Multiple Accurate-Cubic Optical Solitons to the Kerr-Low and Power-low Nonlinear Schrodinger Equation without the Chromatic Dispersion

The main target of this work is implementing multiple accurate cubic optical solitons for the nonlinear Schrödinger equation in the presence of third-order dispersion effects, absence of the chromatic dispersion. The emergence cubic optical solitons of the proposed model are extracted for the kerr-law and power law nonlinearity in the framework of two distinct techniques, the first one is the extended simple equation method (ESEM), while the other is the solitary wave ansatz method (SWAM). These cubic optical solitons for the kerr-law and power law nonlinearity have been extracted successfully at the same time and parallel via these two different techniques. A good comparison not only between our achieved results by these two manners but also with that achieved previously has been extracted .

Recently, few studies have been implemented to discuss the NLSE in the presence of the third order dispersion and absence of the chromatic dispersion [6][7][8], [33][34],while in the absence of third order dispersion and the chromatic dispersion this equation doesn't integrable [35]. According to [18], the suggested model can be proposed in the form, Where R is a function of (,) hxt and its partial derivatives that involve the highest order derivatives and nonlinear terms, according to the transformation 0 ( , ) ( ), hxt h x Ct ζζ = = − equation (2) can be reduced to the following ODE:  (1) If 13 0, aa = = Eq.(5) will be transformed to the Riccati equation [30][31], which has the following solutions; (2) If 03 0, aa = = Eq.(5) will be transformed to the Bernoulli equation [31], which has the following solutions; a Exp a a a Exp a ζζ ϕζ ζζ a Exp a a a Exp a ζζ ϕζ ζζ And the general solution to ansatz equation (5) Where 0 ζ is the constancy of integration?
Finally, via inserting equation (4) into equation (5) and equating the coefficients of different powers of i ϕ to zero, we get a system of algebraic equations, by solving it we obtain the values of the unknown variable mentioned in these equations. Moreover, inserting these achieved variables into equation (5) then the aimed solutions will be extracted.

The cubic optical solitons in the framework of the ESEM
In this section, we will implement the ESEM to construct the cubic optical solitons for the kerr-law and power law nonlinearity of the suggested equation (1) mentioned above,

The cubic optical solitons to the kerr-law nonlinearity,
For the kerr-law nonlinearity Eq.(1) become, According to the suggested method, the solution is, Now, via integrating the imaginary part Eq. (21)with respect to ζ , then it becomes approximately the same as Eq. (20) which represents the real part. For this reason we will implement the suggested method only for the real part which is,  3  33  22  2  2  3  1  01  10  11  0  101   2  22  3  0  1  10  1  1  1  32   3  (3  3  )  (  6  ) 333 , Via inserting the relations (23-27) into Eq. (22) and collecting and equating the coefficients of different powers of i ϕ to zero, lead to a system of algebraic equations and by solving this system the following results will be extracted, These results imply 7-differents solutions, for simplicity we will extract the solutions corresponding to the first and the second result and plot them,  According to the proposed method the solution is, In the same manner, we can plot the other cases.
Via inserting the equations (45-48) into equation (22), hence collecting and equating the coefficients of different powers of i ϕ to zero, we obtain system of algebraic equations, by solving it we get these results, From which we can get 4-various solutions, we will extract the solutions corresponding to the first and third results and plot them.
(3.2.1) for the 1-st case in which 3 3 This result can be simplified to be, a Exp a a a Exp a ζζ ϕζ ζζ    In the same manner, we can plot the other cases.
By the same manner we can implement the first and second family of the suggested method for the imaginary part of the kerr-low nonlinearity Eq. (21) to extract the cubic-solitons.

The cubic optical soliton to the power-law nonlinearity,
For the power-law nonlinearity Eq.(1) is, Via inserting the relations (13)(14)(15)(16)(17)(18)(19) into equation (65), then it will be transformed to the following real and imaginary equations respectively: Now, by applying the homogeneous balance between VV ′′ and 3 V it will lead to 2 N = , hence the solution in the framework of the proposed method must be in the form,

VV V
′′ and 2 V into Eq. (66), by substituting the same values of parameters used above and solving the extracted system we obtain a set of results from which only two are suitable and the others will be refused because either 0 0 a = or 2 0 a = or both.
The two suitable results are,

Description of the SWAM
To construct the solution in the framework of the SWAM [32,34], let us admit this wave transformation, Via simple calculations we can extract the following relations,  (,) t a n h ( ) , ( 1) tanh ( ) 2 tanh ( )

5.1
The bright cubic-solitons for kerr-law nonlinearity Via substituting the relations (97-101) into Eq. (12) mentioned above we obtain, 11 This equation will be divided into the following real and imaginary parts respectively: Via equating the highest order powers 1 sech ( ) i t of the real part we get 1 1 R = and hence, This admits these four solutions which are, Now, let us plot only two of them, say the first and the fourth For similarity we will plot only one solution to each part

9-Conclusion
From the power point of view for two important and powerful distinct techniques new accurate cubic-solitons for the Kerr-low and Power-low NLSE in the presence of third-order dispersion effects, absence of the chromatic dispersion have been extracted. The two techniques have been implemented in the same vein and parallel. The first one is the ESEM which has a successful history in extracting the optical soliton solutions for many nonlinear phenomenas arising in different branches of science. This schema is applied perfectly to introduce new impressive and accurate visions of the cubic solitons for the Kerr-law and Power-law NLSE that involve the third-order dispersion effect and exclude the chromatic dispersion effect, which are obviously through Figures (1-16). In related subject the SWAM has been applied effectively to establish other new accurate perceptions of the cubic solitons for the kerr-law and power law nonlinearity mentioned before, which are obviously through Figures (17-26). Our new achieved pictures of the accurate cubic soliton solutions in the framework of these two various manners which weren't achieved before by any other authors denote to the novelty of these results, especially compared with that achieved previously by [6][7][8] and [33][34] who applied different techniques to study these two cases significantly. Consequently, new distinct and impressive visions to the cubic solitons of these two different versions of this model have been demonstrated. Moreover, the achieved cubic solitons via these two different algorithms will add good future studies not only for this model but also for all related phenomenas.