Analysis of optical solitons solutions of two nonlinear models using analytical technique

: Looking for the exact solutions in the form of optical solitons of nonlinear partial di ﬀ erential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of di ﬀ erent physical phenomena. These solutions are speciﬁc, novel, correct and may be beneﬁcial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are o ﬀ ered.


Introduction
Recently, nonlinear evolution equations (NLEEs) has been developed as specific modules of the class of partial differential equations (PDEs). It is distinguished that investigating exact solutions for NLEEs, via many dissimilar methods shows an active part in mathematical physics and has become exciting and rich zones of research analysis for physicist and mathematicians. Lots of significant dynamic processes and phenomena in biology, chemistry, mechanics and physics can be expressed by nonlinear partial differential equations (NLPDEs). In NLEEs, nonlinear wave phenomena of diffusion, dispersion, reaction, convection and dissipation are very important. It is necessary to define generalized Kudryashov method. In section 3, application of GKM for KNE is presented. Section 4 presents the application of GKM for BAE. Section 5 contains results and discussion. Conclusion of the paper is discussed in section 6.

Description of generalized Kudryashov method
The steps of GKM [54] are as follows Let NLEE in the form W(q, q x , q t , q xt , q tt , q xx , · · · ) = 0, (2.1) where q = q(x, t) is a function.
Step 2. Let (2.3) has the solution in form S (η) = 1 1 + Ae η , (2.5) is the solution in the form where A is constant.
Step 3. Using balancing rule in (2.3) to obtain the values of N and M.
Step 4. Utilizing (2.4) and (2.6) into (2.3), we get an expression in S i , where (i = 0, 1, 2, 3, 4, · · · ). Then collecting all the coefficients of S i with same power(i) and equating to zero, we get a system of alebraic equations in all constant terms. This system of algebraic equations can be solved by Maple to unknown parameters.

Kaup-Newell equation
The governing equation [55] is given as: Here, q(x, t) is a complex valued function, indicates the wave pfofile and rests on variables, space x and time t. It includes the non-Kerr dispersion, evolution and and GVD terms. Also, a is the coefficient of GVD and b is the coefficient of self-steepening term.
Suppose (3.1) has the following solution Here g(η), κ, c and ω are the amplitude, frequency, speed and wave number of the pulse, respectively. Putting (3.2) into (3.1), and splitting into imaginary and real parts.
The imaginary part has the form We can get easily the value of c as under c = −2ακa, and the constraint condition as under The real part has the form Now, balancing the g and non-linear term g 3 in (3.5), we get N = M + 1. So for M = 1, we get N = 2.

Application of GKM
The solution of (3.5) by generalized Kudryashov method as given in (2.4), reduces to the form a 0 , a 1 , a 2 , b 0 and b 1 are constants. Subtitling the (3.6) into (3.5) and also applying (2.6), we get an expression in S (η). Collecting the coefficients of same power of S i and equating to zero, the system of equations is obtained, as follows.
By solving the above system, we get various types of solutions. These solutions are deliberated below. Case 1.
Case 1 corresponds the following solution for Kaup-Newell equation Case 2 corresponds the following solution for Kaup-Newell equation Case 3 corresponds the following solution for Kaup-Newell equation Case 4 corresponds the following solution for Kaup-Newell equation

Biswas-Arshed equation
The BAE with Kerr Law nonlinearity [56] is Here q(x, t) representing the wave form. On the left of (4.1) α 1 and α 2 are the coefficients of GVD and STD, respectively. β 1 and β 2 are the coefficients of 3OD and STD, respectively. On the right of (4.1) µ and θ represents the outcome of nonlinear disperssion and λ represents the outcome of selfsteepening in the nonappearance of SPM.
Let us assumed that the solution of (4.1) is as under Here g(η) shows amplitude, φ(x, t) is phase component. Also κ, v, θ 0 , ω denote the soliton frequency, speed, phase constant and wave number, respectively.
Substituting (4.2) into (4.3) and splitting it into imaginary and real parts. The imaginary part has the form We can get easily the value of v as under v = β 1 β 2 , and the constraints conditions as under The real part has the form Using balancing principal on (4.4), we attain M + 1 = N. So for M = 1, we obtain N = 2.

Application of GKM
Hence, solution of (4.4) by GKM as given in (2.4) will be reduced into the following form a 0 , a 1 , a 2 , b 0 and b 1 are constants. Substituting (4.5) into (4.4) and also applying (2.6), we acquire an expression in S (η). Collecting the coefficients of with same powerS i and equating to zero, the following system equations is got.
(4.6) On solving above system, get various types of solutions. These solutions are deliberated below.
Above these values correspond to the following solution for Biswas-Arshed equation.
Above these values correspond to the following solution for Biswas-Arshed equation Above these values correspond to the following solution for Biswas-Arshed equation Above these values correspond to the following solution for Biswas-Arshed equation Above these values correspond to the following solution for Biswas-Arshed equation

Results and discussions
In this study, we effectively construct novel exact solutions in form of optical solitons for Kaup-Newell equation and Biswas-Arshed equation using the generalized Kudryashov method. This method is considered as most recent scheme in this arena and that is not utilized to this equation earlier. For physical analysis, 3-dim, 2-dim and contour plots of some of these solutions are included with appropriate parameters. These acquired solutions discover their application in communication to convey information because solitons have the capability to spread over long distances without reduction and without changing their forms. Acquired results are novel and distinct from that reported results. In this paper, we only added particular figures to avoid overfilling the document. For graphical representation for KNE and BAE, the physical behavior of (3.8) using the proper values of parameters α = 0.3, a 1 = 0.65, b 1 = 0.85, p = 0.98, q = 0.95, k = 2, A = 3, b = 2, c = 4. and t = 1 are shown in Figure 1, the physical behavior of (3.9) using the appropriate values of parameters α = 0.75, a 0 = 1.5, b 0 = 1.7, b 1 = 0.98, A = 3, b = 1.6, a 0 = 2, c = 2.5. and t = 1 are shown in Figure 2, the physical behavior of (3.11) using the proper values of parameters α = 0.75, a 0 = 1.5, b 0 = 1.7, b 1 = 0.98, A = 3, b = 1.6, a 0 = 2, c = 2.5. and t = 1 are shown in Figure 3, the absolute behavior of (4.9) using the proper values of parameters α = 0.75, a 0 = 1.5, b 0 = 1.7, A = 2.3, b = 1.6, c = 2.5, v = 2.5, θ 0 = 4. and t = 1 are shown in Figure 4.

Conclusions
The study of the exact solutions of nonlinear models plays an indispensable role in the analysis of nonlinear phenomena. In this work, we have constructed and analyzed the optical solitons solutions of the Kaup-Newell equation and Biswas-Arshad equation by using Kudryashove method. The transmission of ultrashort optical solitons in optical fiber is modeled by these equations. We have achieved more general and novel exact solutions in the form of dark, singular and bright solitons. The obtained solutions of this article are very helpful in governing solitons dynamics. The constructed solitons solutions approve the effectiveness, easiness and influence of the under study techniques. we plotted some selected solutions by giving appropriate values to the involved parameters. The motivation and purpose of this study is to offer analytical techniques to discover solitons solutions which helps mathematicians, physicians and engineers to recognize the physical phenomena of these models. This powerful technique can be employed for several other nonlinear complex PDEs that are arising in mathematical physics. Next, the DNLSE classes II and III will be scrutinized via the similar methods to more evaluate them, this definitely will offer a huge understanding of the methods along with the classes of DNLSE. These solutions may be suitable for understanding the procedure of the nonlinear physical phenomena in wave propagation.