Exact Traveling Wave Solutions of the Gardner Equation by the Improved tan(Θ(θ))-Expansion Method and the Wave Ansatz Method

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tan(Θ(θ))-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.


The Improved tan(Θ(ϑ))-Expansion Method
In this section, the main algorithm of the improved tan(Θ(ϑ))-expansion method (ITEM) is explained as follows: Step 1: using a new definition of wave variable (ϑ � μx − θt), a general partial differential equation (PDE) such as is transformed into an ordinary differential equation Step 2: suppose that could be constructed as a solution of equation (2), Taking (5) into account, some solutions are as follows: Category 6: while a � 0 and c � 0, it resulted that Category 7: while b � 0 and c � 0, it resulted that Category 10: while a � c � κ and b � − κ, it resulted that Category 11: while c � a, it resulted that 2 Mathematical Problems in Engineering Category 13: while c � − a, it resulted that Category 15: while b � 0 and a � c, it resulted that Category 16: while a � 0 and b � c, it resulted that Category 17: while a � 0 and b � − c, it resulted that Category 18: while a � 0 and b � 0, it resulted that where (A k , B k (k � 1, 2, · · · , m), a, b) and c are the unknown parameters that need to be calculated. To determine the natural number m, one can use the homogeneous balance rule.
Step 4: solving the algebraic equations in Step 3, it resulted that substituting

Applications of the Gardner Equation via ITEM
In this section, we will examine ITEM for equation (1). To find the traveling solutions for equation (1), we define the wave transformation as u � U(ϑ), where ϑ � μx − θt, μ ≠ 0, and θ ≠ 0 to be determined later. Taking u � u(ϑ) into account allows us to rewrite equation (1) as the following ordinary differential equation: Integrating (25) once with respect to ϑ and neglecting the resulted integration constants, we obtain Now, we apply the ITEM to obtain traveling wave solutions of the Gardner equation (1). According to this method, the solution of equation (26) can be written in the form of equation (4).
Balancing the u ″ and u 3 in (26), by using homogeneous, one has Taking p � 0 in (27), the solution structure is formulated as Substituting equation (28) into equation (26) and following the necessary steps of ITEM, we have the following sets of coefficients for the nontrivial solutions of (1) as follows: where a, b, and c are optional constants, and provided that β < 0.
Setting these values in categories 2, 6, 10, and 14 of Section 2, respectively, we acquire the following solutions: Mathematical Problems in Engineering where a 2 + b 2 − c 2 > 0 and ϑ is given by (32). where where where a, b, and c are optional constants, and provided that β < 0.

Applications of the Wave Ansatz Method
In what follows, and based on the wave ansatz method, several types of soliton wave solutions for the Gardner equation (1) are presented which is based on the wave ansatz method (see the previous study [24]).

Bright Soliton.
To retrieve bright optical solutions of the Gardner equation, we use the following scheme [41]: where where A, B, and ] are disposal parameters.

Dark Soliton.
To retrieve dark solutions of the equation, we use the structure [41] u(x, t) � (A + B tanh τ) n , where where A, B, μ, and ] are unknown parameters. Inserting (59) into (1) gives After some algebra, we conclude that e dark soliton solution of equation (1) is obtained as to exist, from (62), the following restriction is obtained

Singular Soliton.
To extract the singular solitons of the Gardner equation (1), the following structure is examined by [41] u( with τ is defined by (49). Substituting (67) into (1), we obtain Considering the balancing principle indicates (51), vanishing all the coefficients of (coshτ/[D + sinhτ] j ) for j � 2, 3, and 4 to zero in (68), one gets (71) From (70) and (71), one concludes that if holds, the soliton solution 6 Mathematical Problems in Engineering is achieved as a singular solution for the Gardner equation (1). In this solution, A is given by (70), D is shown in (71), and B is an optional constant chosen in such a way that (72) holds.

W-Shaped
Soliton. Now, we explore some exact solutions of the Gardner equation in the form of [41] u(x, t) � A + D sech(τ), where τ is the same as (49). Substituting (74) into (1), we, respectively, obtain − B D sech(τ)tanh(τ) − 6 B 2 c + 3 D 2 β sech 2 (τ) Now, equation (75) holds whenever we have which will be valid for Consequently, the solution (74) with sign "+" in equation (78) is obtained as Moreover, for bright soliton pulse and with sign "− " in equation (78), we obtain for a W-shaped soliton pulse, where B is an optional constant.
e correctness of all given solutions has been confirmed with Maple by substituting them back into the original equation.

Concluding Remarks and Observations
In this research, we exerted the improved tan(Θ(ϑ))-expansion and wave ansatz method as two useful mathematical tools to construct solitary solutions for the Gardner equation. ese two methods for equation (1) have not been reported in the literature so far, to achieve the category of bright, dark, singular, and W-shaped soliton solutions. For a better understanding of the solutions, numerical results have also been included. On the other hand, the results are quite reliable for solving the Gardner equation. e results attest to the efficiency of the proposed method. ese two powerful methods can also be applied to other nonlinear partial differential equations with time-dependent coefficients and their systems.

Data Availability
No data were used to support this study.