A study on the solutions of (1+1)-dimensional Mikhailov-Novikov-Wang equation

The basic principle of this study is to obtain various solutions to the (1+1) dimensional Mikhailov-Novikov-Wang integrable equation (MNWIE). For this purpose, the generalized exponential rational function method (GERFM) is applied to this equation. Thus, several trigonometric functions, hyperbolic functions, and dark soliton solutions to the studied equation are acquired. In this way, some new solutions to the equation that have not been presented before have been obtained. In addition, 2D and 3D graphics of the acquired solutions are drawn for specific values. The obtained results and the graphic drawings of the results have been provided by using Wolfram Mathematica 12


Introduction
In this study, GERFM has been used, the solution methods of nonlinear evolution equations (NLEEs), and this method has been applied to the (1+1)-dimensional MNWIE, which is a variant of NLEEs.NLEEs have very important applications in areas such as mathematical physics, optical fibers, mathematical chemistry, hydrodynamics, fluid dynamics, geochemistry, control theory, meteorology, optics, mechanics, chemical kinematics, biophysics, biogenetics, and so on.A number of methods have been developed by various researchers in order to obtain solutions for NLEEs, which have such important areas of use in the scientific world: Modified direct algebraic, modified Kudryashov and trigonometric-quantic B-spline methods [1], improved Bernoulli sub-equation function method [2], modified extended tanh-function method [3], new extended generalized Kudryashov and generalized new Kudryashov methods [4], new function method [5], exp (−ϕ(ξ)) method [6], double (G ′ /G, 1/G)-expansion method [7], modified extended tanh-expansion based method [8], modified simple equation method [9], Jacobi elliptic function expansion method [10], modified (1/G ′ )-expansion method [11].
(1+1)-dimensional MNWIE is given as [12]: Eq. ( 1) was derived by using the perturbative symmetry approach to the classification of integrable equations.Also, this equation belongs to a hierarchy where the Boussinesq equation is a member of this class.Hence, studying this equation can provide a good understanding of various nonlinear scientific phenomena in physics [12,13].
(1+1)-dimensional MNWIE has been studied by some researchers recently.Ray and Singh got the kink-type multisoliton solutions of the equation with the help of simplified Hirota's method [14].Raza et al. obtained new soliton solutions of the equation using the exp (−ϕ(ξ))-expansion method, singular manifold method, and generalized projective Riccati equations method [15].Akbulut et al. found exact solutions of the equation generalized Kudryashov method, exponential rational function method, and modified extended tanh-function method [16].Bekir et al. obtained new exact soliton solutions of the equation using (G ′ /G)-expansion method [17].This study, which was prepared to specify the solutions of the (1+1)-dimensional MNWIE using GERFM [18][19][20][21][22][23] was designed as follows: In Section 2, GERFM's basic principles are presented.In Section 3, some solutions of (1+1)-dimensional MNWIE have been obtained by applying GERFM.In Section 4, results and discussion are provided and finally, in Section 5, the concluding remarks are given.

Brief description of the GERFM
This section introduces an algorithm of the simplest version of the GERFM method, which is used for finding integrable solutions, whose essential steps are given as follows: Step1: We consider NLPDE given below: We first apply the wave transform given below to Eq. (2); where c values that are not taken into account will be calculated later.Eq. ( 2) is transformed into an ordinary differential equation by using Eq.(3): Step2: Assume that we think that the solutions of Eq. ( 4) as: where Φ(η) = p 1 e q 1 η + p 2 e q 2 η p 3 e q 3 η + p 4 e q 4 η .( Here the value of M is determined through the homogeneous balance principle.p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 are real or complex constants, a 0 , are scalars and will be determined. Step3: If Eq. ( 5) is taken into account in Eq. ( 4), P(e q 1 η , e q 2 η , e q 3 η , e q 4 η ) = 0 equation system is obtained.A system of equations is obtained by equating all coefficients of P to zero.
Step4: If we solve the obtained system of equations and the found values consider in Eq. ( 5), the solutions of the discussed NLPDE are obtained.

Results and discussions
We have obtained seveeral trigonometric function, hyperbolic function and dark soliton solutions of the (1+1)-dimensional MNWIE by applying the GERFM.Several methods were previously applied by some authors to obtain the solutions of the (1+1)-dimensional MNWIE.When we compare the solutions we found with those of previously published papers, our u 1 (x, t) solution is similar to solution (31) given by Raza et al. [15].In addition to our u 3 (x, t) solution is similar to (33) solution given by Raza et al. [15], and solutions ( 23)-(24) given by Akbulut et al. [16].Our u 4 (x, t) solution is similar to ( 27) and (28) solutions given by Akbulut et al. [16].Our u 7 (x, t) solution is similar to solution (24) given by Raza et al. [15].According to our research, our other solutions have not been provided before.Thus, the GERFM appears to be an effective method for finding solutions to NLEEs.

Conclusion
In this study, (1+1)-dimensional MNWIE was investigated.GERFM, which is the solution method of NLEEs, was applied to this equation.Thus, several trigonometric function, hyperbolic function and dark soliton solutions of the equation were obtained.In order to understand the physical appearance of the found solutions, 2D and 3D graphics were drawn.These obtained results can be further extended and investigated to solve other equations of the Boussinesq type due to their significance in making sense of various nonlinear phenomena.In addition, this considered method can be applied to obtain solutions of equations used for various models.The most important advantage of the method used in this study is that a wide variety of solution families can be created.It is a more general method compared to other methods, as it offers a wide variety of solution families.Despite these advantages, since a different algebraic equation system is created for each solution family, the processing density increases.