Optical Solutions of the Date–Jimbo–Kashiwara–Miwa Equation via the Extended Direct Algebraic Method

In this study, the solutions of ( 2 + 1 ) -dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in diﬀerent structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.

In this study, the nonlinear DJKM equation [19] is investigated to construct various solitary wave solutions. In the integrable systems of KP hierarchy, the Jimbo-Miwa equation is the second equation used to explain such interesting (2 + 1)-dimensional waves in physics. e DJKM equation can be used in mathematical physics to model water waves with low surface tension and long wavelengths with weakly nonlinear restoring forces and frequency dispersion. Firstly, Hu and Li [19] applied bilinear Bäcklund transformations and nonlinear superposition formula for nonlinear DJKM equations, and after a gap of more than two decades, Wang et al. [20] used the bell polynomials to study the integrable properties of nonlinear DJKM equations such as Lax system, Bäcklund transformations, and infinite conservation laws along with multishock wave. Yuan et al. [21] presented Grammian-and Wronskian-type solutions by the Hirota method, and other types of solution are also obtained like auxiliary variables, the bilinear Bäcklund transformation, and N-soliton. Pu and Hu [22] employed the sine-Gordon expansion method in finding the traveling wave solutions of nonlinear DJKM equations and obtained hyperbolic, trigonometric, and complex solutions. Singh and Gupta [23] used the direct method and nonlinear selfadjointness to find the Painlevé analysis, symmetric properties, and conservation laws of the nonlinear DJKM equation. Sajid and Akram [24] utilized exp(− Φ(ξ))-expansion method and derived some exact traveling wave solutions including trigonometric, hyperbolic, and rational functions and W-shaped soliton of the DJKM equation. e proposed research analyzes some more new exact solutions such as bell-, V-, and W-shaped multiwave types of the nonlinear DJKM equation which are not yet found in the literature. To our utmost understanding, the DJKM equations were not analyzed using the extended direct algebraic method.
erefore, the benefits of this article included evaluating a wide range of advanced and contextual solutions to the considered wave equations by the use of the extended direct algebraic method. Furthermore, this beneficial and powerful approach can be used to investigate other NLEEs which frequently emerge in different scientific realworld applications. e novelty of this paper lies in the following: (i) complex-valued solutions and solitons are in different shapes and (ii) 3-dimensional figures are first presented by the extended direct algebraic method. e limitations of this work include that the solution methods for the construction of exact solutions to the equation involve various parameters. Such parameters show up in the final precise solution expressions and create hurdles in some physical situations.
ese are resolved with a careful selection of appropriate parametric values which is possible through graphical interpretation and testing of the solution expressions. e structure of this paper is organized as follows: in Section 2, detailed explanation of the extended direct algebraic method has been presented. Section 3 illustrates the method to solve the (2 + 1)-dimensional DJKM equation. In Section 3.1, the physical explanation of the solutions by mechanical features and dynamical characteristics is demonstrated. Finally, conclusion is given in Section 4.

Governing Model.
Considering the governing model, where Φ � Φ(x, y, t) is the real-valued function. e DJKM equation belongs to the well-known KP hierarchy [25,26] which can be obtained from the first two bilinear equations using transformation u � 2(log τ) x . e KP hierarchy is an infinite set of nonlinear PDEs.

Extended Direct Algebraic Method [27]
According to extended direct algebraic method, we have the following.
Step 1. Consider NLEE in three independent variables x, y, and t of the form, as where Φ � Φ(x, y, t) and P is the polynomial in Φ. Using the wave transformation where ω is the wave number. After applying the transformation, equation (2) can be converted into the nonlinear ODE, as where prime denotes the derivatives w.r.t. ξ.
Step 2. Consider that the formal solution of equation (4) has a form, as follows: where b 0 , b 1 , . . . , b N are constants and Q(ξ) satisfies the auxiliary equation, as Family 1. If β 2 − 4ασ < 0 and σ ≠ 0, then the solutions are given as Family 2. If β 2 − 4ασ > 0 and σ ≠ 0, then the solutions are given as

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Family 3. If β � 0, and ασ > 0, then the solutions are given as Family 4. If β � 0 and ασ < 0, then the solutions are given as Family 5. If β � 0 and σ � α, then the solutions are given as Family 6. : If σ � − α and β � 0, then the solutions are given as Family 7. If β 2 � 4ασ, then the solution is given as Family 8. If β � l, σ � 0, and α � ml(m ≠ 0), then the solution is given as Family 9. If β � 0 � σ, then the solution is given as Family 10. If β � α � 0, then the solution is given as Family 11. If β ≠ 0 and α � 0, then the solutions are given as Family 12. If β � l, α � 0, and σ � ml(m ≠ 0), then the solution is given as Remark 1. e generalized triangular functions and hyperbolic functions [28] are defined as follows: where p, q > 0 and ξ is an independent variable.
Step 3. Using homogeneous balancing principle in equation (4), the value of N can be determined. Substituting equation (6) along with equation (5) into equation (4), collecting the coefficients of each power Q j (ξ) (j � 0, 1, 2, . . .), and then setting each coefficient to zero give a system of equations.
Step 4. Unknowns can be found by calculating the system of equations. Putting the unknowns in equation (6), the required solutions of equation (2) are obtained.

Application to the DJKM Equation
e extended direct algebraic scheme is presented to obtain the optical solitons and other solutions to equation (1). After utilizing the transformation Φ(x, y, t) � V(ξ), where ξ � ω(x + μy − kt), to equation (1), we obtain nonlinear ODE as follows: Setting U � V ′ , we obtain Balancing U ‴ with UU ′ in equation (21) gives N � 2. us, the solution can be written as where b 0 , b 1 , and b 2 are constants to be determined. Substituting equations (22) into (21), collecting all terms with the same power of Q(ξ) i (i � 0, 1, 2, 3, 4, 5), and equating the coefficients of each polynomial to zero will yield a set of algebraic equations for b 0 , b 1 , b 2 , and ω as follows: Solving system (23) for b 0 , b 1 , b 2 , and ω gives Five families of traveling wave solutions of the DJKM equation can be obtained, as shown in the following.

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.

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, Family 19. When β 2 � 4ασ, the rational solution is obtained, as .

Conclusion
To investigate the (2 + 1)-dimensional DJKM equation for exact solutions, the extended direct algebraic method is applied. By the extended direct algebraic method, many new exact solitary wave solutions are constructed including the singular, dark, combined dark-bright, periodic-singular, combined dark-singular, combined singular, and rational kinds. Such observations show that the suggested approaches are highly helpful and efficient in solving the NEEs. e complex-valued solutions represent traveling waves in different structures. Even though some are of the well-known forms such as bell-, V-, and W-shaped multiwaves, the shape of some others are completely different from them which were not found in the previous literature. e results of this investigation can be useful in illustrating the physical meaning of the studied model by 3D graphics. e performance of the method is reliable and a computerized mathematical approach to conduct other NLEEs in the field of mathematical physics and applied sciences.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.