Asim Zfr , M.Rheel , M.Asif , Kmyr Hosseini , Mohmm Mirzzeh ,Lnre Akinyemi
a Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan
b Department of Mathematics and Statistics, ISP Multan, Pakistan
c Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
d Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Vajargah P.C.44891-63157, Rudsar, Iran
e Department of Mathematics, Lafayette College, Easton, PA, USA
ABSTRACT The current study deals with exact soliton solutions for Schrödinger-Hirota (SH) equation via two modified integration methods.Those methods are known as the improved (G ′ /G ) -expansion method and the Kudryashov method.This model is a generalized version of the nonlinear Schrödinger (NLS) equation with higher order dispersion and cubic nonlinearity.It can be considered as a more accurate approximation than the NLS equation in explaining wave propagation in the ocean and optical fibers.A novel derivative operator named as the conformable truncated M-fractional is used to study the above mentioned model.The obtained results can be used in describing the Schrödinger-Hirota equation in some better way.Moreover the obtained results are verified through symbolic computational software.Also, the obtained results show that the suggested approaches have broaden capacity to secure some new soliton type solutions for the fractional differential equations in an effective way.In the end, the results are also explained through their graphical representations.
Keywords:Shrödinger-Hirota equation Truncated M-fractional derivative Soliton solutions
In this world, human life affected due to the changes occur on the earth.World with its medley from smooth to persuasive is adequate of interactions.Many natural phenomenon are occurring in this universe.To understand these phenomenon soliton theory is very helpful in various fields.likely, soliton theory have an important role in the field of applied Physics.Solitons are such a waves that has same shape and no energy dispersion throughout the way.Due to these properties of soliton, it has valuable significance in electromagnetism and telecommunication fields.Soliton theory is very important in various areas of science and engineering.In particular, most of the nonlinear PDEs have exact solutions in terms of solitons [1-25] .
There are many analytical schemes have been constructed to find the solutions of the such non-linear PDEs.For instancing, various optical solitons of the new Hamiltonian amplitude equation with the use of Jacobi elliptic functions scheme have been achieved[26] , various solitons of new coupled evolution equation are explained [27] , different solitons of the well-known equation naming Biswas-Arshed model (BAM) have been gained by using the modified extended tanh expansion technique [28 , 29] .There is another important model named as Biswas-Milovic (BM) equation.This model has been solved by different methods such as: different kinds of wave solutions of the Biswas-Milovic equation have been determined by using the tan (φ/ 2) -Expansion scheme [30] ,this model have been solved by applying the Lie Symmetry method[31] , various kinds of soliton solutions of this model has been found by implementing the Adomian decomposition method [32] .
There are many distinct techniques have been applied to gain the exact solitons of the Schrödinger-Hirota (SH) equation like:anstaz and tanh methods are used to obtain the bright, dark 1-soliton, and other soliton solutions [33] , new extended direct algebraic technique is implemented to gain the number of new traveling wave solutions of the nonlinear conformable fractional SH equation [34] , dark and bright optical solitons gained by variable coefficient method [35] , dispersive exact wave solutions are observed by modified simplest equation method [36] , different optical solitons of the perturbed SH equation are achieved by applying the extended trial method [37] , extended sinh-Gordon equation expansion scheme has utilized to obtain different types of optical solitons of truncated M-fractional SH equation [38] , extended auxiliary equation scheme is applied to gain the dispersive optical wave solitons of time-fractional SH equation along power law non-linearity as well as Kerr Law non-linearity [39] , undetermined coefficient method is implemented to gain the distinct kinds of dispersive exact solitons in the presence of several perturbation terms are achieved [40] , with the use of tanh-coth integration algorithm dispersive solitons in optical nanofibers are obtained with constraint conditions [41] , by using the Sine-Cosine function method,different exact solutions are obtained [42] , Bäcklund transformation is used to obtained the optical solitons and solitary wave solutions for SH equation with power law nonlinearity [43] .
There are other two important methods named as: improved(G ′ /G ) -expansion and Kudryashov methods have been utilized to interpret the many models.For example, the optical soliton solutions of Burgers-Fisher equation is approximated with the use of improved (G ′ /G ) -expansion scheme [44] , distinct kinds of solitons of the time-fractional non-linear Biological Population equation and Cahn- Hilliard equation are obtained [45] , hyperbolic and trigonometric functions solitons to the non-linear reaction diffusion equation are achieved [46] .Similarly, with the help of Kudyashov method, many nonlinear partial differential equations have been solved [47-52] .
In this paper, the main work is to investigate some new solitons of Shrödinger-Hirota equation with truncated M-fractioal derivative based on the improved (G ′ /G ) -expansion method and Kudryashov method.
Definition:Suppose g(t) : [0 , ∞ ) → ℜ , then the truncated Mfractional derivative of g of power μis given [53,54] :
where Eβ(.) is a truncated Mittag-Leffler function of one parameter that is defined as:
Properties:Suppose 0 ＜ μ≤1 , β＞ 0 , r, s ∈ ℜ , and g, f are μ-differentiable at a point t ＞ 0 , then:
Consider the nonlinear Schrödinger Hirota equation with Kerr law nonlinearity and in the presence of third order dispersion(3OD) as [55]
where v = v (x, t) shows the wave function while ωis the 3rd order dispersion coefficient.In Eq.(8) , 1st term indicates the evolution term, 2nd term shows the group velocity dispersion (GVD) and 3rd term represents the Kerr law of non-linearity.
By introducing the Lie transformation [55]
Eq.(8) becomes:
After neglecting the higher order terms.This Eq.(10) is called the Schrödinger-Hirota (SH) equation.Eq.(10) is the governing equation that shows the dynamical optical pulse propagation in the presence od 3rd order dispersion.In this study, we consider the following SH equation with general coefficients:
where κ1, κ2, κ3and ρ are the constants.The nonlinear Schrödinger Hirota equation can well explain the effects of nonlinearity and dispersion in the ocean, it is more suitable for describing the deep-sea internal wave propagation and evolution than other mathematical models.Oceanic engineering, solid state physics, fiber optics, biology, astrophysics, solid state physics,chemical physics, and plasma physics are just a few of the areas that are concerned in nonlinear evolutionary equations [56-63] .
3.1.Mathematical analysis of the model:
Consider the SH equation with truncated M-fractional derivative given as:
Let’s us assume the below transformations:
By using the Eqs.(13) -(15) into the Eq.(12) , we get: Real part:
Imaginary part:
From Eq.(17) , we get:
To get the solutions in closed form, we use the new transformation:
By using Eq.(19) into the Eq.(16) , we get:
In this section, we represent the fundamental steps of improved(G ′ /G ) -expansion method [64] .
Step 1:Assuming the below non-linear PDE:
where q is a function of γ, θand t.
Step 2:Now assume the below transformations:
Here ν and κ are the parameters.Putting the Eq. (22) into Eq.(21) , taking the following nonlinear ODE:
Step 3:Assuming the solution of Eq.(23) in the below shape:
In Eq.(24) , α0and αj, (j = 1 , 2 , 3 , ..., m ) are unknown and to be determine later.It is necessary that αj0 .With the help of homogeneous balance technique into Eq.(23) , we get m .
The function G = G (η) fulfil the below Riccati differential equation:
where a, b, and care constants.
Step 4:Consider the Eq.(25) have the below form of solutions:
where a, b, c, C1, and C2are the constants ( Figs.1-8 ).
Step 5:
Substituting Eq.(24) along Eq.(25) into Eq.(23) and summing up the coefficients of same power ofBy taking each coefficient equal to zero, we get the system of algebraic equations involving ν, κ, αj, (j = 0 , 1 , 2 , ..., m ) and other parameters.
Step 6:
By solving the above gained system of algebraic equations with the use of symbolic software MATHEMATICA.
Step 7:
By putting the above obtained solutions into Eq.(24) and we get the trigonometric, hyperbolic trigonometric and rational function soliton type results of NLPD Eq.(21) .
By using the homogeneous balance method between the terms U3and U U ′′ into Eq.(20) , we get m = 2 .For m = 2 , Eq.(23) reduces into:
Here α0, α1and α2are unknowns.By substituting the Eq. (12) and Eq. (25) into Eq. (11) and collecting all the coefficients of same power of （we obtain the algebraic expression involving α0, α1, α2and other parameters.Now we obtain the following solution sets:
Set 1:
Fig.1.The 2D wave profiles of Eq.(35) for Ω=0 .08 , κ1 =0 .2 , κ2 =0 .1 , κ3 =0 .3 , μ=0 .05 , a=0 .5 , β=2 , θ=0 .2 , c =2 , and ρ= 0 .5 .
Fig.2.The 3D wave profile of Eq.(35) for C 1 = 0 .2 , C 2 = 0 .8 , α= 1 , Ω= 0 .08 , κ1 = 0 .2 , κ2 = 0 .1 , κ3 = 0 .3 , μ= 0 .05 , a = 0 .5 , β= 2 , θ= 0 .2 , c = 2 , and ρ= 0 .5 .
Fig.3.The 2D wave profiles of Eq.(35) for Ω=0 .08 , κ1 =0 .2 , κ2 =0 .1 , κ3 =0 .3 , μ=0 .05 , a=0 .5 , α=2 , θ=0 .2 , c =2 , and ρ= 0 .5 .
Fig.4.The 3D wave profile of Eq.(35) for C 1 = 0 .8 , C 2 = 0 .2 , α= 1 , Ω= 0 .08 , κ1 = 0 .2 , κ2 = 0 .1 , κ3 = 0 .3 , μ= 0 .05 , a = 0 .5 , β= 2 , θ= 0 .2 , c = 2 , ρ= 0 .5 .
5.1.Explanation of the Kudryashov method:
Here, in this section we consider the main steps of the this method [48] .The procedure of Kudryashov method is explained in below steps:
Step 1:Assume the Eqs.(21) -(23) .
Step 2:Assuming the solution of Eq.(23) in the below form:
where αj(j = 0 , 1 , 2 , 3 , ..., m ) are the unknowns with αj0 to be found.The positive integer “ m ′′ will becalculated byhomogeneous balance technique.The function φ(η) fulfil the auxiliary equation below:
Eq.(25) yields the below result.
Fig.5.The 2D wave profiles of h given by Eq.(46) for Ω=0 .04 , κ1 =0 .5 , κ2 =0 .2 , κ3 =0 .5 , μ=0 .02 , β=2 , θ=0 .2 , c =2 , and ρ= 0 .5 .
Fig.6.The 3D wave profile of h given by Eq.(46) for a = 0 .7 , b = 0 .3 , α= 1 , Ω= 0 .04 , κ1 = 0 .5 , κ2 = 0 .2 , κ3 = 0 .5 , μ= 0 .02 , β= 2 , θ= 0 .2 , c = 2 , and ρ= 0 .5 .
Step 3: By substituting the Eq. (41) into Eq. (23) with Eq.(42) and collecting all coefficients of the same power of φ(η) .Putting each coefficient equal to zero yields the algebraic expression having αj, μand other.
Step 4: Inserting the determining results of the unknowns with the results of the Eq.(43) , we get the solutions of the non-linear PD Eq.(21) .
By using the homogeneous balance method between the terms U3and U U ′′ into Eq.(20) , we get m = 2 .For m = 2 , Eq.(23) reduces into:
Set 1:
Set 2:
Fig.7.The 2D wave profiles of Eq.(46) for Ω=0 .05 , κ1 =0 .5 , κ2 =0 .2 , κ3 =0 .5 , μ=0 .02 , β=2 , θ=0 .2 , c =2 , and ρ= 0 .5 .
Fig.8.The 3D wave profile of h given by Eq.(46) for a = -1 , b = 1 , α= 1 , Ω= 0 .05 , κ1 = 0 .5 , κ2 = 0 .2 , κ3 = 0 .5 , μ= 0 .02 , β= 2 , θ= 0 .2 , c = 2 , and ρ= 0 .5 .
We have successfully secure some new solitons of the Schrödinger-Hirota equation along with truncated M-fractional derivative by applying the improved (G ′ /G ) -expansion method and Kudryashov method.Several significant oceanic phenomena that correspond to nonlinear shallow or deep water wave propagation have been explained in terms of soliton propagation.Our results might be crucial for further progress in understanding the wave phenomena since the Hirota equation contains terms which may describe waves in the ocean and optical fibers more precisely than the nonlinear Schrödinger equation.Also, all the obtained solutions are new and have been presented for the first time.The gained results are verified and also described with the help of graphs.These results play a valuable role in the area of optical fiber system.Furthermore, our study showed that these methods have their own merits to tackle the nonlinear problems in applied sciences.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.