# New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

Authors: Khalil Ahmad; Khudija Bibi; Muhammad Shoaib Arif; Kamaleldin Abodayeh
Journal: Journal of Function Spaces
 (2023)

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## Abstract

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.


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## Body

## 1. Introduction

Exact solutions of nonlinear partial differential equations play an important role in nonlinear science, especially in nonlinear physical science because they can carry substantial physical information and more understanding into the physical aspects of the problem. [1]. Recently, several methods have been used to find exact solutions of nonlinear model equations like Cahn-Allen equation [2], 2+1-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation [3], Newell-Whitehead-Segel (NWS) equations [4], the Chaffee–Infante equation [5], DNA Peyrard–Bishop equation [6], Burger’s equation [7], the 2+1-dimensional nonlinear Sharma–Tasso–Olver equation [8], and Ablowitz-Kaup-Newell-Segur water wave equation [9]. Recently, a number of concrete techniques have been recognized for finding accurate and comprehensible solutions of nonlinear physical models with the help of computer algebra, such as Maple, MATLAB, and Mathematica. These include power ondex method [7, 9], lie symmetry groups [3, 4], new extended direct algebraic method, and the generalized Kudryashov method [10].The Landau-Ginzburg-Higgs (LGH) equation was introduced by Lev Devidovich Landau and Vitaly Lazarevich Ginzburg having very wide range of applications in radially inhomogeneous plasma having a constant phase relation of ion-cyclotron waves. It demonstrated superconductivity and unidirectional wave propagation in nonlinear media [11]. Several approaches have been adopted to attain the close form and approximate solutions of the LGH equation. Khuri has investigated the close form solutions of LGH, Klein–Gordon, and Sine–Gordon equations by using unified approach [12]. In [13], the new modification method is proved by solving LGH equation and Cahn-Allen equation.
(1)∂2∂t2ux,t−∂2∂x2ux,t−m2ux,t+n2u3x,t=0.In this paper, we will examine the solutions of Landau-Ginzburg-Higgs equation by using polynomial and elementary functions. In Section2, we designate temporarily the PIM and explain index powers by taking first partial derivative of dependent function with respect to time. In Section 3, we apply this method to the LGH equation by introducing new transformations. The PDE (1) is reduced in different form of differential equations. These ODEs are solved by using the elliptic JacobiSN (z, k) function and its differential equation, see (10).


## 2. Power Index Method

Considering the PDE (1), and we want to find its exact solution, we introduce the variables η as
(2)η=qx,t≔k1expa1x+a2t+k2lnb1x+b2t,η=qx,t≔b1x+b2ts,or
(3)η=qx,t≔tanma1x+a2t,and the function transforms
(4)ut,x=b1x+b2trfη,or
(5)ut,x=tanna1x+a2tfη.Now, we differentiate (4) according to PDE (1), and we can find the relation of indexes of x and t in each term as
(6)∂∂tux,t=rb1x+b2tr−1fη+sb1x+b2tr+s−1f′η,if we choose η=b1x+b2ts.We observe the coefficient functions offη and its derivative terms so that the PDE equation may be changed into ODE. In (6), the coefficient function of fη is b1x+b2tr−1, and the coefficient function of f′η is b1x+b2tr+s−1. The optimal indices x and t of the independent variables are chosen such that only two indices are varied at a time and the rest are fixed. We continue this process with different indices x and t so that we find all well-defined transformation. Our objective in this method is to get the required ODE.
(7)Qη,f,ff′,f″,⋯=0.Solve the ODE (7) by using computerized symbolic package like Maple. The exact solution of (1) can be obtained from (4) to (7) after replacing an unknown function fη. Figure 1 shows the procedure of transformation of PDE (1) to ODE (7).Figure 1
Flow chart of the power index method.


## 3. Exact Solutions of Landau-Ginzburg-Higgs Equation

Case 1.
We choose anew variableξ in exponential form and choosing new transformation hξ as
(8)ξ=expa1x+a2t,ut,x=a1a2nhξ.
The PDE (1) reduces to the nonlinear ODE
(9)a12−a22ξ2h″ξ+a12−a22ξh′ξ−a12a22ξh3ξ+m2hξ=0.
Since the ODE (9) is second-order nonlinear ODE, to find the solution of ODE (9), we need the following second-order differential equation.
(10)h″z−2k2h3ξ+1+k2hξ=0.
The elliptic function JacobiSN (z, k) is the solution of (10). The InverseJacobiSNz,k is defined as
(11)InverseJacobiSNz,k=∫0zdη1−η21−k2η2.
Jacobi elliptic functions are named for the famous mathematician Jacobi, and then, Gauss gave some attention to JacobiSN (z, k). Applications of Jacobi elliptic functions include closed-form solutions for nonlinear integrable equations.
If we choosez=α12α2lnξ+C1 and k=α3 with
(12)α1=m2C22a12a22+2m2,α2=2m2−a12a222a12−a22,α3=C2a1a22m2−a12a22and using relations
(13)h′z=ξ2α1α2h′ξ,h″z=ξ22α1α22h″ξ+ξ2α1α22h′ξ,then the ODE (10) transforms to ODE (9), and the solution of ODE (10) transforms to the solution of ODE (9) which has the form
(14)hξ=C2α12JacobiSNα12α2lnξ+C1,α3.
Using (8) and (14), we get the exact solution of the PDE (1) which is
(15)ut,x=a1a2nC2α12JacobiSNα12α2a1x+a2t+C1,α3,where α1,α2,α3 are defined above.Figure2 demonstrates solution (15) of PDE (1). Figure 3 demonstrates contrasts of ux,t in (15).Case 2.
Next, we define a new variableξ in logarithmic function and choosing new transformation hξ as
(16)ξ=lnb1x+b2t,ut,x=hξ.
The PDE (1) can be reduced to the ODE
(17)b12−b22e−2ξh″ξ+b12−b22e−2ξh′ξ−n3h3ξ+m2hξ=0.
Since the ODE (17) can be obtained by using transformation z=α4α5eξ+C1, k=α6C2, and relations,
(18)h′z=e−ξα4α5h′ξ,h″z=e−2ξα4α52h″ξ−e−2ξα4α52h′ξ,in (10). By using the solution of ODE (10), the analytic solution of ODE (17) can be expressed in the form
(19)hξ=C2α4JacobiSNα4α5eξ+C1,α6C2,where
(20)α4=2m2C22−1n2+2m2,α5=2m2−n22b12−b22,α6=n2m2−n2.
Using (19) and (16), we get the exact solution of the PDE (1) that is
(21)ut,x=C2α42JacobiSNα4α5b1x+b2t+C1,α6C2,where α4,α5,α6 are defined above in this case. Figure 4 presents graphical diagram of the analytical solution of (21) which is periodic solution of PDE (1).Case 3.
We choose a new variableξ in polynomial form and choosing new transformation hξ as
(22)ξ=a1x+a2ts,ut,x=a1x+a2trhξ.
The PDE (1) reduces to the algebraic form
(23)a1x+a2tβiFhξ,h′ξ,h″ξ=0,where
(24)βi∈r−2,r+s−2,r+2s−2,r,3r.Figure 2
3D plot of solution (15) of PDE (1).Figure 3
3D plot of solution (15) of PDE (1).Figure 4
3D plot of solution (21) of PDE (1).To convert PDE (1) into nonlinear ODE, the index values of βi must be chosen in such a way so that there are multiples of index s. To select suitable values of βi, its members must have the following relation r=s−1 or r=s/2−1. If we choose the first relation, the algebraic equation (23) can be changed to the following ODE, successfully.
(25)As2−3s+2−m2ξ2/shξ+3sAs−1ξh′ξ+n2ξ2h3ξ+m2s2a22−a12h′′ξ=0,where A=a22−a12. Since the ODE (25) can be achieved by using transformation z=α81/2α8ξ+C1, k=α9C2, and chain rule in (10). Figure 5 presents graphical diagram of the analytical solution of (28).Figure 5
3D plot of solution (28) of PDE (1).The analytic solution of ODE (25) for s=1 can attained from ODE (10) in the form
(26)hξ=C2α7JacobiSNα812α8ξ+C1,α9C2,where
(27)α7=2m2C22−1n2+2m2,α8=22m2−n2a12−a22,α9=n2m2−n2.Using (22) and (26), we get the exact solution of the PDE (1) that is
(28)ut,x=C2α7JacobiSNα812α8a1x+a2t+C1,α9C2.The analytic solution of ODE (25) for s=2 is
(29)hξ=C2α7ξJacobiSNα712α8ξ+C1,α9C2.Using (22) and (25), we get the exact solution of the PDE (1) that is
(30)ut,x=C2α7a1x+a2tJacobiSNα812α8a1x+a2t+C1,α9C2,whereα7,α8,α9 are defined above. If we choose the second relation r=s/2−1, then the transformation (22) reduces of the form
(31)ξ=a1x+a2ts,ut,x=a1x+a2t1/s−1hξ.Figure6 shows graphical representation of solution of (28) which is periodic solution of PDE (1).Figure 6
3D plot of solution (30) of PDE (1).By using (31), the algebraic equation can be changed to the following ODE.
(32)s2a22−a12ξ2h″ξ+s2s−3a22−a12ξh′ξ+a22−a1214s2−32s+2−m2ξ2/shξ+n2ξh3ξ=0.Using transformationz=α71/2α8ξ1/s+C1, k=α9C2 in (1), we found the analytic solution of the ODE (32) in the form
(33)hξ=C2α7ξ2−s/2sJacobiSNα712α8ξ1/s+C1,α9C2.The PDE (1) has the solution
(34)ut,x=C2α7a1x+a2t2−s/2sJacobiSNα712α8a1x+a2t1/s+C1,α9C2.Figure7 is the 3D representation of solution (30).Case 4.
We choose a new variableξ in polynomial form and choosing new transformation hξ as
(35)ξ=tan2a1x+a2t,ut,x=tana1x+a2thξ,(36)a2−b2ξ1+ξ2h′′ξ+52a2−b21+ξξ+3/5h′ξ+12hξ−12n2ξ+a2−b2ξ+1+2=0.
Using suitable transformation in (1), we can get analytic solution of (36) as
(37)hξ=1ξ2C2JacobiSNn2−2α10arctanα11α10ξ+−2α10arctanα11α10ξ+2C11α12,nα13C21α12,where
(38)α10=−2a2n2+2b2n2+16a2−16b2,α11=2a2−b28−n2,α12=2C22n2−n2+8,α13=−n2+8.
The PDE (1) has the solution of the form
(39)ut,x=1tana1x+a2t2C2JacobiSNn2−2α10arctanα11α10tana1x+a2t+1tana1x+a2t−2α10+arctanα11α10tana1x+a2t+2C11α12,nα13C21α12.Figure 7
3D plot of solution (34) of PDE (1).


## 4. Conclusion

In this paper, PIM has been used to construct exact solutions of LGH equation by using function conversion. The exact solutions of the LGH equation that we have obtained in this article are useful for understanding physical phenomena from many aspects. The performance of this method is found to be reliable and effective.



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