Ginzburg Landau equation's Innovative Solution (GLEIS)

A novel soliton solution of the famous 2D Ginzburg-Landau equation is obtained. A powerful Sine-Gordon expansion method is used for acquiring soliton solutions 2D Ginzburg-Landau equation. These solutions are obtained with the help of contemporary software (Maple) that allows computation of equations within the symbolic format. Some new solutions are depicted in the forms of figures. The Sine-Gordon method is applicable for solving various non-linear complex models such as, Quantum mechanics, plasma physics and biological science.


Introduction
Solution of the exact solutions of travelling wave with nonlinear evolution equations plays an important role. The utilization of these solutions are found in various nonlinear physical phenomenons. There are many complex models that are used in many fields of nonlinear sciences. Few of these models are described as biological sciences, quantum mechanics, plasma physics. Obviously, these non linear models requires nonlinear evolutionary equations to express some physical phenomena. Finding out solutions of these equations have direct implementations in the above said fields. For the past two decades, many direct method have been developed to have special solutions with reasonable efficiency. Few of these direct methods are unified Method [1,2], tanh-sech method [3], Bäcklund transformation method [4][5][6], Jacobi elliptic function expansion method [7,8], Hirotas direct method [9], modified auxiliary equation technique [10,11], F-expansion method [12], improved Bernoulli sub-equation function method [13], new extended direct algebraic method [14,15], many other methods are also reported in [1,[16][17][18][19][20][21][22][23][24][25][26][27][28].
In this paper, Ginzburg-Landau equation is presented [29] that appears as a class of nonlinear constant coefficients partial differential equation. As with the passage of time new and innovative solutions are appearing and the research activities in this area is still under voyage, it is proposed to have title matching with the on going train. So, Ginzburg Landau equationʼs Innovative Solution (GLEIS) is expresses the right title to present this research. dynamics, mathematical biology, nonlinear optics, and statistical mechanics is available to discuss different behaviours for this equation. This equation can be used as a generic amplitude equation and researchers were using it since long. This equation can also generate the instabilities that can lead to chaotic dynamics. Various areas of physics fir phase transitions and superconductivity can take benefit from this equation in the theory. Broad range of physical systems that describes evolution phenomena such as physical systems or even extending from physics to optics was considered as remarkable success in past. More recently, a detailed research is carried out for turbulent dynamics as systems model with nonlinear partial differential equations [30]. Ginzburg-Landau equation also found its applications in fluid dynamics by many researchers, however, Poiseuille flow [31] played vital role in such studies. Similarly, the Rayleigh-Bénard problem and Taylor-Couette flow [32][33][34][35] are also explored for the nonlinear growth of convection rolls.

The Sine-Gordon expansion method
In this section the Sine-Gordon Expansion Method (SGEM) is explained to find out general facts.
Sine-Gordon equation can be considered as follows.
Where ⧹{ } Î  n 0 and u = u(x, t). Wave transformation can be utilized to form u = u(x, t) = f(ζ), ζ = α(x − kt) on equation (2), produces the following nonlinear ordinary differential equation (NODE): Where k is travelling wave speed representation and f = f(ζ), ζ is travelling wave amplitude. Integrating equation (3), can also be formulated as follows: Where the integration constant is represented by Q.
Substituting the value of Q = 0, Inserting b = 1 into equation (5), produces: Simplifying equation (6), and reformulating these equations can provide following set of equations Where d is the constant of integration. There is dependency of many parameters for partial differential equation (9) and it also has non linearity; x t 2 Solving the above equation will produce the following equation: equation (10) may be given according to equation (7) and (8) as; Determining m is achieved by balancing the highest power of term having non linearity and adjusting the highest derivative in the transformed NODE. A new set of equations can be produced by considering each summation of the coefficients as zero for the equation , 0 i, j m . So, we have a new set of equations and all these equations are symbolic in nature. We can solve these equations in symbolic manners by the use of modern available computational software. In this research one of these software is used i.e. Wolfram Mathematica 9. By the use of described software we can have values values of all unknown coefficients that appears in these non linear partial differential equations. These includes the unknown coefficients such as A i , B i , μ and c. substituting the values of m along with the other obtained values of coefficients in equation (10) leads to have a new travelling wave solutions which is presented in equation equation (9).

Applications of Sine-Gordon expansion method
In this section, it is described, how SGEM can be used in finding out new possible solution of equation (1).
We can always assume that equation (1) describes exact solution of SGEM. So holding this assumption and continuing with the other equations as function of exponential as described in equation (12) Where p, s are some unknown constants and should be determined, v(x, y, t) is a function having real values in it. Following new equation can be constructed by Substituting equation (12)  Real and imaginary parts of the equation (13) can be separated and it is found that Consider the following equation Where three constants are unknown i.e. k, l, ν. Ordinary differential equations can be obtained by substituting (15) into equations (14), f(ξ) Now, f ″ and f 3 in equation (17) can be balanced with the help of coefficient m, take the value m = 1 for balancing purpose in equation (17).
Following equation can be obtained by using equation (11) and considering m = 1; Putting equations (18) and (19) into equation (17), yields an equation in trigonometric functions. In the above equations, setting the value of zero to each summation for the functions having various coefficients but same powers, provides a collection of new algebraic equations. These equations are solvable and the Solution to these set of equations can be obtained by using any symbolic mathematical solver software like Mathematica or Maple. Values of these coefficients plays important role and different cases can be obtained using different values of these coefficients. Taking different values of the coefficients along with the value m = 1, in equation (10) generates a separate case and hence provides a new solution to equation (1). Each case is highlighted in the subsequent sections. Case-1: There could be many possible set of values for the constants used in equation (1). Each set of values for these constants will generate a different kind of result in form of the solution presented in this paper. One of the possible set of values for these constants is given below. As describes in case-1 and case-2, with the assumption of the above coefficients provide solution to equation (1) in the following format:   figure 3(A), where as 2D surface of equation (25) is shown in figure 3(B). It is also described that figure 3 can only be generated with the assumption of following values of constants. The above coefficients provides a unique solution of (1) and given as below:   for all case. The above coefficients provides a unique solution of (1) and given as below: This ongoing research for finding out different solutions is still under way and getting a faster pace due to the available software available to solve the equations in the symbolic format and hence the title of this paper is suggested as Ginzburg Landau equationʼs Innovative Solution (GLEIS). The solutions obtained from these equations are directly applicable to different field of sciences like microscopic properties of superconductivity, phase transition in superconductivity, quantum field theory and event to string theory.
As a summary, it reiterated that similar solutions were obtained in the literature review and reported in this paper. The solutions obtained in this paper can be applied to the natural travel of waves. These solutions are not only applied to the waves but in general the similar nature real physical phenomena such as heat flow in single or multi-dimensional space. In general, there are three types of solutions for such problems. One soliton is the solution, which is high in the middle and decreasing asymptotically, on both sides. Other solution is known as kinks and this solution is ascending from one asymptotic state to the other asymptotic state. The third form of solution is descending from one asymptotic state to the other asymptotic state. However, the solutions extracted in this paper show entirely new manifestation. This type of solution is known as singular solution which was not obtained in the literature so far. This solutions is also shown pictorially in from of figure 1. Similar solutions are provided in various references such as [48]. The solution provided in [48] are in the form of exponential or exponential with tan functions, In our method the solutions are obtained in the form of exponential function or exponential function with tanh or sech. The proposed solutions manifest another form of traveling wave which was never found in literature.

Discussion
In [48] the first integral method was developed and been utilized in solving the 2D Ginzburg-Landau equation and various solutions in hyperbolic functions form were obtained. Secondly, the well-known ( ) ¢ G G -Some exact hyperbolic and trigonometric function were obtained by using the method of expansion as given in [58]. It is also observed that same solution structures also exists in literature when these two methods are used. However, the results are new even when the structures are same. These solutions and implementation on waves are novel and different from the existing solutions. On the other hand, 3D and 2D surfaces of u 1 is singular soliton surfaces by observing figures 1. The solution can be extended in figures 2 for periodic 3D and 2D of u 1 surface.

Conclusion
It is evident from the simulation results that the existing solutions were not the only possible solutions. New solutions are possible that can find the applications in wave propagation of physical phenomenons. The singular solution obtained in figure 1 is novel.
The results extracted in this paper show that SGEM is effective tool which can provide various different solutions. These solutions have importance especially for nonlinear evolution equations. More interesting solutions are expected in future.