The Application of the exp (−Φ(𝜉)) -Expansion Method for Finding the Exact Solutions of Two Integrable Equations

In the exp (−Φ(𝜉)) method is connected to search for new hyperbolic, periodic, and rational solutions of (1 + 1) dimensional fifth-order nonlinear integrable equation and (2 + 1) -dimensional Date-Jimbo-Kashiwara-Miwa equation. The obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved.


Introduction
e present paper organized as follows: In Section , description of the exp(−Φ( )) method for nding the exact traveling wave solutions of NLEEs is presented. Section illustrates the method to solve the (1 + 1)-dimensional horder nonlinear integrable equation and (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. Results and discussion are presented in Section . Finally, in Section , some conclusions are given.

Traveling Wave Hypothesis
Consider a NLEE in two independent variables and of the form as . . e exp(− ( )) Expansion Method: Quick Recapitulation. e key steps of exp(−Φ( )) method are given as Step . According to exp(−Φ( )) method, the wave solution can be expressed as where ( ̸ =0 )are constants to be determined and Φ( ) satis es the auxiliary ODE given as ( ) e auxiliary Eq. ( ) has the general solutions given as follows.
Step . Substituting , ], , ,a n d ,av a r i e t yo fe x a c t solutions of Eq.( ) can be constructed.

Mathematical Problems in Engineering
where the integration constant is taken as zero. Balance the linear term of highest order ὔὔὔ with the highest order nonlinear term ὔ in Eq. ( ), to get balancing number as =2. usthesolutionis Substituting Eq. ( ) into Eq. ( ) and collecting the coecient of each power of exp(−Φ( )) and then setting each of coe cients to zero give a system of algebraic equations as ( ) Solving the above system of equations yields Consequently, the following di erent cases are obtained for the exact solutions of (1 + 1)-dimensional h-order nonlinear integrable equation.

Results and Discussion
In this manuscript, several traveling wave solutions are developed. A new kind of W-shaped soliton solution is demonstrated and the other obtained solutions consist of trigonometric, hyperbolic, rational functions which are also new. On comparing our results with the well-known results obtained in [ , ], it may be concluded that the obtained results for Mathematical Problems in Engineering N-soliton solutions to Eq. ( ) using Hirota method and auxiliary variables. In both equations for case 1,theabsolute behavior of the solution | 1 ( , )| and | 6 ( , )| is shown in Figures and , respectively. e graphical representation of the other solutions ( 2 , 3 , 4 , 5 )forthe(1 + 1)-dimensional h-order nonlinear integrable equation and ( 7 , 8 , 9 , 10 ) for the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation is shown by Figures -and -, respectively.

Conclusion
e exp(−Φ( )) method is used to investigate the exact solutions of the (1 + 1)-dimensional h-order nonlinear integrable equation and (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. With the implementation of the exp(−Φ( )) method, many exact traveling wave solutions are obtained including trigonometric, hyperbolic, rational, and W-shaped soliton. e reported solutions in this article may be useful in explaining the physical meaning of the studied models and other nonlinear models arising in the eld of ber optics and other related elds. ese results indicate that the proposed method is very useful and e ective in performing solution to the NLEEs.
Data Availability e data used to support the ndings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no con icts of interest. [ ] M. Li, T. Xu, and D. Meng, "Rational solitons in the parity-timesymmetric nonlocal nonlinear Schrödinger model, " Journal of the Physical Society of Japan,vol. ,no. ,p. , . [ ]T .X u ,H .L i ,H .Z h a n g ,M .L i ,a n dS .L a n ," D a r b o u xt r a n sformation and analytic solutions of the discrete PT-symmetric nonlocal nonlinear Schrödinger equation