# New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups

Authors: Khudija Bibi; Khalil Ahmad
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, new exact solutions of2+1-dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2+1-dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) have attractive applications in engineering and many physical sciences [1–6]. The soliton and rational solutions for some NLEEs have been discussed and constructed in different ways by many researchers, e.g., see in [7–13]. Methods for deriving the soliton solutions for the NLEEs such as Painleve technique [14], the inverse scattering transformation [15], Backlund transformation [16], Hirota method [17–19], Darboux transformation [20–24], binary-Bell-polynomial scheme [25, 26], first integral method [27], homotopy asymptotic [28], G′/G expansion [29, 30], and Exp-function method [31, 32] have been suggested. Many researchers have considered the following integrable equation [33, 34],(1)wxxxxy+4wxxywx+2wxxxwy+6wxywxx+wyyy−2wxxt=0,to analyze the (2 + 1)-dimensional DJKM equation.Lie group method initiated by Sophus Lie has a powerful role to discover the solutions of partial differential equations [35–37]. Lie groups of transformations can be used in several ways to analyze the solutions of partial differential equation, see [38–42]. Exact solutions of partial differential equations express dynamic mathematical and physical phenomena. Lie symmetry generators corresponding to Lie groups are also used to find the group invariant solutions. These group invariant solutions lead to the many exact solutions of partial differential equations. Many researchers have applied this method to find the solutions of the different partial differential equations, e.g., see [40, 43–49]. In this article, exact solutions of DJKM equation are explored by utilizing the Lie group method.In the current work, using the similarity variables obtained from Lie symmetry generators,2+1-dimensional DJKM equation is transformed into a linear PDE, and afterwards by using the Lie symmetries of this linear PDE, new exact solutions of DJKM equation are obtained.This paper is arranged into four sections as follows: in Section2, Lie symmetry generators of DJKM equation are presented. In Section 3, DJKM is transformed into a linear PDE by considering the similarity variables which are obtained by taking linear combination of two Lie symmetry generators of DJKM equation while Lie symmetry generators of obtained linear PDE are also presented in this section. The group invariant solutions of the obtained linear PDE which lead to the exact solutions of DJKM are discussed in Section 4. Finally, in Sections 5 and 6, the summary and conclusion of the present work are discussed.


## 2. Lie Symmetries of Date Jimbo Kashiwara Miwa Equation

The main purpose of this paper is to investigate the exact solutions of the DJKM equation given by (1) by using its Lie symmetry analysis.The method to obtain the Lie symmetries of the partial differential equations has been discussed in many books, e.g., see in [35, 36, 38, 41]. Let(2)X=ϕ1x,y,t,w∂x+ϕ2x,y,t,w∂y+ϕ3x,y,t,w∂t+η1x,y,t,w∂w,be the vector field that generates the symmetry group of equation (1). By applying the second order prolongation of X to equation (1), following determining equations are obtained(3)η1w=−14ϕ3t,η1x=−12ϕ2t,η1y=−ϕ1t,ϕ1w=0,ϕ1x=14ϕ3t,ϕ1y=0,ϕ2w=0,ϕ2x=0,ϕ2y=12ϕ3t,ϕ3w=0,ϕ3x=0,ϕ3y=0.The infinitesimals obtained by solving the above system are(4)η1=−14C8w−12C7x−C6y+C5t+C4,ϕ1=14C8x+C6t+C3,ϕ2=12C8y+C7t+C2,ϕ3=C8t+C1.Thus, the Date Jimbo Kashiwara Miwa equation (1) is spanned by the following vector fields:(5)X1=∂t,X2=∂y,X3=∂x,X4=∂w,X5=t∂w,X6=t∂x−y∂w,X7=t∂y−12x∂w,X8=14x∂x+12y∂y+t∂t−14w∂w,which form eight dimensional Lie algebras.


## 3. Conversion of DJKM Equation into a Linear PDE by Using Similarity Variables

In this section, DJKM equation (1) is transformed to a linear PDE with two independent variables and the symmetry generators of this PDE are also presented here.The similarity variables corresponding to(6)X5+X7=t∂w+t∂y−12x∂w,are(7)ϕ1=x,ϕ2=t,uϕ1,ϕ2=12−2yt+xy+2wtt,which imply that solution of (1) is in the form,(8)wx,y,t=uϕ1,ϕ2+y1−x2t.By using similarity variables given in (7), equation (1) is transformed into the following PDE:(9)21−ϕ12ϕ2uϕ1ϕ1ϕ1−3ϕ2uϕ1ϕ1−2uϕ1ϕ1ϕ2=0.For simplicity, we putϕ1=x and ϕ2=t so that the above PDE takes the following form:(10)21−x2tuxxx−3tuxx−2uxxt=0.The PDE given in (10) is spanned by the following vector fields:(11)X1=∂u,X2=u∂u,X3=t∂u,X4=x∂u,X5=x∂x+t∂t,X6=t2∂u,X7=tx∂u,X8=t∂x−12t∂t,X9=1+u∂u,X10=t+u∂u,X11=x+u∂u,X12=t∂x+∂u,X13=t∂x+t∂u,X14=t∂x+x∂u,X15=x+2t∂x+∂u,X16=x+2t∂x+t∂u,X17=x+2t∂x+x∂u,X18=u+tx∂u,X19=t∂x+tx∂u,X20=x+2t∂x+tx∂u,X21=32x+t∂x+t∂t−12u∂u,X22=−t+12x∂x+t∂t+12u∂u,X23=12x−t+t∂x+t∂t−12u∂u,X24=−1+12xt∂x+∂t+1−12tu∂u,X25=x2t−1+t∂x+∂t−u2t∂u,X26=121+2ttx−1+2t∂x+∂t−u2t∂u.


## 4. Exact Solutions of Date Jimbo Kashiwara Miwa Equation

In this section, various exact solutions of (1) by means of the group invariant solutions of PDE given in (10) which are obtained by considering different linear combinations of the Lie symmetry generators of (10) are presented.(1)
For(12)X5+X6=x∂x+t∂t+t2∂u,similarity variables are(13)ν=tx,μ=−12t2+u,which convert PDE (10) into the following ODE:(14)ν22ν+1μννν+ν5+12νμνν+41+3νμν=0,with solution(15)μ=C1ν+2ln2ν+1ν+ln2ν+1−2lnνν−lnν−1νC2+2ν+1νC3.So,(16)ux,t=C1x+2C2ln2t+x/xt+C2ln2t+x/xx−2C2lnt/xt−C2lnt/xx−C2x+2C3t+C3xt+12t2is a solution of (10) and thus(17)wx,y,t=2t+xlnx+2t/xC2−2t+xlnt/xC2+C1−C2+C3x+2C3tt+t22+y1−x2tis a solution of (1).(2)
For(18)X13=32x+t∂x+t∂t−12u∂u,similarity variables are(19)ν=x+2tt3/2,μ=ut,which convert PDE (10) into the following ODE:(20)2νμννν+4μνν=0,with solution(21)μ=C1+C2ν+C3lnν.So,(22)ux,t=C1t3/2+2C2x/2+t+C3lnx+2t/t3/2t3/2t2,is a solution of (10) and thus(23)wx,y,t=C1t3/2+2C2x/2+t+C3lnx+2t/t3/2t3/2t2+y1−x2tis a solution of (1).(3)
For(24)X14=−t+12x∂x+t∂t+12u∂u,similarity variables are(25)ν=x+2tt,μ=wt,which convert PDE (10) into the following ODE:(26)μνν=0,with solution(27)μ=C1+C2ν.So,(28)ux,t=C1t+2C2x2+t,is a solution of (10) and thus(29)wx,y,t=C1t+2C2x2+t+y1−x2t,is a solution of (1).(4)
For(30)X15=12x−t+t∂x+t∂t−12u∂u,similarity variables are(31)ν=x+2t−tlntt,μ=wt,which convert PDE (10) into the following ODE:(32)μννν=0,with solution(33)μ=C1+C2ν+C3ν2.So, solution of (10) is(34)ux,t=4C1x+C1lnt2−2C2lnt+2C3t3/2+4C1t5/2+C1x2t−4C1lnt−C2x/2+tt2t2,and hence(35)wx,y,t=4C1x+C1lnt2−2C2lnt+2C3t3/2+4C1t5/2+C1x2t−4C1lnt−C2x/2+tt2t2+y1−x2t,is a solution of (1).(5)
For(36)X16=−1+12xt∂x+∂t+1−12tu∂u,similarity variables are(37)ν=x+2tt,μ=wtexpt,which convert PDE (10) into the following ODE:(38)μνν=0,with solution(39)μ=C1+C2ν.So, solution of (10) is(40)ux,t=C1t+2C2x/2+ttexp−t,and thus(41)wx,t,y=C1t+2C2x/2+ttexp−t+y1−x2tis a solution of (1).(6)
For(42)X17=x2t−1+t∂x+∂t−u2t∂u,similarity variables are(43)ν=x+2t−t3/2t,μ=wt,which convert PDE (10) into ODE:(44)μννν=0,with solution(45)μ=C1+C2ν+C3ν2.So,(46)ux,t=4C1x+2C3t3/24C1−2C2t5/2+C1t7/2+C1x2t−4−C2+C1tx/2+tt2t2,is a solution of (10) and hence(47)wx,y,t=4C1x+2C3t3/2+4C1−2C2t5/2+C1t7/2+C1x2t−4−C2+C1tx/2+tt2t2+y1−x/2t,is a solution of (1).(7)
For(48)X18=121+2ttx−1+2t∂x+∂t−u2t∂u,similarity variables are(49)ν=x+2texp−tt,μ=ut,which convert PDE (10) into the following ODE:(50)νμννν+2μνν=0,with solution(51)μ=C1+C2ν+C3lnν.So,(52)ux,t=C1t+2C2x/2+texp−t+C3lnx+2texp−t/ttt,is a solution of (10). Thus,(53)wx,y,t=C1t+2C2x/2+texp−t+C3lnx+2texp−t/ttt+y1−x2t,is a solution of (1).Graphical representations for exact solutions of (1), obtained in equations (6)–(10) are presented in Figures 1–5, respectively, to show the significance of new exact solutions of the considered equation.Figure 1
Graph ofw given by (6) for C1=C2=C3=1.Figure 2
Graph ofw given by (7) for C1=C2=C3=1.Figure 3
Graph ofw given by (8) for C1=C2=1.Figure 4
Graph ofw given by (9) for C1=C2=C3=1.Figure 5
Graph ofw given by (10) for C1=C2=1.


## 5. Results and Discussion

In [50], Lie symmetry generators of DJKM equation involving arbitrary functions of t have been discussed through which exact solutions are difficult to find, while in [51], exact solutions of DJKM equation using exponential method are discussed. But, in the current article, first, using the Lie symmetry generators, DJKM equation given in (1) is transformed into a linear PDE (10), and afterwards using the invariant solutions of (10), new and significant exact solutions of (1) are presented.Graphs for different solutions of DJKM equation show the various behaviors of the considered model for different values ofx and t and can be helpful to understand the physical meaning of the DJKM equation.


## 6. Conclusion

In this paper, Lie symmetry analysis is applied to2+1-dimensional DJKM equation through which several new exact solutions of this equation are obtained.In this article, considering a linear combination of only two symmetry generators, DJKM equation is transformed into a linear partial differential equation, but by using different linear combinations of the other symmetry generators, considered equation can be transformed to ODEs and solutions of these ODEs can lead to other exact solutions.



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