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.
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], -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 -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.
In this paper, we will examine the solutions of Landau-Ginzburg-Higgs equation by using polynomial and elementary functions. In Section 2, 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 (, ) 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 as or and the function transforms or
Now, we differentiate (4) according to PDE (1), and we can find the relation of indexes of and in each term as if we choose .
We observe the coefficient functions of and its derivative terms so that the PDE equation may be changed into ODE. In (6), the coefficient function of is , and the coefficient function of is . The optimal indices and 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 and so that we find all well-defined transformation. Our objective in this method is to get the required ODE.
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 . Figure 1 shows the procedure of transformation of PDE (1) to ODE (7).
3. Exact Solutions of Landau-Ginzburg-Higgs Equation
Case 1. We choose anew variable in exponential form and choosing new transformation as
The PDE (1) reduces to the nonlinear ODE
Since the ODE (9) is second-order nonlinear ODE, to find the solution of ODE (9), we need the following second-order differential equation.
The elliptic function JacobiSN (, ) is the solution of (10). The is defined as
Jacobi elliptic functions are named for the famous mathematician Jacobi, and then, Gauss gave some attention to JacobiSN (, ). Applications of Jacobi elliptic functions include closed-form solutions for nonlinear integrable equations.
If we choose and with
and using relations
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
Using (8) and (14), we get the exact solution of the PDE (1) which is
where are defined above.
Figure 2 demonstrates solution (15) of PDE (1). Figure 3 demonstrates contrasts of in (15).
Case 2. Next, we define a new variable in logarithmic function and choosing new transformation as The PDE (1) can be reduced to the ODE Since the ODE (17) can be obtained by using transformation , , and relations, in (10). By using the solution of ODE (10), the analytic solution of ODE (17) can be expressed in the form where Using (19) and (16), we get the exact solution of the PDE (1) that is where 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 variable in polynomial form and choosing new transformation as The PDE (1) reduces to the algebraic form where
To convert PDE (1) into nonlinear ODE, the index values of must be chosen in such a way so that there are multiples of index . To select suitable values of , its members must have the following relation or . If we choose the first relation, the algebraic equation (23) can be changed to the following ODE, successfully. where . Since the ODE (25) can be achieved by using transformation , , and chain rule in (10). Figure 5 presents graphical diagram of the analytical solution of (28).
The analytic solution of ODE (25) for can attained from ODE (10) in the form where
Using (22) and (26), we get the exact solution of the PDE (1) that is
The analytic solution of ODE (25) for is
Using (22) and (25), we get the exact solution of the PDE (1) that is
where are defined above. If we choose the second relation , then the transformation (22) reduces of the form
Figure 6 shows graphical representation of solution of (28) which is periodic solution of PDE (1).
By using (31), the algebraic equation can be changed to the following ODE.
Using transformation , in (1), we found the analytic solution of the ODE (32) in the form
The PDE (1) has the solution
Figure 7 is the 3D representation of solution (30).
Case 4. We choose a new variable in polynomial form and choosing new transformation as Using suitable transformation in (1), we can get analytic solution of (36) as where The PDE (1) has the solution of the form
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.
Data Availability
Please contact the authors for data requests.
Conflicts of Interest
The authors declare that they have no competing interest.
Authors’ Contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgments
The authors would like to acknowledge the support of Prince Sultan University for paying the Article Processing Charges (APC) of this publication through the Theoretical and Applied Sciences Lab. First and third authors are also grateful to Air Marshal Javed Ahmed, HI(M) (Retd), Vice-Chancellor, Air University, Islamabad, Pakistan, for providing the excellent research facilities.