Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

College of Mathematics and Statistics, Cangzhou Normal University, Cangzhou 061001, China Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov Str., Lankaran, Azerbaijan Department of Statistics, Faculty of Physical Sciences, Shah Abdul Latif University Khairpur, Sindh, Pakistan Banking University, Ho Chi Minh City, Vietnam International University of Japan, Niigata, Japan Dai Nam University, Vietnam

In this paper, we consider the ð3 + 1Þ-dimensional generalized Hirota-Satsuma-Ito (HSI) shallow water wave equation which will be read [28] as by using the following bilinear transformation Equation (1) is transformed to the bilinear form as below: in which Firstly, Hirota and Satsuma introduced the ð1 + 1Þ -dimensional Hirota-Satsuma shallow water wave equation as a model equation describing the unidirectional propagation of shallow water waves [29], where we can write as x D t + D 2 x À Á f:f = 0, ð5Þ by applying the bilinear transformation Ψ = 2ðln f Þ xx . Also, by introducing the Hirota bilinear method in integrability of nonlinear systems, the ð2 + 1Þ-dimensional Hirota-Satsuma shallow water wave equation [30] was studied as by using the bilinear transformation Ψ = 2ðln f Þ x . Chen and coauthors proposed the ð3 + 1Þ-dimensional Hirota-Satsuma-Ito-like equation to describe the wave motion in fluid dynamics and shallow water [31]. Also, Liu et al. [32] investigated the N-soliton solution to construct the ð2 + 1Þ -dimensional generalized Hirota-Satsuma-Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons, and their interactions. Kuo and Ma [33] studied on resonant multisoliton solutions to the ð2 + 1Þ-dimensional Hirota-Satsuma-Ito equations and the existence and nonexistence of solutions. Kaur and Wazwaz [34] used the bilinear form to the new reduced form of the ð3 + 1Þ-dimensional generalized BKP equation and obtained lump solutions with sufficient and necessary conditions. A variety of lump solutions, generated from quadratic functions, for the ð3 + 1Þ-dimen-sional BKP-Boussinesq equation have been obtained by using the Hirota bilinear form in [35]. Also, the same authors obtained the optical soliton solutions to the Schrödinger-Hirota equation [36]. The authors of [37] obtained the lump solutions by making use of Hirota bilinear form to the ð3 + 1Þ -dimensional generalized KP-Boussinesq equation. By using Hirota's bilinear form and the extended Ansatz function method, Liu and Ye got the new exact periodic cross-kink wave solutions for the ð2 + 1Þ-dimensional KdV equation [38]. Liu and Xiong [39] obtained abundant multiwave, breather wave, and lump solutions by using the three wave method, the homoclinic breather approach, and the Hirota bilinear method for A variable-coefficient Boiti-Leon-Manna-Pempinelli (BLMP) equation. Also, Liu and He [40] utilized the Hirota bilinear form and concluded abundant lump solutions and lump-kink solutions of the new ð3 + 1Þ -dimensional generalized KP equation. Some three-wave solutions including kinky periodic solitary wave, periodic soliton, and kink solutions have been obtained to the ð3 + 1Þ -dimensional BLMP equation by the extended three-wave approach and the Hirota bilinear method in [41].
The outline of this paper is organized as follows. In Section 2, the new periodic solutions and multiple wave solutions of the ð2 + 1Þ-dimensional generalized HSI equation will be obtained by applying the Hirota bilinear method; in addition, the corresponding three-dimensional, contour, and density plots vividly show the physical structure of the periodic wave solutions. In Section 3, carrying the bilinear method to the cross-kink wave solutions will be obtained via choosing the specified function. In addition, we will plot several groups of maps to illustrate the crosskink of the corresponding solutions by symbolic computation. We gave three cases of solitary solutions with the semi-inverse variational principle in Section 4. Finally, the improved tan ðχðξÞÞ method and its application are given and investigated in Section 5. A few conclusions and outlook will be given in the final section.

New Periodic Wave Solutions for Generalized HSI Eq
By employing Hirota operator [42] for Equation (1), we have
where a 2 , a 3 , a 4 , Ω 2 , Ω 4 , Ω 9 are arbitrary values. Also, we need to satisfy the condition Ω 7 ≠ 0, δ 1 δ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in parallel to the x axis in Figure 1.
where a 2 , a 3 , a 4 , Ω 1 , Ω 5 , Ω 9 are arbitrary values. Also, we need to satisfy the condition Ω 1 ≠ 0, δ 1 δ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in the line of x − y axis in which they intersect at one point in Figure 2.
3 Advances in Mathematical Physics where a 2 , a 3 , a 4 , Ω 2 , Ω 6 , Ω 7 are arbitrary values. Also, we need to satisfy the condition Ω 7 ≠ 0, δ 1 δ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in the line of x − y axis in which they intersect at one point in Figure 3.
where a 2 , a 3 , a 4 , Ω 6 , Ω 7 , Ω 8 are arbitrary values. Also, we need to satisfy condition 4Ω 2 4 − δ 4 ≠ 0, δ 4 < 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in parallel to the y axis in Figure 4.
where a 2 , a 3 , a 4 , Ω 4 , Ω 7 are arbitrary values. Also, we need to satisfy the condition ð2 Ω 4 2 − 2 Ω 7 2 − δ 4 Þδ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in the line of x − y axis in which they intersect at one point in Figure 5.
Advances in Mathematical Physics where a 2 , a 3 , a 4 , Ω 1 , Ω 2 , Ω 4 are arbitrary values. Also, we need to satisfy the condition Ω 1 δ 5 ðΩ 1 2 − 3 Ω 4 2 + δ 4 Þ ≠ 0. By assigning particular values of the parameters, we can eas-ily observe the wave motion as periodic waves in the line of x − y axis in which they intersect at one point in Figure 6.
Ξ 9 = 1 12 7 Advances in Mathematical Physics where a 2 , a 3 , a 4 , Ω 1 , Ω 2 , Ω 4 are arbitrary values. Also, we need to satisfy condition 3 Ω 4 2 − δ 4 > 0. By assigning particular values of the parameters, we can easily observe the wave motion as periodic waves in the line of x − y axis in which they intersect at one point in Figure 7.
where a 2 , a 3 , a 4 , Ω 2 , Ω 4 , Ω 9 are arbitrary values. Also, we need to satisfy the condition Ω 7 ≠ 0, δ 1 δ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in parallel to the x axis in Figure 8.
where a 2 , a 3 , a 4 , Ω 2 , Ω 6 , Ω 7 are arbitrary values. Also, we need to satisfy the condition Ω 7 ≠ 0, δ 1 δ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in the line of x − y axis in which they intersect at one point in Figure 10. where a 2 , a 3 , a 4 , Ω 5 , Ω 6 , Ω 7 are arbitrary values.
11 Advances in Mathematical Physics where a 2 , a 3 , a 4 , Ω 6 , Ω 7 , Ω 8 are arbitrary values. Also, we need to satisfy the condition 4Ω 2 4 − δ 4 ≠ 0, δ 4 < 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in the line of x − y axis in which they intersect at one point in Figure 11.
where a 2 , a 3 , a 4 , Ω 4 , Ω 7 are arbitrary values. Also, we need to satisfy the condition ð2 Ω 4 2 − 2 Ω 7 2 − δ 4 Þδ 5 ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in the line of x − y axis in which they intersect at one point in Figure 12.
where a 2 , a 3 , a 4 , Ω 1 , Ω 2 , Ω 4 are arbitrary values. Also, we need to satisfy the condition Ω 1 δ 5 ðΩ 1 2 − 3 Ω 4 2 + δ 4 Þ ≠ 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in the line of x − y axis in which they intersect at one point in Figure 13.
13 Advances in Mathematical Physics where a 2 , a 3 , a 4 , Ω 1 , Ω 2 , Ω 4 are arbitrary values. Also, we need to satisfy the condition 3 Ω 4 2 − δ 4 > 0. By assigning particular values of the parameters, we can easily observe the wave motion as cross-kink waves in the line of x − y axis in which they intersect at one point in Figure 14.

Application of SIVP for Equation (1)
By utilizing ξ = kðx + ay − ctÞ in Equation (1), one becomes By points of Refs. [16,17,43] and by multiplying Equation (57) with Ψ′, we get Case 1. With selection the below solution function then the stationary integral changes to With the help of below Advances in Mathematical Physics two nonlinear algebraic systems will be concluded as By solving the above cases, one gets ffiffiffiffi ffi cS p , The domain of definition is

Advances in Mathematical Physics
By solving the above cases, one gets The domain of definition is then the stationary integral changes to With the help of below,

Advances in Mathematical Physics
By solving the above cases, one gets The domain of definition is Then, the dark wave solution will be obtained as

The Improved Rational tan ðχðξÞ/2Þ-Expansion Method
The application of the improved tan ðχðξÞ/2Þ-expansion technique will be studied where for the first time is given here. We first discuss the mathematical analysis of nonlinear partial differential equations (NPDEs). Hence, we consider the NPDEs in the following way.
Step 1. Assume a nonlinear partial differential equation is given in the general form as follows: After simple algebraic operations, this equation is transformed into an ordinary differential equation (ODE) with the below transformation:
Step 3. To determine the positive integer η, we usually balance linear terms of the highest order in the resulting equation with the highest order nonlinear terms appearing in Equation (80).
Step 4. We collect all the terms with the same order of tan ðχðξÞ/2Þ k ðk = 0, 1, 2, ⋯Þ together. Equating each coefficient of the polynomials of i to zero yields the set of algebraic equations for ζ k , θ k ðk = 1, 2, ⋯, σÞ, α 1 , α 2 , and α 3 with the aid of the Maple.

Conclusion
In this study, the periodic, cross-kink, solitary, bright, and dark wave solutions of the ð2 + 1Þ-dimensional generalized Hirota-Satsuma-Ito equation have been achieved. From the bilinear form of this equation, one test function or ansatz has been chosen. Through Maple, the evolution phenomenon of these waves is seen in Figures 1-14, respectively. Mainly, by choosing specific parameter constraints, all cases of 2D and 3D in solitons can be captured from the periodic and cross-kink wave solutions. Also, the improved tan ðχðξ ÞÞ method on the generalized nonlinear wave equation studied and four sets of solutions were obtained. The obtained solutions are extended with numerical simulation to analyze graphically, which results into multiwave and cross-kink wave solutions. Moreover, we studied the solitary, bright, and dark soliton wave solutions of the generalized HSI equation by help of SIVP in the previous section. Finally, literature is full of nonlinear evolution that rich soliton structures are still to be constructed while applying these methods. Further investigations deserve to be made in order to ameliorate the improved tan ðχðξÞÞ scheme, so that it may be possible to provide all the different solutions to a nonlinear system. These questions will constitute future works.

Data Availability
The datasets supporting the conclusions of this article are included in the article.