∂∂x−∂∂x′a∂∂y−∂∂y′b∂∂t−∂∂t′cF1x,y,tF2x′,y′,t′x=x′,y=y′,t=t′.With the help of transformation (4), the general lump solutions of Eq. (3) can be given. We introduce ansatz f and g as follows:(6)f=xTΣx+f0,g=xTΛx+g0,0.with,(7)Σ=ε00ε01ε02ε04ε10ε11ε12ε13ε20ε21ε22ε23ε30ε31ε32ε33,Λ=λ00λ01λ02λ04λ10λ11λ12λ13λ20λ21λ22λ23λ30λ31λ32λ33,xT=1,x,y,t,in which Σ,Λ∈R4×4 are the symmetric matrixes and f0 and g0 are the positive constants. Then,(8)f=∑i≤j=03εijxixj+f0=ε11x2+ε22y2+ε33t2+2ε12xy+2ε13xt+2ε23yt+2ε01x+ε02y+2ε03t+ε00+f0,g=∑i≤j=03λijxixj+g0=λ11x2+λ22y2+λ33t2+2λ12xy+2λ13xt+2λ23yt+2λ01x+λ02y+2λ03t+λ00+g0,or we can write the custom form as below:(9)f=δ1ε1x+ε2y+ε3t+ε42+δ2λ1x+λ2y+λ3t+λ42+ε0,g=δ3ε1x+ε2y+ε3t+ε42+δ4λ1x+λ2y+λ3t+λ42+λ0.Substituting relations (10) and (11) into Eq. (3) and collecting all the coefficients of x,y,t, we can get twenty determining equations. Solving these resulting equations, we have the following cases:Case 1.
(10)ϵ2=−23ϵ1α,ϵ3=43ϵ13βσ−3β−σα2,ϵ4=−23λ4ϵ1λ2α,λ1=−32λ2α,λ3=−23βσ−3β−σλ2α.
Under the transformationu=−2/αlng/fx, the corresponding general lump solutions is read as follows:(11)ulump=−2αgfdf/dxg−fdg/dxg2,in which,(12)f=δ143tϵ13βσ−3β−σα2+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+δ2−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+ε0,g=δ343tϵ13βσ−3β−σα2+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+δ4−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+λ0.Case 2.
(13)ϵ0=9α2δ2λ22+4δ1ϵ12λ09α2δ4λ22+4δ3ϵ12,ϵ2=−23ϵ1α,ϵ3=−23ϵ1λ3λ2α,ϵ4=−23λ4ϵ1λ2α,λ1=−32λ2α.
Under the transformationu=−2/αlng/fx, the corresponding general lump solutions read as follows (see Figure 1):(14)ulump=−2αgfdf/dxg−fdg/dxg2,in which(15)f=δ1−23tϵ1λ3λ2α+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+δ2tλ3−32xαλ2+yλ2+λ42+9α2δ2λ22+4δ1ϵ12λ09α2δ4λ22+4δ3ϵ12,g=δ3−23tϵ1λ3λ2α+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+δ4tλ3−32xαλ2+yλ2+λ42+λ0.Figure 1
Behavior of lump solutionux,y,t in Eq. (13) with the selected amounts α=1,β=1,σ=1,ϵ0=1,ϵ1=2,λ0=2,ϵ1=2,λ2=3,λ4=1.3,δ1=1,δ2=2.5,δ3=2,δ4=−2,t=2.


## 3. Lumpoff Solution

In the section, the lumpoff solutions of the Gardner equation, a solution with the interaction between lump wave and stripe soliton wave, are investigated. On the basis offlump and glump of the general lump solutions, the lumpoff solutions can be written as follows:(16)f=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ1eθ1x+θ2y+θ3t+θ4+ε0,(17)g=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ2eθ1x+θ2y+θ3t+θ4+λ0.We can easily detect that it consists of two parts: lump wave part and solitary wave part. It is worth noting that the solitary wave part is dominant whenθ1x+θ2y+θ3t+θ4>0. Otherwise, the lump solution only arises when θ1x+θ2y+θ3t+θ4<0. To summarize, the existence of the soliton is based on the existence of the lump. Then, substituting (16) and (17) into Eq. (3), the selected constants of flumpoff and glumpoff are obtained as follows:Case 3.
(18)ϵ2=−23ϵ1α,ϵ3=43ϵ13βσ−3β−σα2,λ1=−32λ2α,λ3=−23βσ−3β−σλ2α,θ3=2σθ23β−1α.
Under the transformationu=−2/αlng/fx, the corresponding general lumpoff solutions read as follows:(19)ulumpoff=−2αglumpoffflumpoffdflumpoff/dxglumpoff−flumpoffdglumpoff/dxglumpoff2,in which,(20)flumpoff=43tε13βσ−3β−σα2+xε1−23yε1α−23λ4ε1λ2α2+−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42−δ2e−2σθ23β−1t/α+yθ2+θ4+ε0,glumpoff=δ343tε13βσ−3β−σα2+xε1−23yε1α−23λ4ε1λ2α2+δ4−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+δ2e−2σθ23β−1t/α+yθ2+θ4+λ0.
Figure2 presents the overtaking interactions between lump wave and stripe soliton wave containing 3D plot, density plot, and 2D plot, respectively, x=−1,0,1.Figure 2
Behavior of lumpoff solutionux,y,t in Eq. (25) with the selected amounts α=1,β=1,σ=1,ϵ0=1,ϵ1=2,λ0=2,ϵ1=2,λ2=3,λ4=1.3,δ1=1,δ2=2.5,δ3=2,δ4=−2,θ2=1.5,θ4=2,t=2.


## 4. Rogue Waves with Predictability

In the section, we will explore the rogue waves with predictability for the (2 + 1)-dimensional Gardner equation. The so-called predictability is that the emerging time and place of the rogue waves can be described by some particular expressions. Let us consider a more general ansatz as follows:(21)f=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ1coshθ1x+θ2y+θ3t+θ4+ε0,(22)g=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ2coshθ1x+θ2y+θ3t+θ4+λ0.Substituting (21) and (22) into Eq. (3) and collecting all relevant coefficients of x,y,t,cosh, and sinh, a series of equations have been obtained. Solving these equations, we have the following case:Case 4.
(23)ϵ2=−23ϵ1α,ϵ3=43ϵ13βσ−3β−σα2,λ1=−32λ2α,λ3=−23βσ−3β−σλ2α,θ3=2σθ23β−1α.
Under the transformationu=−2/αlng/fx, the corresponding special rogue solution is read as follows:(24)urogue=−2αgroguefroguedfrogue/dxgrogue−froguedgrogue/dxgrogue2,in which,(25)frogue=43tε13βσ−3β−σα2+xε1−23yε1α−23λ4ε1λ2α2+−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42−δ2cosh−2σθ23β−1tα+yθ2+θ4+ε0,grogue=δ343t∫13βσ−3β−σα2+xε1−23yε1α−23λ4ε1λ2α2+δ4−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+δ2cosh−2σθ23β−1tα+yθ2+θ4+λ0.
Actually, the hyperbolic function cosh part that predominates in frogue and grogue led to the lump waves which are always disappeared. Figure 3 presents the overtaking interactions between lump wave and stripe soliton wave containing 3D plot, density plot, and 2D plot, respectively, x=−1,0,1.Figure 3
Behavior of rogue wave solution for I and II and singular solution for III ofux,y,t in Eq. (31) with the selected amounts α=1,β=1,σ=1,ϵ0=1,ϵ1=2,λ0=2,ϵ1=2,λ2=3,λ4=1.3,δ1=1,δ2=2.5,δ3=2,δ4=−2,θ2=1.5,θ4=2,t=2.


## 5. Lump-Periodic Solutions

In the section, we will explore the lump-periodic solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:(26)f=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ1cosθ1x+θ2y+θ3t+θ4+ε0,(27)g=ε1x+ε2y+ε3t+ε42+λ1x+λ2y+λ3t+λ42+δ2cosθ1x+θ2y+θ3t+θ4+λ0.Substituting (26) and (27) into Eq. (3) and collecting all relevant coefficients of x,y,t,cos, and sin, a series of equations have been obtained. Solving these equations, we have the following case:Case 5.
(28)ϵ2=−23ϵ1α,ϵ3=43ϵ13βσ−3β−σα2,λ1=−32λ2α,λ3=−23βσ−3β−σλ2α,θ3=−18θ227α4θ22+48βσ−48β−16σα.
Under the transformationu=−2/αlng/fx, the corresponding special lump-periodic solution can be read as follows:(29)uLP=−2αgLPfLPdfLP/dxgLP−fLPdgLP/dxgLP2,in which,(30)fLP=43tϵ13βσ−3β−σα2+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+δ2cos−18θ227α4θ22+48βσ−48β−16σtα−32xαθ2+yθ2+θ4+ε0,gLP=δ343tϵ13βσ−3β−σα2+xϵ1−23yϵ1α−23λ4ϵ1λ2α2+δ4−2t3βσ−3β−σλ2α−32xαλ2+yλ2+λ42+δ2cos−18θ227α4θ22+48βσ−48β−16σtα−32xαθ2+yθ2+θ4+λ0.
Figure4 presents the overtaking interactions between lump wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, x=−1,0,1.Figure 4
Behavior of lump-periodic solutionux,y,t in Eq. (37) with the selected amounts α=1,β=1,σ=1,ϵ0=1,ϵ1=2,λ0=2,ϵ1=2,λ2=3,λ4=1.3,δ1=1,δ2=2.5,δ3=2,δ4=−2,θ2=1.5,θ4=2,t=2.


## 6. Double Breather Solutions

In the section, we will explore the breather solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:(31)f=e−ε1x−ε2y−ε3t−ε4+δ1eε1x+ε2y+ε3t+ε4+δ2sinλ1x+λ2y+λ3t+λ4+δ3sin2λ1x+2λ2y+2λ3t+2λ4,(32)g=e−ε1x−ε2y−ε3t−ε4+δ4eε1x+ε2y+ε3t+ε4+δ5sinλ1x+λ2y+λ3t+λ4+δ6sin2λ1x+2λ2y+2λ3t+2λ4.Substituting (31) and (32) into Eq. (3) and collecting all relevant coefficients of exp,cos, and sin, a series of equations have been obtained. Solving these equations, we have the following case:Case 6.
(33)ϵ3=−α2ϵ13−3α2ϵ1ϵ2+6αβσϵ2−2ασϵ2−2αϵ12+4βϵ1α2,λ3=λ23αϵ1−6βσ+2σα.
Under the transformationu=−2/αlng/fx, the corresponding special breather solution is read as follows:(34)uB=−2αgBfBdfB/dxgB−fBdgB/dxgB2,in which,(35)fB=e−Ω+δ2sinΞ+δ3sin2Ξ,gB=e−Ω,Ω=tα2ε13−3α2ε1ε2+6αβσε2−2ασε2−2αε12+4βε1α2−xε1−yε2−ε4,Ξ=λ23α∫1−6βσ+2σtα+λ2y+λ4.
Figure5 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, y=−1,0,1.Figure 5
Behavior of breather solutionux,y,t in Eq. (43) with the selected amounts α=1,β=1,σ=1,ϵ1=1,ϵ2=2,ϵ4=1.5,λ2=2,λ4=1.2,δ1=1.3,δ2=2,δ3=1.5,δ4=3,x=1.5.Case 7.
(36)δ1=δ32δ4δ62,δ2=δ5δ3δ6,ϵ2=13ϵ13αδ3ϵ1+3αδ6ϵ1−2δ3+2δ6αδ3−δ6,ϵ3=23ϵ1ωδ3−δ62α2,λ3=λ23αδ3ϵ1+3αδ6ϵ1−6βσδ3+6βσδ6+2σδ3−2σδ6αδ3−δ6,ω=3α2δ32ϵ12+12α2δ3δ6ϵ12+3α2δ62ϵ12−9αβσδ32ϵ1+9αβσδ62ϵ1+3ασδ32ϵ1−3ασδ62ϵ1+6βσδ32−12βσδ3δ6+6βσδ62−6βδ32+12βδ3δ6−6βδ62−2σδ32+4σδ3δ6−2σδ62.
Under the transformationu=−2/αlng/fx, the corresponding special breather solution is read as follows:(37)uB=−2αgBfBdfB/dxgB−fBdgB/dxgB2,in which,(38)fB=e−Ω+δ32δ4δ62eΩ+δ5δ3δ6sinΞ+δ3sin2Ξ,gB=e−Ω+δ4eΩ+δ5sinΞ+δ6sin2Ξ,Ω=23tε1ωδ3−δ62α2+xε1+13yε13αδ3ε1+3αδ6ε1−2δ3+2δ6αδ3−δ6+ε4,Ξ=λ23αδ3ε1+3αδ6ε1−6βσδ3+6βσδ6+2σδ3−2σδ6tαδ3−δ6+λ2y+λ4.
Figure6 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, y=−1,0,1.Figure 6
Behavior of breather solutionux,y,t in Eq. (45) with the selected amounts α=1,β=1,σ=1,ϵ1=1,ϵ2=2,ϵ4=1.5,λ2=2,λ4=1.2,δ1=1.3,δ2=2,δ3=1.5,δ4=3,x=1.5.Case 8.
(39)δ1=δ33δ6δ3+δ62,δ4=δ3δ63δ3+δ62,ϵ3=−4λ13αδ3λ2+3αδ6λ2+2δ3λ1+2δ6λ1αδ3−δ6,λ3=22α2λ13−3αβσλ2+ασλ2−2βλ1α2.
Under the transformationu=−2/αlng/fx, the corresponding special breather solution is read as follows:(40)uB=−2αgBfBdfB/dxgB−fBdgB/dxgB2,in which,(41)fB=e−Ω+δ32δ4δ62eΩ+δ3sin2Ξ,gB=e−Ω+δ4eΩ+δ6sin2Ξ,Ω=4tλ13αδ3λ2+3αδ6λ2+2δ3λ1+2δ6λ1αδ3−δ6,Ξ=22α2λ13−3αβσλ2+ασλ2−2βλ1tα2+λ1x+λ2y+λ4.
Figure7 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, t=−0.01,0,0.01.Figure 7
Behavior of breather solutionux,y,t in Eq. (49) with the selected amounts α=1,β=1,σ=1,ϵ1=1,ϵ2=2,ϵ4=1.5,λ2=2,λ4=1.2,δ1=1.3,δ2=2,δ3=1.5,δ4=3,x=1.5.


## 7. Periodic Cross-Kink Solutions

In the section, we will explore the periodic cross-kink solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:(42)f=e−ε1x−ε2y−ε3t−ε4+δ1eε1x+ε2y+ε3t+ε4+δ2sinλ1x+λ2y+λ3t+λ4+δ3sinh2λ1x+2λ2y+2λ3t+2λ4,(43)g=e−ε1x−ε2y−ε3t−ε4+δ4eε1x+ε2y+ε3t+ε4+δ5sinλ1x+λ2y+λ3t+λ4+δ6sinh2λ1x+2λ2y+2λ3t+2λ4.Substituting (42) and (43) into Eq. (3) and collecting all relevant coefficients of exp,cos,sin,cosh, and sinh, a series of equations have been obtained. Solving these equations, we have the following cases:Case 9.
(44)ϵ3=−α2ϵ13−3α2ϵ1ϵ2+6αβσϵ2−2ασϵ2−2αϵ12+4βϵ1α2,λ3=−13ωα2δ52λ1,λ2=−133αδ62ϵ1ϵ2+12αδ4ϵ1ϵ2+2δ52λ12+2δ62ϵ12+8δ4ϵ12αδ52λ1,θ2=−133αϵ2+4ϵ1α,ω=9α2δ52ϵ12λ12−3α2δ52λ14−18αβσδ62ϵ1ϵ2−72αβσδ4ϵ1ϵ2+6ασδ62ϵ1ϵ2−12βσδ52λ12−12βσδ62ϵ12+24ασδ4ϵ1ϵ2−48βσδ4ϵ12+12βδ52λ12+4σδ52λ12+4σδ62ϵ12+16σδ4ϵ12.
Under the transformationu=−2/αlng/fx, the corresponding special periodic cross-kink solution is read as follows:(45)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(46)fCK=e−Ω+δ4eΩ−δ5sinΞ+δ3sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ3sinhΛ,Ω=−tα2∫13−3α2∫1λ12+6αβσ∫2−2ασ∫2+4β∫1α2+xε1+yε2+ε4,Ξ=−13ωtα2δ52λ1+λ1x−13y3αδ62ε1ε2+12αδ4ε1ε2+2δ52λ12+2δ62ε12+8δ4ε12αδ52λ1+λ4,Λ=−133α2ε13−9α2ε1λ12−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2+x∫1−13y3αε2+4ε1α+θ4.Case 10.
.(47)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(48)fCK=e−Ω−14δ62eΩ−δ5sinΞ+δ6sinhΛ,gCK=e−Ω−14δ62eΩ+δ5sinΞ+δ6sinhΛ,Ω=−tα2∫13+6αβσ∫2−2ασ∫2+4β∫1α2+xε1+yε2+ε4,Ξ=−2σλ23β−1tα+λ2y+λ4,Λ=−133α2ε13−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2+x∫1−13y3α∫2+4∫1α+θ4.Case 11.
(49)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(50)fCK=e−Ω+δ4eΩ−δ5sinΞ+δ6sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ6sinhΛ,Ω=−13tε13α2ε12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−2σλ23β−1tα+λ2y+λ4,Λ=−13t∫13α2ε12−12βσ+12β+4σα2+xε1−23yε1α+θ4.Case 12.
(51)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(52)fCK=e−Ω+δ4eΩ+δ2sinΞ+δ6sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ6sinhΛ,Ω=−13tε13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−139α2ε12−3α2λ12−12βσ+12β+4σλ1tα2+λ1x−23yλ1α+λ4,Λ=−13t∫13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+θ4.Case 13.
(53)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(54)fCK=e−Ω+δ2sinΞ+δ6sinhΛ,gCK=e−Ω+δ5sinΞ+δ6sinhΛ,Ω=112tε181α4λ22−12α2ε12+48βσ−48β−16σα2+xε1−23yε1α+ε4,Ξ=−18λ227α4λ22−36α2ε12+48βσ−48β−16σtα−32αλ2x+λ2y+λ4,Λ=−112t∫181α4λ22−12α2ε12+48βσ−48β−16σα2−xε1+23yε1α+θ4.Case 14.
(55)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(56)fCK=e−Ω+δ4eΩ−δ5sinΞ+δ6sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ6sinhΛ,Ω=−tα2ε13−3α2ε1λ12+6αβσε2−2ασε2+4βε1α2+x∫1+y∫2+∫4,Ξ=−13ωtα2δ52λ1+λ1x−13y3αδ62ε1ε2+12αδ4ε1ε2+2δ52λ12+2δ62ε12+8δ4ε12αδ52λ1+λ4,Λ=133α2ε13−9α2ε1λ12−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2−x∫1+13y3αε2+4ε1α+θ4,ω=9α2δ52ε12λ12−3α2δ52λ14−18αβσδ62ε1ε2−72αβσδ4ε1ε2+6ασδ62ε1ε2−12βσδ52λ12−12βσδ62ε12+24ασδ4ε1ε2−48βσδ4ε12+12βδ52λ12+4σδ52λ12+4σδ62ε12+16σδ4ε12.
Figure8 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, y=−1,0,1.Figure 8
Behavior of cross-kink solutionux,y,t in Eq. (70) with the selected amounts α=1,β=1,σ=1,∫1=1,∫2=1.5,∫4=2.5,λ1=1,λ4=1.2,δ4=2,δ5=3,δ6=2.5,θ2=1.5,θ4=2,x=2.Case 15.
(57)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(58)fCK=e−Ω+δ4eΩ+δ2sinΞ+δ6sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ6sinhΛ,Ω=−13tε13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−139α2ε12−3α2λ12−12βσ+12β+4σλ1tα2+λ1x−23λ1yα+λ4,Λ=13t∫13α2ε12−9α2λ12−12βσ+12β+4σα2−xε1+23yε1α+θ4.
Figure9 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, y=−1,0,1.Figure 9
Behavior of cross-kink solutionux,y,t in Eq. (73) with the selected amounts α=1,β=1,σ=1,ϵ1=1,ϵ2=1.5,ϵ4=2.5,λ1=1,λ4=1.2,δ2=1.5,δ4=2,δ5=3,δ6=2.5,θ2=1.5,θ4=2,x=2.


## 8. Periodic Wave Solutions

In the section, we will explore the periodic wave solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:(59)f=e−ε1x−ε2y−ε3t−ε4+δ1eε1x+ε2y+ε3t+ε4+δ2cosλ1x+λ2y+λ3t+λ4+δ3cosh2λ1x+2λ2y+2λ3t+2λ4,(60)g=e−ε1x−ε2y−ε3t−ε4+δ4eε1x+ε2y+ε3t+ε4+δ5cosλ1x+λ2y+λ3t+λ4+δ6cosh2λ1x+2λ2y+2λ3t+2λ4.Substituting (59) and (60) into Eq. (3) and collecting all relevant coefficients of exp,cos,sin,cosh, and sinh, a series of equations have been obtained. Solving these equations, we have the following cases:Case 16.
(61)∫3=−α2ε13−3α2ε1λ12+6αβσε2−2ασε2+4βε1α2,λ3=−13ωα2δ52λ1,λ2=−133αδ62ε1ε2+12αδ4ε1ε2+2δ52λ12+2δ62ε12+8δ4ε12αδ52λ1,θ2=−133α∫2+4∫1α,ω=9α2δ52ε12λ12−3α2δ52λ14−18αβσδ62ε1ε2−72αβσδ4ε1ε2+6ασδ62ε1ε2−12βσδ52λ12−12βσδ62ε12+24ασδ4ε1ε2−48βσδ4ε12+12βδ52λ12+4σδ52λ12+4σδ62ε12+16σδ4ε12,θ3=−133α2ε13−9α2ε1λ12−18αβσε2+6ασε2−24βσε1+12βε1+8σε1α2.
Under the transformationu=−2/αlng/fx, the corresponding special periodic wave solution is read as follows:(62)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(63)fCK=e−Ω+δ4eΩ−δ5sinΞ+δ3sinhΛ,gCK=e−Ω+δ4eΩ+δ5sinΞ+δ3sinhΛ,Ω=−tα2ε13−3α2ε1λ12+6αβσε2−2ασε2+4βε1α2+xε1+yε2+ε4,Ξ=−13ωtα2δ52λ1+λ1x−13y−3αδ62ε1ε2+12αδ4ε1ε2+2δ52λ12−2δ62ε12+8δ4ε12αδ52λ1+λ4,Λ=−133α2ε13−9α2ε1λ12−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2+xε1−133αε2+4ε1yα+θ4.Case 17.
(64)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(65)fCK=e−Ω+14δ62eΩ−δ5cosΞ+δ6coshΛ,gCK=e−Ω+14δ62eΩ+δ5cosΞ+δ6coshΛ,Ω=−tα2ε13+6αβσε2−2ασε2+4βε1α2+xε1+yε2+ε4,Ξ=−2σλ23β−1tα+λ2y+λ4,Λ=−133α2ε13−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2+xε1−13y3αε2+4ε1α+θ4.Case 18.
(66)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(67)fCK=e−Ω+δ4eΩ−δ5cosΞ+δ6coshΛ,gCK=e−Ω+δ4eΩ+δ5cosΞ+δ6coshΛ,Ω=−13t∫13α2ε12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−2σλ23β−1tα+λ2y+λ4,Λ=−13t∫13α2ε12−12βσ+12β+4σα2+xε1−23yε1α+θ4.Case 19.
(68)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(69)fCK=e−Ω+δ4eΩ+δ2cosΞ+δ6coshΛ,gCK=e−Ω+δ4eΩ+δ5cosΞ+δ6coshΛ,Ω=−13tε13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−139α2ε12−3α2λ12−12βσ+12β+4σλ1tα2+λ1x−23yλ1α+λ4,Λ=−13t∫13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+θ4.Case 20.
(70)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(71)fCK=e−Ω+δ2cosΞ+δ6coshΛ,gCK=e−Ω+δ5cosΞ+δ6coshΛ,Ω=112tε181α4λ22−12α2ε12+48βσ−48β−16σα2+xε1−23yε1α+ε4,Ξ=−18λ227α4λ22−36α2ε12+48βσ−48β−16σtα−32αλ2x+λ2y+λ4,Λ=−112t∫181α4λ22−12α2ε12+48βσ−48β−16σα2−xε1+23yε1α+θ4.Case 21.
(72)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(73)fCK=e−Ω+δ4eΩ−δ5cosΞ+δ6coshΛ,gCK=e−Ω+δ4eΩ+δ5cosΞ+δ6coshΛ,Ω=−tα2ε13−3α2ε1λ12+6αβσε2−2ασε2+4βε1α2+xε1+yε2+ε4,Ξ=−13ωtα2δ52λ1+λ1x−13y3αδ62ε1ε2+12αδ4ε1ε2+2δ52λ12+2δ62ε12+8δ4ε12αδ52λ1+λ4,Λ=133α2ε13−9α2ε1λ12−18αβσε2+6ασε2−24βσε1+12βε1+8σε1tα2−xε1+13y3αε2+4ε1α+θ4,ω=9α2δ52ε12λ12−3α2δ52λ14−18αβσδ62ε1ε2−72αβσδ4ε1ε2+6ασδ62ε1ε2−12βσδ52λ12−12βσδ62ε12+24ασδ4ε1ε2−48βσδ4ε12+12βδ52λ12+4σδ52λ12+4σδ62ε12+16σδ4ε12.
Figure10 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, t=1,2,3.Figure 10
Behavior of periodic wave solutionux,y,t in Eq. (72) with the selected amounts α=1,β=1,σ=1,∫1=1,∫2=1.5,∫4=2.5,λ1=1,λ4=1.2,δ4=2,δ5=3,δ6=2.5,θ2=1.5,θ4=2,x=2.Case 22.
(74)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(75)fCK=e−Ω+δ4eΩ+δ2cosΞ+δ6coshΛ,gCK=e−Ω+δ4eΩ+δ5cosΞ+δ6coshΛ,Ω=−13tε13α2ε12−9α2λ12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=−139α2ε12−3α2λ12−12βσ+12β+4σλ1tα2+λ1x−23λ1yα+λ4,Λ=13tε13α2ε12−9α2λ12−12βσ+12β+4σα2−xε1+23yε1α+θ4.
Figure11 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, y=−1,0,1.Figure 11
Behavior of periodic wave solutionux,y,t in Eq. (74) with the selected amounts α=1,β=1,σ=1,∫1=1,∫2=1.5,∫4=2.5,λ1=1,λ4=1.2,δ2=1.5,δ4=2,δ5=3,δ6=2.5,θ2=1.5,θ4=2,x=2.


## 9. Multiwave Solutions

In the section, we will explore the multiwave solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:(76)f=δ1coshε1x+ε2y+ε3t+ε4+δ2cosλ1x+λ2y+λ3t+λ4+δ3cosh2λ1x+2λ2y+2λ3t+2λ4,(77)g=δ4coshε1x+ε2y+ε3t+ε4+δ5cosλ1x+λ2y+λ3t+λ4+δ6cosh2λ1x+2λ2y+2λ3t+2λ4.Substituting (76) and (77) into Eq. (3) and collecting all relevant coefficients of exp,cos,sin,cosh, and sinh, a series of equations have been obtained. Solving these equations, we have the following cases:Case 23.
(78)δ2=δ1δ5δ4,δ3=δ1δ6δ4,ε2=−23ε1α,ε3=−13ε13α2ε12−12βσ+12β+4σα2,λ1=−32αλ2.
Under the transformationu=−2/αlng/fx, the corresponding special multiwave solution is read as follows:(79)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(80)fCK=δ1coshΩ+δ1δ5δ4cosΞ+δ1δ6δ4coshΛ,gCK=δ4coshΩ+δ5cosΞ+δ6coshΛ,Ω=−13t∫13α2ε12−12βσ+12β+4σα2+xε1−23yε1α+ε4,Ξ=λ3t−32αλ2x+λ2y+λ4,Λ=tθ3+θ4.
Figure12 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, contour plot, and 2D plot, respectively, t=−0.5,0,0.5.Figure 12
Behavior of multiwave solutionux,y,t in Eq. (78) with the selected amounts α=1,β=1,σ=1,∫1=1,∫4=2.5,λ2=1,λ3=2,λ4=1.2,δ1=2.3,δ4=2,δ5=3,δ6=2.5,θ3=1.5,θ4=2,y=2.Case 24.
(81)δ2=δ1δ5δ4,δ3=δ1δ6δ4,ε2=−23ε1α,ε3=−112∫181α4θ22+12α2ε12−48βσ+48β+16σα2,θ1=−32αθ2.
Under the transformationu=−2/αlng/fx, the corresponding special multiwave solution is read as follows:(82)uCK=−2αgCKfCKdfCK/dxgCK−fCKdgCK/dxgCK2,in which,(83)fCK=δ1coshΩ+δ1δ5δ4cosΞ+δ1δ6δ4coshΛ,gCK=δ4coshΩ+δ5cosΞ+δ6coshΛ,Ω=−112tϵ181α4θ22+12α2ϵ12−48βσ+48β+16σα2+xϵ1−23yϵ1α+ϵ4,Ξ=tλ3+λ4,Λ=θ3t−32αθ2x+θ2y+θ4.
Figure13 presents the overtaking interactions between two stripe waves and periodic wave containing density plot and 2D plot, respectively, t=0.0,0.01,0.1.Figure 13
Behavior of multiwave solutionux,y,t in Eq. (79) with the selected amounts α=1,β=1,σ=1,∫1=1,∫4=2.5,λ3=.5,λ4=1,δ1=2.3,δ4=2,δ5=3,δ6=2.5,θ2=.5,θ3=.5,θ4=2,y=2.


### 9.1. Solitary Wave Solutions

Here, we will consider the solitary wave solution by selecting the below function in which (1) has been taken as follows:(84)f=δ1expε1x+ε2y+ε3t+ε4+δ2expλ1x+λ2y+λ3t+λ4+δ3,(85)g=δ4expε1x+ε2y+ε3t+ε4+δ5expλ1x+λ2y+λ3t+λ4+δ6.Plugging (84) and (85) into Eq. (3) and collecting all relevant coefficients of expΩ and expΞ, a series of equations have been obtained. Solving these equations, we have the following solutions:


#### 9.1.1. First Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(86)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(87)fS=δ1expΩ+δ2expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ3δ42δ1δ5∫1+δ1δ5λ1+δ2δ4λ1δ1δ1δ5λ1+2δ2δ4∫1+δ2δ4λ1,Ω=−tω1δ1δ5−δ2δ4α2+xϵ1+yϵ2+ϵ4,Ξ=−13ω2λ1tδ1δ5−δ2δ42α2−λ1x−13λ13αδ1δ5∫1+3αδ1δ5λ1+3αδ2δ4∫1+3αδ2δ4λ1−2δ1δ5+2δ2δ4yαδ1δ5−δ2δ4−λ4,ω1=α2δ1δ5ε13−α2δ2δ4ε13−3α2δ1δ5ε1ε2−3α2δ1δ5ε2λ1−3α2δ2δ4ε1ε2−3α2δ2δ4ε2λ1+6αβσδ1δ5ε2−6αβσδ2δ4ε2−2ασδ1δ5ε2+2ασδ2δ4ε2−2αδ1δ5ε12−2αδ1δ5ε1λ1−2αδ2δ4ε12−2αδ2δ4ε1λ1+4βδ1δ5ε1−4βδ2δ4ε1,ω2=9α2δ12δ52ε1λ1+6α2δ12δ52λ12+36α2δ1δ2δ4δ5ε12+54α2δ1δ2δ4δ5ε1λ1+24α2δ1δ2δ4δ5λ12+9α2δ22δ42ε1λ1+6α2δ22δ42λ12−18αβσδ12δ52ε1−18αβσδ12δ52λ1+18αβσδ22δ42ε1+18αβσδ22δ42λ1+6ασδ12δ52ε1+6ασδ12δ52λ1−6ασδ22δ42ε1−6ασδ22δ42λ1+12βσδ12δ52−24βσδ1δ2δ4δ5+12βσδ22δ42−12βδ12δ52+24βδ1δ2δ4δ5−12βδ22δ42−4σδ12δ52+8σδ1δ2δ4δ5−4σδ22δ42.


#### 9.1.2. Second Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(88)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(89)fS=δ1expΩ+δ1δ5δ4expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ6,Ω=−tω1α2δ1δ6−δ3δ4+xε1+yε2+ε4,Ξ=−ω2tα2δ1δ6−δ3δ4+xε1−λ2y−λ4,ω1=α2δ1δ6ε13−α2δ3δ4ε13−3α2δ1δ6ε1ε2−3α2δ3δ4ε1ε2+6αβσδ1δ6ε2−6αβσδ3δ4ε2−2ασδ1δ6ε2+2ασδ3δ4ε2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1,ω2=α2δ1δ6ε13−α2δ3δ4ε13+3α2δ1δ6ε1λ2+3α2δ3δ4ε1λ2−6αβσδ1δ6λ2+6αβσδ3δ4λ2+2ασδ1δ6λ2−2ασδ3δ4λ2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1.


## 9.1. Solitary Wave Solutions

Here, we will consider the solitary wave solution by selecting the below function in which (1) has been taken as follows:(84)f=δ1expε1x+ε2y+ε3t+ε4+δ2expλ1x+λ2y+λ3t+λ4+δ3,(85)g=δ4expε1x+ε2y+ε3t+ε4+δ5expλ1x+λ2y+λ3t+λ4+δ6.Plugging (84) and (85) into Eq. (3) and collecting all relevant coefficients of expΩ and expΞ, a series of equations have been obtained. Solving these equations, we have the following solutions:


### 9.1.1. First Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(86)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(87)fS=δ1expΩ+δ2expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ3δ42δ1δ5∫1+δ1δ5λ1+δ2δ4λ1δ1δ1δ5λ1+2δ2δ4∫1+δ2δ4λ1,Ω=−tω1δ1δ5−δ2δ4α2+xϵ1+yϵ2+ϵ4,Ξ=−13ω2λ1tδ1δ5−δ2δ42α2−λ1x−13λ13αδ1δ5∫1+3αδ1δ5λ1+3αδ2δ4∫1+3αδ2δ4λ1−2δ1δ5+2δ2δ4yαδ1δ5−δ2δ4−λ4,ω1=α2δ1δ5ε13−α2δ2δ4ε13−3α2δ1δ5ε1ε2−3α2δ1δ5ε2λ1−3α2δ2δ4ε1ε2−3α2δ2δ4ε2λ1+6αβσδ1δ5ε2−6αβσδ2δ4ε2−2ασδ1δ5ε2+2ασδ2δ4ε2−2αδ1δ5ε12−2αδ1δ5ε1λ1−2αδ2δ4ε12−2αδ2δ4ε1λ1+4βδ1δ5ε1−4βδ2δ4ε1,ω2=9α2δ12δ52ε1λ1+6α2δ12δ52λ12+36α2δ1δ2δ4δ5ε12+54α2δ1δ2δ4δ5ε1λ1+24α2δ1δ2δ4δ5λ12+9α2δ22δ42ε1λ1+6α2δ22δ42λ12−18αβσδ12δ52ε1−18αβσδ12δ52λ1+18αβσδ22δ42ε1+18αβσδ22δ42λ1+6ασδ12δ52ε1+6ασδ12δ52λ1−6ασδ22δ42ε1−6ασδ22δ42λ1+12βσδ12δ52−24βσδ1δ2δ4δ5+12βσδ22δ42−12βδ12δ52+24βδ1δ2δ4δ5−12βδ22δ42−4σδ12δ52+8σδ1δ2δ4δ5−4σδ22δ42.


### 9.1.2. Second Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(88)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(89)fS=δ1expΩ+δ1δ5δ4expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ6,Ω=−tω1α2δ1δ6−δ3δ4+xε1+yε2+ε4,Ξ=−ω2tα2δ1δ6−δ3δ4+xε1−λ2y−λ4,ω1=α2δ1δ6ε13−α2δ3δ4ε13−3α2δ1δ6ε1ε2−3α2δ3δ4ε1ε2+6αβσδ1δ6ε2−6αβσδ3δ4ε2−2ασδ1δ6ε2+2ασδ3δ4ε2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1,ω2=α2δ1δ6ε13−α2δ3δ4ε13+3α2δ1δ6ε1λ2+3α2δ3δ4ε1λ2−6αβσδ1δ6λ2+6αβσδ3δ4λ2+2ασδ1δ6λ2−2ασδ3δ4λ2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1.


## 9.1.1. First Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(86)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(87)fS=δ1expΩ+δ2expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ3δ42δ1δ5∫1+δ1δ5λ1+δ2δ4λ1δ1δ1δ5λ1+2δ2δ4∫1+δ2δ4λ1,Ω=−tω1δ1δ5−δ2δ4α2+xϵ1+yϵ2+ϵ4,Ξ=−13ω2λ1tδ1δ5−δ2δ42α2−λ1x−13λ13αδ1δ5∫1+3αδ1δ5λ1+3αδ2δ4∫1+3αδ2δ4λ1−2δ1δ5+2δ2δ4yαδ1δ5−δ2δ4−λ4,ω1=α2δ1δ5ε13−α2δ2δ4ε13−3α2δ1δ5ε1ε2−3α2δ1δ5ε2λ1−3α2δ2δ4ε1ε2−3α2δ2δ4ε2λ1+6αβσδ1δ5ε2−6αβσδ2δ4ε2−2ασδ1δ5ε2+2ασδ2δ4ε2−2αδ1δ5ε12−2αδ1δ5ε1λ1−2αδ2δ4ε12−2αδ2δ4ε1λ1+4βδ1δ5ε1−4βδ2δ4ε1,ω2=9α2δ12δ52ε1λ1+6α2δ12δ52λ12+36α2δ1δ2δ4δ5ε12+54α2δ1δ2δ4δ5ε1λ1+24α2δ1δ2δ4δ5λ12+9α2δ22δ42ε1λ1+6α2δ22δ42λ12−18αβσδ12δ52ε1−18αβσδ12δ52λ1+18αβσδ22δ42ε1+18αβσδ22δ42λ1+6ασδ12δ52ε1+6ασδ12δ52λ1−6ασδ22δ42ε1−6ασδ22δ42λ1+12βσδ12δ52−24βσδ1δ2δ4δ5+12βσδ22δ42−12βδ12δ52+24βδ1δ2δ4δ5−12βδ22δ42−4σδ12δ52+8σδ1δ2δ4δ5−4σδ22δ42.


## 9.1.2. Second Solution

Under the transformationu=−2/αlng/fx, the corresponding special solitary wave solution is read as follows:(88)uS=−2αgSfSdfS/dxgS−fSdgS/dxgS2,in which,(89)fS=δ1expΩ+δ1δ5δ4expΞ+δ3,gS=δ4expΩ+δ5expΞ+δ6,Ω=−tω1α2δ1δ6−δ3δ4+xε1+yε2+ε4,Ξ=−ω2tα2δ1δ6−δ3δ4+xε1−λ2y−λ4,ω1=α2δ1δ6ε13−α2δ3δ4ε13−3α2δ1δ6ε1ε2−3α2δ3δ4ε1ε2+6αβσδ1δ6ε2−6αβσδ3δ4ε2−2ασδ1δ6ε2+2ασδ3δ4ε2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1,ω2=α2δ1δ6ε13−α2δ3δ4ε13+3α2δ1δ6ε1λ2+3α2δ3δ4ε1λ2−6αβσδ1δ6λ2+6αβσδ3δ4λ2+2ασδ1δ6λ2−2ασδ3δ4λ2−2αδ1δ6ε12−2αδ3δ4ε12+4βδ1δ6ε1−4βδ3δ4ε1.


## 10. Conclusion

In this paper, the lump solutions, lumpoff solutions, and rogue waves with predictability of the (2 + 1)-dimensional Gardner equation are investigated with more arbitrary autocephalous parameters. It is not hard to see that the general lump solution is an algebraically localized wave decayed in all space directions and exists in all time. We constructed a general quadratic function because of deriving the general lump solution for this equation. Hence, the lumpoff solutions are shown with more free autocephalous parameters, in which the lump solution localized in all directions in space. Moreover, when the lump solution is cut by twin-solitons, the special rogue waves are also introduced. Furthermore, we obtain new sufficient solutions containing cross-kink, periodic-kink, multiwaves, and solitary wave solutions. From the acquired results, it can be concluded that the procedures followed in this analysis can be implemented in a simple and straightforward manner to create new exact solutions of a lot of other nonlinear partial differential equations in terms of Hirota operator.



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*Source: 9360263-2022-10-14.xml*