Extended Auxiliary Equation Method to the perturbed Gerdjikov-Ivanov Equation

We apply utilized the extended form of the auxiliary equation method to obtain extensively reliable exact travelling wave solutions of perturbed Gerdjikov–Ivanov equation (GIE) that is widely used as a model in the field theory of quanta and non-linear optics. The method is based on a simple first order second degree ODE. The new form of the approach gives more solutions to the governing equation efficiently.


Introduction
Optical soliton perturbation is studied in both physical and experimental phenomena modelled in to linear and non-linear PDEs such as; non-linear optics, electro-magnetics, plasma and solid state physics, water wave propagation in shallow media, fluid problems, and more.However, different approaches have been proposed over the years to handle the analytical solution of non-linear partial differential equations (NPDEs), see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].The perturbed GIE with full non-linearity is given in [13] of the form herein a, b and c are constants with real values and m denotes all non-linearity effects, α is the dispersion coefficient, μ is another dispersion term with higher effect as λ is the coefficients of short pulses that steepen themselves.It is remarkable to mention that, quite number of methods have been set for finding the solutions of GIE.Structure in Hamiltonian forms, algebro-geometric solutions approach, tangent expansion technique, solution in Wronskian-types, transformations similar to Darboux.For more details see [19][20][21][22][23][24].In this paper, we will propose another scheme for the exact and soliton solutions of GIE with aid of extended form of the auxiliary equation method (EAEM) [25].
The remaining article is designed as follows.In the next, we described methodology of the our scheme using EAEM in [25].Then, in Section 3, some solutions are set to the GIE using the aforementioned methodology.Eventually, a brief conclusion is given in the last section.

Description of the method:
Suppose that we have the following PDE ( , , , , ) 0, where ( , ) u x t u = is the dependent variable, x and t are dependent variables, F is a polynomial-like in terms of u and its derivatives.

EAEM mainly consists of three steps:
Step 1: The first step is the reduction of the governing equation by a compatible wave transformation such as ( , ) ( ), , where v is arbitrary constant.This transform can be changed to some complex-valued transforms in order to the governing equation into a non-linear ODE with respect to the variable  of the form ( , , , ), where the prime represents classical derivative for the independent variable  .
Step 2. Assume that the predicted solution to Eq. (2.3) is of the form of finite series as 0 ( ) ( ) are all real constants to be determined, N is a positive integer which can be determined by classical balancing procedure between the highest order derivative terms and the highest degree non-linear terms in Eq. (2.3).One should note that ()   satisfies the following new first order second degree auxiliary ODE where m1, m2, m3 are real parameters to be found.Equation (2.5) gives the following solutions:


of the soliton, the other constants , , , kv  are frequency, the number of wave, the constant of wave and soliton speed, respectively.Putting (3.1) into (2.2),followed by an uncoupled system of equations derived from the real and imaginary parts of the equation.The imaginary component reads: which anticipates the soliton speed.By setting m=1 in Eq. (3.2) and equating the coefficients of linearly independent functions to zero forces: The real component reads: WU  = ( )  Solving this set of algebraic equations using Maple, we found our solutions to be

Physical Illustrations
In this section, we illustrate some particular forms of the solutions determined above for particular choices of parameters satisfying given constraints.
The real and imaginary components of the solution ) , (

5.Conclusion:
In this paper, we found exact travelling wave solutions for the non-linear perturbed GIE with the help of new EAEM.The complex valued solutions represent traveling waves in various forms.Even though some are of the form of well-known bell shaped multi waves, the shape of some others are completely different from them.Eventually, the scheme is simple, direct, efficient and robust for solving perturbed Gerdjikov-Ivanov non-linear equation and it kind that usually arises experimental and mathematical physics.
profile.Moreover Eq. (3.4) is reduced to following ODE, by employing dependent variable change as Fig 1a-1b for some particular choice of the parameters as  = 1,  = −1,  = −1,  = 1,  = 1,  = 1,  = 1,  = 0 in the finite domain of independent variables  ∈ [−10,10] ×  ∈ [0,2].Fig 1c represent the modulus of the same solution for the same parameters and in the same domain.Both the real imaginary components represent some periodic solutions as the modulus is a single solitary wave propagating to the right along the space axis.domain  ∈ [−10,10] ×  ∈ [0,2] for the parameters  = 1,  = −1,  = −1,  = 1,  = 1,  = 1,  = 1,  = 0.The real and imaginary components of the solutions have some positive and negative explosions at some points of space and time due to their functional nature, Fig 2a-b.The modulus of ) with huge positive amplitudes time to time owing to the behaviours of the real and imaginary components of the solutions.

Fig
Fig 3 demonstrates the solution