Rational wave solutions to a generalized (2+1)-dimensional Hirota bilinear equation

A generalized form of (2+1)-dimensional Hirota bilinear (2D-HB) equation is considered herein in order to study nonlinear waves in fluids and oceans. The present goal is carried out through adopting the simplified Hirota’s method as well as ansatz approaches to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions. Some figures corresponding to a series of rational wave structures are provided, illustrating the dynamics of the obtained solutions. The results of the present paper help to reveal the existence of rational wave structures of different types for the 2D-HB equation.

The basic goal of this paper is to study a generalized 2D-HB equation describing nonlinear waves in fluids and oceans as follows [34] or which is a generalization of the Hirota bilinear equation studied in [35]. The Hirota bilinear form corresponding to the above nonlinear model (1) is under the specific transformation = 2(ln ) .
In order to advance the studies on the generalized 2D-HB equation (1); in this paper, the simplified Hirota's method as well as different ansatz approaches are utilized formally to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions.

The generalized 2D-HB equation and its rational waves solutions
In this section, a number of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions for the generalized 2D-HB equation are derived using the simplified Hirota's method as well as ansatz approaches.

Multiple soliton solutions
In order to procure multiple soliton solutions of the generalized 2D-HB equation, first the expressions are substituted into the linear terms of the generalized 2D-HB equation; and then, the resulting equation is solved for obtaining the dispersion relation . This leads to and therefore, the phase variables , 1 ≤ ≤ can be written as Now, by inserting the logarithmic transformations = (ln ) , = (ln ) , into (2) where the auxiliary function ( , , ) is defined as we find that = 2.
Based on this result, a single soliton solution can be obtained as in which the phase variables , = 1,2 are as before and the phase shift 12 is an unknown. After performing some calculations, a double soliton solution is gained as where the phase shift is Finally, a triple soliton solution to the generalized 2D-HB equation is derived as in which the auxiliary function, the phase shifts, and the phase variables are defined as It is easy to show that the following multiple complex soliton solutions to the generalized 2D-HB equation can be constructed in which and The plots of single, double, and triple soliton solutions have been provided in Figures 1-3, illustrating the dynamics of the multiple solutions. Figure 1 shows a bright soliton wave whereas Figure 2 demonstrates the interaction of two bright soliton waves. Furthermore, the interaction of three bright soliton waves including two strong and one weak waves has been illustrated in Figure 3.

Breather and rational solutions
In the present subsection, first, the breather solution of the generalized 2D-HB equation is acquired by adopting a specific ansatz approach which is a combination of exponential functions and a trigonometric function as follows in which and , 0 , ℎ, 1 and , 1 ≤ ≤ 8 are constants to be computed. Inserting Eq. (26) into Eq. (3) and performing some calculations, yields the following nonlinear algebraic system Now, a breather solution is obtained as where = − ( 1 + 2 + 3 + 4 ) + 0 cos(ℎ( 5 + 6 + 7 + 8 )) + 1 and 3 , 7 , and 1 have been defined in (33).
The plots of the breather solution (34) have been depicted in Figure 4, demonstrating its dynamical behavior. It is evident that Figure 4 presents the interaction of bright and dark waves with a periodic property.
The above rational solution is plotted graphically in Figure 5

Complexiton solutions
This subsection deals with complexiton solutions of the generalized 2D-HB equation. For this purpose, a test function is exerted as in which and 1 and 2 satisfy ( , , ) = � � , ̅ , �� = 0, Here, the polynomial has been defined owing to the Hirota operator (3). It is easy to prove that The unknown 12 also can be gained as Now, a complexiton solution to the generalized 2D-HB equation is obtained as in which It is worth mentioning that the following complex complexiton solution to the generalized 2D-HB equation can be extracted in which The above complexiton solution has been portrayed graphically in Figure 6 for suitable parameters.

Hyperbolic solutions
In this subsection, hyperbolic solutions of the generalized 2D-HB equation are obtained with the use of ansatz methods. To start, let's consider the solution of the generalized 2D-HB equation as By setting Eqs. (58) and (59) in the system (2) and performing some calculations, we find Therefore, the following hyperbolic solution to the generalized 2D-HB equation is obtained = ( 0 − 2 2 tanh( + + (8 1 3 − 6 0 1 − 2 ) ) 2 ).
The plots of the hyperbolic solution (68) have been provided in Figure 7. Obviously, Figure 7 shows a bright soliton wave solution.

Concluding remarks
A generalized (2+1)-dimensional Hirota bilinear equation which describes nonlinear waves in fluids and oceans was comprehensively explored, in this paper. The current goal was performed by exerting the simplified Hirota's method and ansatz approaches as reliable techniques. Different classes of rational wave