# New Exact Solutions and Modulation Instability for the Nonlinear (2+1)-Dimensional Davey-Stewartson System of Equation

Authors: Kwasi Boateng; Weiguo Yang; Wilson Osafo Apeanti; David Yaro
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

The Davey-Stewartson Equation (DSE) is an equation system that reflects the evolution in finite depth of soft nonlinear packets of water waves that move in one direction but in which the waves’ amplitude is modulated in spatial directions. This paper uses the Generalized Elliptic Equation Rational Expansion (GEERE) technique to extract fresh exact solutions for the DSE. As a consequence, solutions with parameters of trigonometric, hyperbolic, and rational function are achieved. To display the physical characteristics of this model, the solutions obtained are graphically displayed. Modulation instability assessment of the outcomes acquired is also discussed and it demonstrates that all the solutions built are accurate and stable.


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## Body

## 1. Introduction

Nonlinear partial differential equations (NLPDEs) are used in multiple study areas to define significant phenomena. Exact NLPDEs solutions play an important part in the research of physics, applied mathematics, and engineering, including solid state physics, fluid mechanics, population ecology, plasma physics, plasma waves, biology, optical fibres, propagation of shallow waves, heat flow, quantum mechanics, and wave propagation phenomena. Current studies are underway to find new techniques to extract traveling wave solution for NLPDEs. These techniques include the tanh-sech method [1], the Jacobi elliptic function method [2], the homotopy perturbation method [3], the homogeneous balance method [4], sine-cosine method [5, 6], the extended Weierstrass transformation method [7–10], the expansion method [11, 12], the Ba¨cklund transformation method [13], the inverse scattering method [14], modified extended mapping method [15], Darboux transformation [16], the extended tanh-coth method [17], Hirota bilinear method [18], modified extended direct algebraic method [19], lumped Galerkin method [20], and auxiliary equation method [21]. A lot of these methods are dependent on the type of problem and may or may not be suitable for other different problems [22–31].The Davey-Stewartson Equation (DSE) [32] is an equation system that reflects the evolution in finite depth of soft nonlinear packets of water waves traveling in one direction but in which the amplitude of waves is modulated in spatial directions. This equation also defines long wave–short wave resonances and the development of a 3-dimensional wave packet on finite depth water [33, 34]. A few solutions have been acquired for this equation [35–38]. The DSE, a two-dimensional system that coexists with both short waves and long waves, and a precise representation of two-dimensional modulation of nonlinear waves should require both short wave and long wave modes. A pair of coupled nonlinear DSE in two dependent variables can be decreased to the (1+1)-dimensional nonlinear Schrödinger (NLS) equation by carrying out a suitable dimensional reduction. A number of analytical methods have been created to be used for NLPDEs such as the DSE, which in recent years have specific types of solutions, such as growing and decaying solutions [39–41], dromions, breathers, instantaneous, propagating, and regular wave patterns, and fluid flow and heat transfer over a stretching or shrinking sheet in a porous medium [34, 42, 43].The Generalized Elliptic Equation Rational Expansion (GEERE) technique developed by Wan, Song, Yin, and Zhang [44] is applied in this paper. This technique is a very strong technique that can be used to obtain several particular solutions including rational formal regular triangular, Weierstrass doubly regular, rational formal solitary wave, and rational formal Jacobi solutions [44]. In addition, the modulation instability assessment of the solutions obtained will be explored. Modulation instability is universal, and the basis for the formation of soliton solutions results from the interaction between dispersion and nonlinearity in the spatial and time domain [45–47]. It is therefore of interest to derive explicit DSE equation solutions using the GEERE method. This paper is structured as follows: in Section 2 new soliton solutions are built for DSE. DSE assessment of modulation instability is shown in Section 3. Section 4 also presents the outcomes and discussion. Finally, in Section 5, the conclusion is provided.


## 2. Nonlinear (2+1)-Dimensional Davey-Stewartson Equation

The Davey-Stewartson system [32] is of the form(1)i∂Z∂t+12α2∂2Z∂x2+α2∂2Z∂y2-νZ2Z+Z∂Q∂x=0,∂2Q∂x2-α2∂2Q∂y2-2ν∂Z2∂x=0.Here z(x,y,t) is a complex value function, x is the dimensionless variable, y is the propagation coordinate, t is the time, ν=±1, and α2=±1. The case where α=1 is called the DSE I equation, while that where α=i is the DSE II. The ν parameter denotes the focusing or defocusing case. The DSE has the following types of soliton solutions: conventional line, algebraic, periodic, and lattice solution. Conventional line soliton has a structure that is essentially one-dimensional. The solitons of periodic, algebraic, and lattice have a structure of two dimensions. The DSE I and II are common examples of two-dimensional integrable equations that emerge as a higher-dimensional generalization of the nonlinear Schrödinger equation (NLSE). DSE has several applications that include the description of gravity-capillarity surface wave packets within the shallow water boundary. Several strong techniques have been created to obtain explicit solutions for (1), such as the technique of homotopy analysis [48], the sine-cosine method [49], and the technique of variational iteration [50]. To obtain fresh soliton solutions for (1) using the following traveling wave equation, we apply the widely discussed GEERE technique in [44]:(2)zx,y,t=Zξeiγ,qx,y,t=Qξ,ξ=x+y-ct,γ=k1x+k2y+k3t,wherec,k1,k2, and k3 are real constants. Substituting (2) into (1) gives the real and imaginary part:(3)c=α2k1+α2k2,(4)α21+α2Z′′ξ+2νZ3ξ-2Q′ξ+k3+α2k12+α2k22Zξ,(5)1-α2Q′′ξ-2νZ2ξ′=0.Integrating (5) with respect to ξ and setting the constant of integration to zero, we have(6)Q′ξ=2ν1-α2Z2ξ.Putting (6) into (4) gives(7)α2α4-1Z′′ξ+2να2+1Z3ξ-α2-12k3+α2k12+α2k22Zξ=0.Introducing a new ansatz of finite rational expansion in the form:(8)Zξ=a0+∑i=1Naiϕiξ+biϕi-1ξϕ′ξμϕξ+1i,the new parameterϕ=ϕ(ξ) and(9)ϕ′=∑j=04ljϕj,where lj,a0,ai, and bi(j=0,1,2,…;i=1,2,3,…,N) are constant real values to be computed. Using homogeneous balance principle by balancing the highest nonlinear terms and the highest order partial derivative terms in (7) we have N=1; hence(10)Zξ=a0+a1ϕξ+b1ϕ′ξμϕξ+1,putting (10) and (9) in (7) we have a system of algebraic equations in parameters, l1,l2,l3,l4,a0,a0,a1,b1,μ. The system of algebraic equations after solving gives the following family of solutions with respect to different cases.Case 1.
l 4 = 0 , l 3 = 0.
Family 1.1. a0=±-α2α2-1l2/κ/2, a1=a1, b1=a1, k3=-1/4α22k22σ2+2k12+α2l2+l2, μ=0(11)Z11ξ=±-α2α2-1l2/κ2+a1el2-ct+x+y-l12l2±a1l2el2-ct+x+y-l12l22+l1el2-ct+x+y-l12l2+l0×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(12)Q11ξ=2ν31-α2,Z11ξ3,l0=l124l2,l2>0.(13)Z12ξ=±-α2α2-1l2/κ2+a1l2l12sin2l2--ct+x+y-1/l2+l1sinl2--ct+x+y-12l2×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(14)Q12ξ=2ν31-α2Z12ξ3,l0=0,l2<0.(15)Z13ξ=±-α2α2-1l2/κ2+a1l2l12sin2hl2-ct+x+y-1/l2+4l0+l1sinhl2-ct+x+y-12l2×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(16)Q13ξ=2ν31-α2Z13ξ3l0=0,l2>0.
Family 1.2. a0=0, a1=a1, b1=2α2α2-1(-κ)l22l12-4l0l22/κl13-4κl0l1l2±a1, k3=1/2α22l2α2+2l2+k22σ2+k12, μ=0(17)Z14ξ=a1el2-ct+x+y-l12l2±a1+l12l2el2-ct+x+y-l1/2l2+l12κl13-4κl0l1l2×l2el2-ct+x+y-l12l22+l1el2-ct+x+y-l12l2+l0-α2α2-1κl22l12-4l0l22×expik1x+k2y+12α22l2α2+2l2+k22σ2+k12,(18)Q14ξ=2ν31-α2Z14ξ3,l0=l124l2,l2>0.(19)Z15ξ=cscl2ct-x-y2κl13l2a1κl13l1sinl2ct-x-y-1-2l2+2l2-α2α2-1κl14l22×-l12cos2l2ct-x-yl2×expik1x+k2y+12α22l2α2+2l2+k22σ2+k12,(20)Q15ξ=2ν31-α2Z15ξ3,l0=0,l2<0.(21)Z16ξ=csch-l2ct-x-y2κl13l2a1κl13l1sinh-l2ct-x-y-1-2l2+2l2-α2α2-1κl14l22×-l12cos2h-l2ct-x-yl2×expik1x+k2y+12α22l2α2+2l2+k22σ2+k12,(22)Q16ξ=2ν31-α2Z16ξ3,l0=0,l2<0.Case 2.
l 1 = 0 , l 3 = 0.
Family 2.1. a0=--α2α2-1l2/κ/2, a1=a1, b1=a1, k3=-1/4α2t2k22σ2+2k12+α2l2+l2, μ=0(23)Z21ξ=a1κl0l42l23tan2l2-ct+x+y2l2l43tan2l2-ct+x+y2+4+16l0+2l2l4l0a1κtanl2-ct+x+y2-2l0-α2α2-1l22×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(24)Q21ξ=2ν31-α2Z213ξ3,l0=l124l2,l2>0,l4<0.(25)Z22ξ=a1κl0l42l23tan2hl2-ct+x+y2l2l43tan2hl2-ct+x+y2+4+16l0+2l2l4l0a1κtanhl2-ct+x+y2-2l0-α2α2-1l22×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(26)Q22ξ=2ν31-α2Z223ξ3,l0=l124l2,l2>0,l4>0.
Family 2.2. a0=0, a1=a1, b1=α4κl0l2-α2κl0l2-2a1κl0/2κl0, k3=-1/4α2t2k22σ2+2k12+α2l2+l2, μ=-il2/l0(27)Z23ξ=κl016l016l0-8il2l2l4tanl2-ct+x+y/2α2α2-1κl0l2-2a1κl0×l42l23tan2l2-ct+x+y2l2l43tan2l2-ct+x+y2+4+16l0+4a1l2l4κl0tanl2-ct+x+y2×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(28)Q23ξ=2ν31-α2Z233ξ3,l0=l124l2,l2>0,l4>0.(29)Z24ξ=κl016l016l0-8il2l2l4tanh-l2-ct+x+y/2α2α2-1κl0l2-2a1κl0×l42l23tan2-l2-ct+x+y2l2l43tanh2-l2-ct+x+y2+4+16l0+4a1l2l4κl0tan-l2-ct+x+y2×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(30)Q24ξ=2ν31-α2Z243ξ3,l0=l124l2,l2<0,l4>0.Case 3.
l 3 = 0 , l 4 = 0 , l 0 = 0.
Family 3.1. a0=0, a1=a1, b1=κh22α2-κl22α4-2a1κl1/2κl1, k3=-1/4σ2l2α2+l2+2k22α2+2k12, μ=l2/l1(31)Z31ξ=2a1κl1l1sinl2--ct+x+y-1-l2l12sin2l2--ct+x+y-1/l22κl1l2sinl2--ct+x+y+1+2κl1l2sinl2--ct+x+y+1+l2-α2α2-1κl22l12sin2l2--ct+x+y-1/l22κl1l2sinl2--ct+x+y+1×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(32)Q31ξ=2ν31-α2Z313ξ3,l2<0.(33)Z32ξ=2a1κl1l1sinhl2-ct+x+y-1-l2l12sin2hl2-ct+x+y-1/l22κl1l2sinhl2-ct+x+y+1+2κl1l2sinhl2-ct+x+y+1+l2-α2α2-1κl22l12sin2hl2-ct+x+y-1/l22κl1l2sinhl2-ct+x+y+1×expik1x+k2y-14α2t2k22σ2+2k12+α2l2+l2,(34)Q32ξ=2ν31-α2Z323ξ3,l2>0.
Family 3.2. a0=0, a1=a1, b1=2κh22α2-κl22α4-a1κl1/κl1, k3=-1/2α22l2α2+2l2+k22α2+k12, μ=2l2/l1(35)
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.Case 4.
l 0 = 0 , l 1 = 0 , l 4 = 0.
Family 4.1. a0=0, a1=a1, b1=α2κl32-α4κl32/l2-2a1κ/2κ, k3=-1/2α2t(k2α2+l2-k22α2-l12), μ=l3/l2(39)Z41ξ=a1l2csc21/2l2--ct+x+yl3-12κl2-α2α2-1κl32-2a1κl2cot212l2--ct+x+yl22sec41/2l2--ct+x+yl3-l22sec21/2l2--ct+x+yl33×expik1x+k2y-12α2tk2α2+l2-k22α2-l12,(40)Q41ξ=2ν31-α2Z413ξ3,l2<0.(41)Z42ξ=a1l2csch21/2l2-ct+x+yl3-12κl2-α2α2-1κl32-2a1κl2cot2h12l2-ct+x+yl22sec4h1/2l2-ct+x+yl3-l22sec2h1/2l2-ct+x+yl33×expik1x+k2y-12α2tk2α2+l2-k22α2-l12,(42)Q42ξ=2ν31-α2Z423ξ3,l2>0.Family 4.2. a0=0, a1=a1, b1=-2a1κ+i2μ/2κ, k3=1/2k12-k22, μ=0(43)Z43ξ=12κi2μ+2ia1κl22sec41/2l2--ct+x+yl3-l22sec21/2l2-ct+x+yl33+a1l2sec21/2l2--ct+x+yl3×expik1x+k2y+12tk12-k22,(44)Q43ξ=2ν31-α2Z433ξ3,l2<0.(45)Z44ξ=12κi2μ+2ia1κl22sech41/2l2-ct+x+yl3-l22sech21/2l2-ct+x+yl33+a1l2sec2h1/2l2-ct+x+yl3×expik1x+k2y+1/2tk12-k22,(46)Q44ξ=2ν31-α2Z443ξ3,l2>0.Case 5.
l 0 = 0 , l 1 = 0.
Family 5.1. a0=0, a1=a1, b1=a1-κl3-4i2l4/κl3, k3=1/2k12-k22, μ=4l4/l3(47)Z51ξ=14l2l4sec2-l2ξ/2/l32-l2l4tan1/2-l2ξ+l3+1a1l2sec2-l2ξ/22-l2l4tan1/2-l2ξ+l3+a1-κl3-4i2l4κl3×l4l24sec8-l2ξ/22-l2l4tan1/2-l2ξ+l34+l3l23sec6-l2ξ/22-l2l4tan1/2-l2ξ+l33+l23sec4-l2ξ/22-l2l4tan1/2-l2ξ+l321/2×expik1x+k2y+12tk12-k22,(48)Q51ξ=2ν31-α2Z513ξ3,l2<0.(49)Z52ξ=14l2l4sec2l2ξ/2/l32l2l4tan1/2l2ξ+l3+1a1l2sec2l2ξ/22l2l4tan1/2l2ξ+l3+a1-κl3-4i2l4κl3×l4l24sec8l2ξ/22l2l4tan1/2l2ξ+l34+l3l23sec6l2ξ/22l2l4tan1/2l2ξ+l33+l23sec4l2ξ/22l2l4tan1/2l2ξ+l321/2×expik1x+k2y+12tk12-k22,(50)Q52ξ=2ν31-α2Z523ξ3,l2>0.


## 3. Modulation Instability of Davey-Stewartson Equation

Equation (1) reveals a certain formal connection with the defocusing nonlinear Schrödinger equation in 1+1 dimension [51, 52], given below(51)i∂Z∂t+12α2∂2Z∂x2-νZ2Z=0.Most nonlinear higher-order NLPDEs demonstrate instability leading to the research of steady-state modulation as a result of interaction between nonlinear and dispersive impacts. The modulation instability analysis of (51) is carried out by using the standard linear stability analysis [53, 54] to check how weak and time dependent perturbations develop along the propagation distance [53, 54]. The 1+1 dimension defocusing nonlinear Schrödinger equation (51) of the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation (1) has a state solution of the form(52)Zx,y,t=P+ϕx,y,teiγ,γt=Pβt,where the optical power P is normalized and β is a constant. The perturbation ϕ(x,y,t) is examined by utilizing linear stability analysis. Putting (52) into (51) and linearizing, we obtain(53)i∂ϕ∂t+12α2∂2ϕ∂x2+P3β+ν-3Pνϕ∗=0,where ∗ is a complex conjugate. The solution of (53) is considered in the form giving below(54)ϕx,y,t=ψ1eih1x+h2y-tλ+ψ2e-ih1x+h2y-tλ,where λ is the frequency of perturbation and h1,h2 normalized wave numbers. The dispersion relation determines how spatial oscillations eih1x,eih2y are linked to time oscillations eiλt of a wave number; substituting (54) into (53), we obtain the dispersive relation as(55)λ=±α2ψ1h12+6ψ2νP2ψ1,ψ1≠0.The dispersion relation in (55) above demonstrates that the steady-state stability depends on the self-phase modulation, stimulated scattering, group velocity dispersion, and self-phase modulation. The expression of (55) shows that the value of the frequency λ is real for all values of h except when ψ=0, which means that the steady-state solution is stable.


## 4. Results and Discussion

Rational, periodic, and solitonic solutions for the DSE have been successfully obtained using the GEERE method. The constructed solutions by this method are novel and unique from other solutions obtained using different methods [48–50]. Different forms of solutions such as trigonometric, hyperbolic trigonometric, and rational functions are obtained because the assumed solution of GEERE is different from other methods. Physical properties of some obtained results are demonstrated graphically using suitable parameters. Figure 1 evaluates the rational solitary wave solution (11), periodic solitary wave solution (13), and soliton solitary wave solution (15) plotted at (a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=0.1;y=0.8), (a1=1;α=0.6;l0=0;l1=9;l2=-30;κ=-0.7;c=0.1;y=0.8), and (a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=0.1;y=0.8), respectively. The rational solitary wave solution (17), periodic solitary wave solution (19), and soliton solitary wave solution (21) are evaluated in Figure 2 plotted at (a1=1;α=0.6;l0=0;l1=9;l2=20;κ=0.7;c=-0.1;y=0.8), (a1=1;α=0.6;l0=0;l1=9;l2=-30;κ=-0.7;c=-0.1;y=0.8), and (a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=-0.1;y=0.8), respectively. Figure 3 shows a periodic solitary wave solution (23) and a dark solitary wave solution (25) plotted at (a1=1;α=0.6;l0=0.7;l4=9;l2=30;κ=-0.7;c=0.1;y=0.8) and (a1=1;α=0.6;l0=0.7;l4=9;l2=-10;κ=0.7;c=0.1;y=0.8), respectively. Periodic solitary wave solution (27) and dark solitary wave solution (29) plotted at (a1=1;α=0.6;l0=0.7;l4=9;l2=30;κ=-0.7;c=0.1;y=0.8) and (a1=1;α=0.6;l0=0.7;l4=9;l2=-10;κ=0.7;c=0.1;y=0.8), respectively, are evaluated in Figure 4. Figure 5 evaluates bright solitary wave solution (31) and soliton solitary wave solution (33) plotted at (a1=1;α=0.6;l1=9;l2=-10;κ=0.7;c=0.1;y=0.8) and (a1=1;α=0.6;l1=9;l2=10;κ=0.7;c=0.1;y=0.8), respectively. Periodic solitary wave solution (35) and soliton solitary wave solution (37) plotted at (a1=1;α=0.6;l1=9;l2=-10;κ=0.7;c=0.1;y=0.8) and (a1=9;α=0.6;l1=9;l2=10;κ=0.7;c=0.1;y=0.8), respectively, are evaluated in Figure 6. Figure 7 shows periodic solitary wave solution (39) and dark solitary wave solution (41) plotted at (a1=9;α=0.6;l2=-1;l3=-1;κ=0.7;c=0.1;y=0.8) and (a1=9;α=0.6;l2=9;l3=1;κ=0.7;c=0.1;y=0.8), respectively. Figure 8 shows periodic solitary wave solution (43) and dark solitary wave solution (45) plotted at (a1=9;α=0.6;l2=-1;l3=-1;κ=0.7;c=0.1;y=0.8) and (a1=1;α=0.6;l2=1;l3=1;κ=0.7;c=0.1;y=0.8), respectively. Also periodic solitary wave solution (47) and dark solitary wave solution (49) plotted at (a1=9;l2=-3;l3=-3;l4=1;κ=0.7;c=0.1;y=0.8) and (a1=1;l2=5;l3=5;l4=0;κ=0.7;c=0.1;y=0.8), respectively, are evaluated in Figure 9. The dispersion relation among frequency (λ) and wave numbers (h1,h2) of (55) is plotted in Figure 10. This relation determines how spatial oscillations eih2x,eih2y are linked to time oscillations eiλt of a wave number of obtained solutions for the DSE.Figure 1
Plot of the exact traveling wave solutions of (a) rational solitary wave solution (11) with parameters a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=0.1;y=0.8; (b) periodic solitary solution (13) plotted at a1=1;α=0.6;l0=0;l1=9;l2=-30;κ=-0.7;c=0.1;y=0.8; and (c) soliton solitary solution (15) plotted at a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=0.1;y=0.8.
(a) (b) (c)Figure 2
Plot of the exact traveling wave solutions of (a) rational solitary wave solution (17) with parameters a1=1;α=0.6;l0=0;l1=9;l2=20;κ=0.7;c=-0.1;y=0.8; (b) periodic solitary wave solution (19) plotted at a1=1;α=0.6;l0=0;l1=9;l2=-30;κ=-0.7;c=-0.1;y=0.8; and (c) soliton solitary wave solution (21) plotted at a1=1;α=0.6;l0=0.7;l1=9;l2=20;κ=0.7;c=-0.1;y=0.8.
(a) (b) (c)Figure 3
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (23) with parameters a1=1;α=0.6;l0=0.7;l4=9;l2=20;κ=0.7;c=0.1;y=0.8; (b) dark solitary wave solution (25) plotted at a1=1;α=0.6;l0=0;l4=9;l2=-30;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 4
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (27) with parameters a1=1;α=0.6;l0=0.7;l4=9;l2=30;κ=-0.7;c=0.1;y=0.8; (b) dark solitary wave solution (29) plotted at a1=1;α=0.6;l0=0.7;l4=9;l2=-10;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 5
Plot of the exact traveling wave solutions of (a) bright solitary wave solution (31) with parameters a1=1;α=0.6;l1=9;l2=-10;κ=0.7;c=0.1;y=0.8; (b) soliton solitary wave solution (33) plotted at a1=1;α=0.6;l1=9;l2=10;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 6
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (35) with parameters a1=1;α=0.6;l1=9;l2=-10;κ=0.7;c=0.1;y=0.8; (b) soliton solitary wave solution (37) plotted at a1=9;α=0.6;l1=9;l2=10;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 7
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (39) with parameters a1=9;α=0.6;l2=-1;l3=-1;κ=0.7;c=0.1;y=0.8; (b) dark solitary wave solution (41) plotted at a1=9;α=0.6;l2=9;l3=1;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 8
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (43) with parameters a1=9;α=0.6;l2=-1;l3=-1;κ=0.7;c=0.1;y=0.8; (b) dark solitary wave solution (45) plotted at a1=1;α=0.6;l2=1;l3=1;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 9
Plot of the exact traveling wave solutions of (a) periodic solitary wave solution (47) with parameters a1=9;l2=-3;l3=-3;l4=1;κ=0.7;c=0.1;y=0.8; (b) dark solitary wave solution (49) plotted at a1=1;l2=5;l3=5;l4=0;κ=0.7;c=0.1;y=0.8.
(a) (b)Figure 10
Diagram of dispersion relation among frequency (λ) and wave number (h1).


## 5. Conclusion

By implementing the strong GEERE technique, this article has effectively built fresh solitonic and reasonable regular solutions for the DSE. Some freshly obtained solitary wave solutions are provided graphically to display their physical characteristics and also provide information on the circumstances for the creation of solitons of light and kink. The solutions extracted have theoretical and experimental applications that can be anticipated to occur in other associated revolutionary equations such as the nonlinear Schrödinger equation to explain the Bose-Einstein condensation. We contrasted our solution to current existing solution and claimed that there are many new solutions. The equation recognize the huge variety of possible solutions for only values of a small subset of parameters, which helps to comprehend the physical phenomena of this equation. Modulation instability assessment is used to verify the stability of the solutions acquired. We conclude that the modulation instability analytical expression is gained, which demonstrates that all built solutions are accurate and stable. The effectiveness and simplicity of the suggested GEERE technique show that it can be applied to distinct kinds of separate nonlinear models in different nonlinear science fields.



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