Travelling wave solutions and modulation instability analysis of the nonlinear Manakov-system

In this paper, the accurate closed-form solutions of the Manakov-system are extracted via the extended -expansion method, the -expansion method and generalized Kudryashov method. The solutions are obtained in the form of trigonometric, hyperbolic and rational functions. The dynamical behaviour of obtained solutions is discussed by plotting 3D graphs, 2D contour graphs and density graphs. The symbolic softwares such as Maple and Mathematica are used to plot the graphs of retrieved solutions. The modulation instability (MI) of Manakov-system is also discussed in this paper.


Introduction
The Lorentz force law and a set of coupled partial differential equations known as Maxwell's equations form the basis of classical electromagnetism, classical optics and electric circuits.The coupled nonlinear Schrödinger equations are obtained by converting Maxwell's equations into cylindrical coordinates and accounting for an optical fibre's boundary conditions.When the inverse scattering transform is applied to the resultant equations, the Manakov-system is obtained.
Manakov-system [1,2] has been studied many researchers in recent years and novel results have been achieved.It can be written as where ϑ(x, t) and (x, t) are complex valued functions and x i 's and y i 's are constants.The Manakov-system describes the wave propagation in two mode optical fibres and photo refractive materials [3].Multicolour solitons are vector solitons that maintain their form throughout propagation and have two or more polarization components coupled to each other.Only coupled nonlinear Schrödinger equations can explain the behaviour of both vector and multicolour solitons.Manakov initially presented the vector soliton in 1974, when he deduced the one-soliton solution and performed an asymptotic analysis for the two-soliton solution using the inverse scattering transform (IST) approach.The auxiliary equation method is used to obtain the V-shaped, periodic singular, periodic bright, etc., soliton solutions of Manakov-system [2].The physical interpretation of Manakov-system is confirmed in 1996 in the experimental observation of Manakovsolitons [4].The Manakov-system is a special case of coupled nonlinear Schrödinger equation [5][6][7].
Many physical phenomena and dynamical processes are modelled by nonlinear evolution equations.The graphical demonstration and stability of NLEEs urged many researchers to find exact solutions using different exact methods.The multiple lump and rogue waves solutions of some nonlinear coupled equations are obtained by different exact methods.For example, the solution of nematic liquid crystals model is obtained by Kerr law [8], the Hirota bilinear approach is used to find the solution of timefractional resonant nonlinear Schrödinger equation [9], the solution of Zabolotskaya-Khokholov model is obtained by new modified extended direct algebraic [10], the solution of (3 + 1)-dimensional generalized Kortewegde-VriesZakharovKuznetsov equation is obtained by new modified extended direct algebraic method [11], to obtain some chirped periodic waves with self-steeping phenomena we investigate the resonant nonlinear Schrödinger equation [12], the soliton solution of nonlinear telegraph equation is obtained by Hirota bilinear method [13], the solution of chlorite iodide malonic acid chemical equation is obtained by residual power series method [14], the chirped periodic wave type solutions of cubic-quintic nonlinear Schrödinger equation is obtained by Jacobi elliptic function [15], the solution of Ablowitz-Kaup-Newell-Segur water waves equation is obtained by painlevé analysis [16], the solution of nonlinear Biswas-Milovic coupled system is obtained in [17] and the solution of time-fractional ion sound and Langmuir waves system is obtained in [18].
The Manakov-system is investigated in the article by employing three most favourite, reliable and authentic integrating strategies.The three integrating techniques are the exp(−φ( ))-expansion method, ( G G 2 )expansion method and the generalized Kudryashov method.In order to utilize the proposed techniques, the nonlinear PDEs are first converted into nonlinear ODEs by applying travelling wave transformation.The proposed techniques efficiently extract trigonometric function solutions, hyperbolic function solutions and rational solutions.Trigonometric function solutions can be characterized as periodic solutions.Hyperbolic function solutions can be characterized as dark soliton, kink soliton, bright soliton, singular soliton and complexitons.
Integration schemes applied in this article are most efficient and reliable, but the three have their own limitations.In fact, it is important to state here that every analytical method for nonlinear evolution equations has to satisfy certain constraint conditions.The existence and validity of the obtained solutions depend on these constraint conditions.For example, the classic inverse scattering transform cannot be extended to the cases of power law linearity, dual-power law or even log-law nonlinearity.The method of semi-inverse variational principle is restricted to the retrieval of bright soliton solutions only.Otherwise, for dark or singular solitons, the corresponding stationary integral would be rendered divergent and that prevents universal applicability.
This paper has seven sections.Section 2 presents demarcation of Manakov-system.Suggested are elaborated in Section 3. Suggested methods are applied to extract the solutions of Manakov-system in Section 4. The modulation instability of Manakov-system is explained in Section 5.The work is summarized in Section 6.

Demarcation of Manakov-system
Considering the following travelling wave transformations: where where q i (r i ) and i (x, t) for i = 1, 2 are the amplitude and phase component, respectively.Substituting Equation (3) into Equations ( 1) and ( 2), following real and imaginary parts are achieved as For imaginary part Real part takes the following form: By applying principle of balance, then Equation ( 7) is transformed as

Description of methods
In order to utilize the proposed techniques, the nonlinear PDEs are first converted into nonlinear ODEs by applying travelling wave transformation.

The extended ( G G 2 )-expansion method
The exact solution of transformed ODE is considered as where In Equations ( 10) and ( 11), d 0 , d i 's, e i 's are unknown constants and = 1 and χ = 0 are integers.The formal solutions of Equation ( 11) are given in [57].

The exp(−φ( ))-expansion method
According to exp(−φ( ))-expansion method, solution of ODE is considered as where v i 's are constants to be determined.In Equation (12), φ( ) satisfies the following ODE: where and χ are arbitrary constants.The general solutions of Equation ( 13) are given in [58].

The generalized Kudryashov method
The following general solution of Kudryashov method is used to obtained the exact solution of transformed ODE where g i (i = 0, 1, 2, . . ., N) and h j (j = 0, 1, 2, . . ., M) are constant such that g N = 0 and h M = 0. Consider the ODE where u = u( ) satisfied.The solution of Equation ( 15) is considered as where D is an integrating function.

Determination of travelling wave solutions
This section provides novel solutions of the proposed model via suggested integrating techniques.

The extended ( G G 2 )-expansion method
The exact solution of Equation ( 9) is investigated in Equation (17).We obtained N = 1 by using homogenous balance principle.The solution is considered in Equation ( 17) as follows: where in Equation ( 17) the unknown constants are d 0 , d 1 and e 1 .Equations ( 17) and ( 11) are substituted in Equation ( 9).The following system is obtained by collecting all the coefficients of ( G G 2 ) i along with i = −3, −2, −1, 0, 1, 2, 3 and putting them equal to zero By solving these algebraic system of equations simultaneously using Mathematica, the following sets of solutions are obtained: Family 1: According to S1 the trigonometric, hyperbolic and rational functions solution are obtained.
If ( χ > 0), then following periodic solutions are obtained: The graphical simulation of |ϑ a (x, t)| for choosing arbitrary values of parameters as If ( χ < 0), then the following dark soliton solutions are obtained: The graphical simulation of Family 2: According to S2 the trigonometric, hyperbolic and rational functions solution are obtained.If ( χ > 0), then the following periodic solutions are obtained: +n 2 sin( √ χr 1 ) The graphical simulation of If ( χ < 0), then following singular soliton solutions are obtained: If = 0 and χ = 0, then following rational solutions are obtained: Family 3: According to S3 the trigonometric, hyperbolic and rational functions solution are obtained.

If
= 0 and χ = 0, then following rational solutions are obtained:

The exp(−φ( ))-expansion method
According to exp(−φ( ))-expansion method, solution of Equation ( 9) is considered as where the unknown constants are v 0 and v 1 .Equations ( 19) and ( 13) are substituted in Equation ( 9).The following system of algebraic equations is obtained by collecting all the coefficients of exp(−φ( )) i with i = −3, −2, −1, 0 and putting them equal to zero The following set of solution is obtained by solving these algebraic equations using mathematica:

The generalized Kudryashov method
In this section, Equation ( 9) is investigated by generalized Kudryashov method.Using homogenous balance principle we obtained N = M + 1. Choosing M = 1 we obtain N = 1.The general solution is assumed as where g 0 , g 1 , g 2 , h 0 and h 1 are constant.Equation ( 21) is substituted in Equation ( 9), we attain a polynomial in u( ).By collecting all the coefficient of u( ) of same power and putting them equal to zero, the following system is obtained: The following set of solutions are obtained by solving above algebraic system of equations: , x i (2h 0 − h 1 ) , Family 6: The hyperbolic function solution of Manakov-system is obtained corresponding to S6. 1 (x, t) = Family 7: The hyperbolic function solution of Manakov-system is obtained corresponding to S7.
Family 8: The hyperbolic function solution of Manakov-system is obtained corresponding to S8. (28)

Graphical representation
The presented graphs show the diverse wave behaviour corresponding to the Manakov-system.Figure 1 represents the periodic solution ϑ a (x, t).Periodic solutions are the solutions expressed in terms of trigonometric functions.The graphical simulations in Figure 2 represent a dark soliton.Physically, dark soliton is a localized surface soliton that causes a temporary decrease in an associated wave amplitude.The periodic solution ϑ d (x, t) is represented in Figure 3.A bright soliton is a localized surface soliton that causes a temporary increase in an associated wave amplitude (Figure 4).Thus, Figure 5 shows a dark-bright soliton.The graphical illustrations presented in Figure 6 exhibit a kink soliton.Figures 7 and 8 represent singular solitons.

Modulation instability
The modulation instability (MI) of Equations ( 1) and ( 2) is finalized in this section on the basis of linear stability technique.The steady-state solution of Equations ( 1) and ( 2) is given as the instability occurred and grows exponentially when the wave number is imaginary.Consequently, we must have the following condition for modulation instability to exist: The MI gain spectrum is retrieved as where It can be noticed that the modulation instability gain spectrum is dependent on group velocity dispersion and incident power (           decreasing curve if the value of p increases and the value of f decreases.Similarly, the results are increasing curve if the value of p decreases and the value of f increases.

Conclusion
The three proposed methods namely, the extended ( G G 2 )-expansion method, the exp(−φ( ))-expansion   method and the generalized Kudryashov method are used to extract the travelling wave solutions of Manakov-system.Symbolic software, specifically Maple and Mathematica, is used to plot the 3D graphs, density graphs and 2D contour graphs of certain derived solutions.The obtained exact solutions are in the form of dark soliton, dark-singular combo soliton, singular soliton, Kink and anti-Kink soliton, periodic solutions and rational solutions.It is important to mention here that the solutions obtained in this paper by the application of three state of the art integrating techniques are novel and reliable.The dark-singular combo soliton solutions, Kink soliton and rational solutions have been retrieved for the first time in this article as compared with [1,2].Moreover, the modulation instability analysis has been carried out for the very first time in this paper as compared with [1,2].On the basis of linear stability analysis, the MI analysis of Manakov-system is investigated.To identify the region for the occurrence of instability, a dispersion relation is derived.The dispersion relation reflects stability of steady-state nature.
The unstable solution arises when the wave number is complex, because the perturbation grows exponentially.Moreover, the stable solution occurs when the wave number is real.The MI gain is significantly affected by the incident power.Due to novelty of results, this article becomes a good contribution in the theory of soliton in future.
Figures 9 and 10) The MI gain is plotted in Figures 9-16 with different parameter values.The results are increasing curve by decreasing the values of p and f and vice versa.The results are

Figure 9 .
Figure 9. Increasing value of f gives decreasing curve.

Figure 10 .
Figure 10.Decreasing values of f results in increasing curve.

Figure 11 .
Figure 11.Increasing in p results in decreasing curve.

Figure 12 .
Figure 12.Decrease in p results in increasing curve.

Figure 13 .
Figure 13.Increasing in both p and f results in decreasing curve.

Figure 14 .
Figure 14.Decreasing in both p and f both results in increasing curve.

Figure 15 .
Figure 15.Increase in p and decrease in f result in decreasing curve.

Figure 16 .
Figure 16.Decrease in p and increase in f result in increasing curve.