Optical solitons and modulation instability analysis to the quadratic-cubic nonlinear Schrödinger equation

This paper obtains the dark, bright, dark-bright, dark-singular optical and singular soliton solutions to the nonlinear Schrödinger equation with quadratic-cubic nonlinearity (QC-NLSE), which describes the propagation of solitons through optical fibers. The adopted integration scheme is the sine-Gordon expansion method (SGEM). Further more, the modulation instability analysis (MI) of the equation is studied based on the standard linear-stability analysis, and the MI gain spectrum is got. Physical interpretations of the acquired results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the PNSE.


Introduction
Nonlinear Schrödinger equations (NLSEs) appear in various areas of engineering sciences, physical and biological sciences.In particular, the NLSEs appears in fluid dynamics, nonlinear optics, plasma and nuclear physics [1,30,56].The previous studies provide a test bed to investigate soliton solutions in presence of many different nonlinearities [16].These days QC-NLSE has gathered significant attention.This paper will study a newly proposed form of nonlinearity called the quadratic-cubic law.The law first appeared in 2011 [22].The model supports soliton solutions (both temporal and spatial) with applications in soliton lasers, optical communications, ultra fast soliton switches and logic gate devices [47].Recently, we observe many new progresses in the field of nonlinear optics [2-15, 17-26, 28, 29, 31-46, 48-55, 57-60].The NLSE with quadraticcubic nonlinearity is given by [6,19,25,27,47,54]: In Eq. ( 1), t and x are the independent variables representing the temporal and spatial variables, respectively.ψ(x, t) is the dependent variable.The real-valued constant a represents group velocity dispersion (GVD), while b 1 and b 2 are real-valued constants.The chaotic phenomena of the equation was studied in [27].In [47], the analytical self-similar wave solutions of the equation were constructed.In [54], the method of undetermined coefficients was adopted to extract the soliton solutions and the conservation laws of the equation were reported.In [19], the He's semi-inverse variational principle was adopted to study the equation.In [6], the Jaccobi elliptic function ansatz method was used to study the equation.In [25], the extended trial equation method was utilized to retrieve some soliton solutions of the equation.This paper determines the dark, bright, dark-singular and a new dark-bright soliton solutions to the model by SGEM [23,57].The SGEM is a very strong approach to to retrieve the soliton solutions of nonlinear models.It has been used to study several NLSEs by several authors in [23,34,[36][37][38][39]57].One advantage of this approach is that it has the capability to retrieve dark-bright or combined optical solitons and combined singular solitons, which are special types of soliton solutions combining the futures of bright and dark optical solitons in one expression, and dark-singular and singular solitons in another expression.The subsequent section gives a full description of the integration scheme that will be applied retrieve the soliton solutions of the governing equation.Finally, the MI of the equation will be studied using the standard linear-stability analysis [1,2,36,48].

Description of the sine-Gordon expansion method
Consider the following sine-Gordon equation: where α is a non-zero constant.We apply the transformation where v is the traveling wave velocity.Substituting Eq. (3) into Eq.( 2), we obtain Equation ( 4) can be simplified to give where K is a constant of integration.By letting K = 0, w(ξ) = u(ξ)/2 and f 2 = α/(vη 2 ), Eq. ( 5) reduces to and in a more simplified form gives Setting f = 1 in Eq. ( 7), we get Equation ( 8) has the following solutions: and sin w(ξ) = i csch ξ or cos w(ξ)r = coth ξ.
To obtain the solution of the nonlinear partial differential equation of the form we use the following series solution: From Eqs. ( 9) and ( 10), the solution of Eq. ( 12) can be written as and To obtain the value of n, we use the balancing principle.Substituting n into Eq.( 12) and putting of the result into the reduced ordinary differential equation using Eq.(8) give a system of algebraic equations.Equating the coefficients of sin i w and cos i w to zero and solving the result gives the values of A i , B i and v. Subsequently, one can retrieve the soliton solutions of Eq. (11).
https://www.mii.vu.lt/NATo solve Eq. ( 1), we apply the transformation where In Eq. ( 16), φ(x, t) represents the phase component, k is the frequency, ω represents the wave number, θ represents the phase constant.In Eq. ( 15), v is the velocity, while K represents the width of the traveling wave [54].Putting Eq. ( 15) in Eq. ( 1) and separating into real and imaginary components, a pair of equation is acquired.The imaginary part yields the speed of the soliton stays the same irrespective of the type of soliton in question real part, therefore, Balancing the terms of u 3 and u in Eqs. ( 18) and (19) gives n = 1.Substituting n = 1 into Eq.( 12), we obtain Substituting Eq. ( 19) and the necessary derivatives into Eq.( 18) using Eq. ( 8) substituting trigonometric identities where necessary, we obtain the following algebraic expression: Equating each summation of the coefficients of trigonometric functions having the same power to zero, we obtain the following independent parametric equations.cos w: constants: Nonlinear Anal.Model.Control, 24(1):20-33 sin w cos w: sin w: sin 2 w cos w: Solving Eqs. ( 21)-( 27), we obtain the following families.
With the aid of Mathematica, from Eqs. ( 21)-( 27) we can get

Dark optical soliton
From the coefficients in case 1 above we obtain the dark optical solitary solution

Bright optical soliton
From the coefficients in case 2 above we obtain the bright optical solitary solution https://www.mii.vu.lt/NA

Dark-bright optical soliton
From the coefficients in case 3 above we obtain the dark optical solitary solution

Singular solitons
From the coefficients in case 1 we obtain the dark-singular solution while the coefficients in case 2 give the dark-singular solitary wave

Combined singular soliton
From the coefficients in case 3 above we acquire the dark optical solitary wave 3 Physical expressions, discussion and comparative study In this section, we compare the obtained solutions in this manuscript with the existing results in [6,19,25,54].We observe that some of the solutions in this manuscript are newly constructed solutions.It is observed that our solutions are related to the physical features of optical and singular soliton solutions, which play a vital role in understanding various physical phenomena in nonlinear systems.The dark optical soliton Eq. ( 28) and the bright optical soliton Eq. ( 28) are similar to the ones obtained in [6,19,25,54].Singular soliton solutions similar to Eq. ( 28) and Eq. ( 28) have also been reported in [47].
Figure 1.The 3D and 2D evolution of dark soliton Eq. ( 28) for some chosen parameters mentioned in the text.Finally, the dark-bright optical soliton Eq. ( 30) and the combined singular soliton Eq. ( 33) are, to the best of our knowledge, new forms of solutions introduced into the literature for the first time in this work.To understand the physical nature and evolution of the optical soliton solutions Eqs. ( 28)-( 33), we consider Figs.1-5 for the dispersion parameters k, K, b 1 and b 2 .For the dark optical soliton Eq. ( 28) and the dark-singular soliton Eq. ( 28), we choose k = 0.2, K = 1, b 1 = 0.1 and b 2 = 0.4.Plotting the 3D profiles of the bright optical soliton Eq. ( 28) and the singular soliton Eq. ( 28), we choose k = 0.2, K = 1, b 1 = 0.2 and b 2 = 0.1.On close observation, the intensity profiles of the optical soliton solutions Eqs. ( 28)-( 33) are demonstrated in Figs.1-5.We have found that the wave number k, associated with the phase component, plays a significant role in defining the trajectory of the pulses as they propagate through the waveguide.This may find applications in some optical devices such as counter, amplifier, etc.Moreover, we have verified that the solutions obtained in this paper are indeed solutions of Eq. ( 1), this is carried out with the assistance of Wolfram Mathematica 9.
M. Inc et al.

Modulation instability analysis
In the previous section, optical solitons to Eq. ( 1) are got by employing the complex envelope function ansatz.Now we will study the MI.We apply the linear stability analysis technique.We suppose that Eq. ( 1) has the perturbed steady-state solution of the form where P 0 represent the incident power.We investigate the evolution of the perturbation ρ(x, t) using the concept of linear stability analysis.Substituting Eq. ( 34) into Eq.( 1) and linearizing the result in ρ(x, t), we acquire The linear equation Eq. ( 35) can be solved in the frequency domain easily.But because of the ρ * component, the Fourier terms at frequencies Ω and −Ω are coupled.So, we seek for where K is the wave number, Ω is the frequency of the perturbation, respectively.Eqs.(35) and (36) give a set of two homogeneous equations in a 1 and a 2 .Substituting Eq. ( 36) into Eq.( 35), we get the following system of equations for a 1 and a 2 upon separating the coefficients of e i(Kx−Ωt) and e −i(Kx−Ωt) : From Eq. ( 37) one can easily obtain the following coefficient matrix of a 1 and a 2 : The coefficient matrix Eq. ( 38) has a nontrivial solution if the determinant vanishes.By expanding the determinant, we obtain the following dispersion relation: The dispersion relation Eq. ( 39) has the following solution: The stability of the steady state is determined by Eq. (40).If the wave number K has an imaginary part, the steady-state solution is unstable since the perturbation grows exponentially.But if the wave number K is real, the steady state is stable against small perturbations.Thus, the necessary condition necessary for the existence of modulation instability to occur from Eq. ( 36) is when either The MI gain is significantly affected by the incident power P 0 .From Fig. 3 it can be seen that the MI growth rates increases with increase in incidence power values.The main reason is due to increase in the gain along the fiber length.It is observed that two different side bands appear in the MI gain, but the intensity of the MI gain remains closely the same with increase of incident power.

Figure 2 .
Figure 2. The 3D and 2D evolution of intensity of the bright soliton Eq. (29) for some chosen parameters mentioned in the text.

Figure 3 .Figure 4 .
Figure 3.The 3D and 2D evolution of intensity of the dark-bright soliton Eq. (30) for some chosen parameters mentioned in the text.

Figure 5 .
Figure 5.The 3D and 2D evolution of intensity of the dark-singular soliton Eq. (32) for some chosen parameters mentioned in the text.