Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation

Qiaoling Yang; Jalil Manafian; Khaled H. Mahmoud; Abdullah Aldurayhim

doi:10.1515/phys-2024-0027

Article Open Access

Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation

Qiaoling Yang , Jalil Manafian , Khaled H. Mahmoud and Abdullah Aldurayhim

Published/Copyright: June 22, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this work, the exact solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation are reported by adopting the He’s variational direct technique (HVDT). The analytic findings of solutions were obtained by semi-inverse scheme, and six form of supposed studies reveal that the solutions belong to soliton groups. The modulation instability is considered. The tan ( Π ( ξ ) ) scheme on the suggested model is employed to study new rational solutions. The investigated properties of solutions were determined by graphic studies, which shows significantly values of the parameters and susceptibility of abundant solutions. The obtained results in this work are expected to open new perspectives for the traveling wave theory. For the aforementioned wave solutions, we graphically describe their dynamical properties. It is worth mentioning that our results not only enable us to understand the dynamic properties of such equations more intuitively but also provide some ideas for researchers to facilitate more in depth exploration. It is important to mention that our proposed method is highly effective, consistent, and impacting and can be utilized to solve different physical models.

Keywords: traveling wave solutions; He’s variational direct technique; The rational tan(Π(ξ)) scheme; generalized Hirota–Satsuma–Ito equation; modulation instability

Lead Paragraph This article studies the dynamics of semi-inverse scheme and six forms of supposed studies reveals that the solutions belongs to soliton. The modulation instability is analyzed. It is also found that the tan ( ϕ ∕ 2 ) scheme on the suggested model is employed. And properties of solutions were determined by graphic studies.

1 Introduction

Nonlinear partial differential equations (NPDEs) play an important role in describing various natural phenomena arising in physics, chemistry, biology, medical, and so on. The (1+1)-dimensional Hirota–Satsuma shallow water wave equation first introduced by Hirota and Satsuma as a model equation describing the unidirectional propagation of shallow water waves [1] as follows:

(1.1) Ψ t − Ψ x x t − 3 Ψ Ψ t + 3 Ψ x ∫ x ∞ Ψ t d x + Ψ x = 0 ,

by employing the following bilinear relation

(1.2) Ψ = 2 ( ln f ) x x .

Eq. (1.1) changed to the bilinear model as follows:

(1.3) ( D x D t − D x 3 D t + D x 2 ) f . f = 0 .

Also, (2+1)-dimensional Hirota–Satsuma shallow water wave equation [2] will be read as follows:

(1.4) Ψ x x x t + 3 ( Ψ x Ψ t ) x + Ψ y t + Ψ x x = 0 .

Moreover, the (3+1)-dimensional generalized Hirota–Satsuma–Ito (HSI) shallow water wave equation [3] will be read as follows:

(1.5) Ψ x x x t + 3 ( Ψ x Ψ t ) x + δ 1 Ψ y t + δ 2 Ψ x x + δ 3 Ψ x y + δ 4 Ψ x t + δ 5 Ψ y y = 0 .

Nowadays, a number of novel analytical techniques were focused by the research group around the world, and among that He’s variational direct technique is one of the most widely interested schemes due to its susceptibility. However, the effective usage of techniques depends on the application and quality, but obtaining a better quality than other methods is still not easy owing to the formation of method process. Hence, the techniques to generate and detect the solutions with different forms to solve some nonlinear partial differential equations (PDEs) include nonlinear fractional partial differential equations [4], the variable coefficients fourth-order parabolic PDEs [5], the Biswas–Milovic equation for Kerr law nonlinearity [6], the back-propagation neural network technique [7], the multimodal vision-language learning scheme [8], the complete language-vision interaction network method [9], the multimodal hybrid parallel network technique [10], the multi-scale channel-spatial scheme [11], the Internet of things technique [12], the novel quaternary surfactant scheme [13], the developed multifunctional adsorbent method [14], the Fourier decomposition scheme [15], the novel fiber-supported model [16], an improved particle swarm optimization technique [17], the symmetrical printed structures method [18], the blockchain technology [19], and the finite element multi-physics method [20]. In recent years, the rapid development of applied sciences in many field includes the higher-order finite dimensional algebra technique [21], the impulse switched singular systems [22], the compositional optimization and structural techniques [23], a robust observer based on nonlinear descriptor systems [24], a novel three-phase quasi-Z-source inverter [25] and light-absorbing and shadowing of graphene nanoplatelets [26], a fuzzy logic path planning algorithm [27], the longitudinal wave equation [28], He’s variational direct methods for the KP-BBM equation [29], a novel bistable origami flexible gripper [30], the (3+1)–dimensional variable-coefficient nonlinear wave equation [31], the generalized Hietarinta equation [32], the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon-Kotera–Sawada equation [33], a stratified random sampling technique [34], separation between the scattering and the intrinsic attenuation [35], reverse roll coating for a non-isothermal third-grade fluid [36], numerical study on maneuvering a container ship in shallow water waves [37], an imperative role of studying existing battery datasets and algorithms [38], estimation of viscoelastic attenuation of real seismic data [39], contradictory cyber-physical power systems [40], a new biomass-based hybrid energy system [41], the theoretical model and verification of electric field [42], algorithm based on random sampling [43], fuzzy event-triggered sliding mode depth control [44], multilegged robot smooth motion [45], a machine learning study on superlattice electron blocking layer [46], low-resistance asymmetric method [47], the two-dimensional electron gas [48], modulation at last quantum barrier [49,50] and other important topics [51,52]. The importance of the nonlinear phenomena has grown in a number of newly established categories, including branch of applied mathematics and mathematical physics [53–57]. The investigations on various types of wave propagation generated by some unavoidable geophysical disturbance are parts of extensive mathematical modeling and analysis [58–62].

In the last few decades, numerous researchers have investigated (analytically and numerically) the differential equations for difference equations and differential-difference equations. A fourth-order time-fractional partial differential equation with Riemann–Liouville definition was analyzed using the general method of separation of variables to transform the original equation and the trial equation method to obtain its integral form [63]. The fault diagnosis method for improved residual shrinkage network of rolling bearings was proposed in [64]. The authors of [65] presented the complete analysis of traveling wave solutions to a special kind of nonlinear Schrödinger equation with logarithmic nonlinearity. The Greenberger-Horne-Zeilinger state was studied of macroscopic quantum effects and for applications in quantum metrology and quantum information [66]. The nonlinear Zakharov system was investigated through a comprehensive analysis that integrates logarithmic ransformations and the symbolic structures of exponential functions [67]. The exact soliton solutions and different wave structures to the (2+1) dimensional Chaffee-Infante equation have obtained in [68]. An extended version of the Ginzburg -Landau equation was investigated to describe bifurcations and chaotic behavior of the model with applications of the motion of particles in a plasma [69]. The photonic in-memory computing architecture based on phase change materials was increasingly attracting widespread attention due to its high computational efficiency and low power consumption [70]. In the study by Gao et al. [71] the composite meta-absorber with broadband sound absorption was studied utilizing the genetic algorithm to optimize the geometric dimensions under normal incidence and free field conditions. Most of the series solutions methods were applied to steady state heat transfer in extended surfaces. In most of the previous works, the analytical methods applied to the nonlinear problems are limited to linear problems, while numerical methods are used for nonlinear transient heat transfer problems.

The best of the authors’ knowledge, the analytical solutions for the generalized Hirota–Satsuma–Ito equation has not been presented in literature with the mentioned and explained methods. Such solutions provide proper physical insights and effective predictions to water waves. The developed exact analytical solutions to the steady-state thermal analysis of the extended surfaces are not explicitly expressed. Alternatively, most of the nonlinear models have been solved numerically. However, the recent developments in nonlinear analysis have welcome the use of various approximate analytical method for symbolic solutions of nonlinear equations. The developments of symbolic solutions to the problems often lead to the applications of series solution techniques. Unfortunately, these techniques require high mathematical skills, and the eventual solutions give series solutions with huge expressions and large number of terms that are not appealing to learners and readers. Also, applying such series solution techniques to transient nonlinear problems make the series solutions to be so huge for convenient especially for practical applications. Moreover, such analysis comes with complex mathematical analysis involving high computational cost and time. However, the solutions of the numerical methods are numeric in nature, and the methods have inherent high computational cost and time. Therefore, the need and the obvious advantages of developing highly accurate symbolic analytical solutions to the nonlinear problems cannot be overemphasized. In addition, the obvious advantages of generating exact analytical solutions to the nonlinear problems are very much important. There have been few attempts where exact analytical solutions were developed for the nonlinear models, but the developed solutions are implicit in nature which add to the computational efforts, time, and costs. Consequently, over the years, recourse has been made to approximate analytical, semi-analytical, semi-numerical, and numerical methods for the thermal analysis of the passive devices as shown in the reviewed studies.

The new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and (2+1)-dimensional breaking soliton equations via a generalized exponential rational function method [72]. A new inverse ( G ′ ∕ G )-expansion method for extracting novel soliton solutions in the context of the (2+1)-dimensional generalized Benjamin–Ono equation was applied [73]. The behavior of optical soliton solutions in optical fiber theory was described by the Schrödinger–Hirota equation based on the modified generalized Riccati equation mapping approach [74].

The exact solutions of nonlinear Schrödinger equation with third-order dispersion in constant potential were achieved using Bernoulli ( G ′ ∕ G )-expansion method [75]. The soliton solutions of the nonlinear Schrödinger equation with x-spatially varying coefficients involving Woods–Saxon potential were acquired by using the extended ( G ′ ∕ G )-expansion method [76]. The new optical soliton solutions of nonlinear Schrödinger equation with time-dependent coefficients, which describes the dispersion decreasing fiber were obtained using Bernoulli ( G ′ ∕ G )-expansion method [77].

Three-wave type, lump wave, rogue wave, and hybrid solutions of different forms were studied by the bilinear neural network method to the (3+1)-dimensional Ito equation [78]. The (2+1)-dimensional Ito equation was analyzed for exact analytic solutions using two reliable techniques involving the simplest equation and modified simplest equation algorithms [79]. The conformable derivative and adequate fractional complex transform were implemented to discuss the fractional higher-dimensional Ito equation analytically using the Jacobi elliptic function method and Riccati equation mapping method [80].

This article emphasizes on applications of the HVDT, modulation instability analysis, and rational tan ( Z ( ξ ) ) method. After reviewing the literature, it is concluded that the governing model is not solved yet by the proposed techniques. By provoking this, the utmost focus of this work is to retrieve optical and other soliton solutions of given model by employing the HVDT, modulation instability analysis, and rational tan ( Z ( ξ ) ) method.

This dispersion results from the waves’ interactions with the surface tension, water depth, and waves. For waves with large amplitudes propagating over great distances, the dynamics of water waves can become nonlinear. Solitons are intricate patterns that can develop due to nonlinear wave dynamics. Waves may get steeper and more unstable as they approach shallow water, eventually breaking into choppy whitecaps. This phenomenon is most noticeable near coastlines. Water waves, in general, show a wide variety of features, making them a significant subject of interest and research in fluid dynamics, oceanography, and other related fields.

The main idea of this current modified method is to take advantage of the exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation and reduce it to a second-order linear ordinary differential equations (ODEs) through some suitable transformations. This second-order linear ODE can easily be solved, and it is known to have three cases of possible solutions in its general form. In addition, we will demonstrate that the He’s variational direct technique and the tan ( Π ( ξ ) ) scheme can be used to acquire more traveling wave solutions to a variety of NLEEs. These solutions often play a crucial role in understanding and describing various nonlinear phenomena, such as solitary, solitons, shock waves, or other localized structures. There are the general strengths and weaknesses for each method – the strengths of the mentioned method is founding different forms of exact solutions and the weaknesses of this method is where able to find only three categories of solutions and cannot find logarithmic and polynomial solutions.

The rest of this article is structured as follows: the He’s variational direct method has been summarized in Section 2. In Section 3, the HVDT is employed to make the travel wave solutions for the generalized Hirota–Satsuma–Ito equation. In Section 4, the modulation instability analysis is investigated. Moreover, in Sections 5 and 6, the rational tan ( Z ( ξ ) ) scheme and application on the generalized nonlinear wave equation are given. Finally, we present the conclusion in Section 7.

2 He’s variational direct technique

Equation (1.5) is a nonlinear partial differential equation, which is very difficult to solve analytically. The common recourse is to make use of the numerical method. However, the numerical methods give no physical insights into the general properties of the solution. Consequently, approximate or semi-analytical methods have been extensively used for the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation, but these solutions are in power series. Also, there have been few cases in literature where exact analytical solutions were developed for the nonlinear mentioned models of the extended surfaces. but these solutions are inherently implicit, which increases the computational efforts, time, and costs. Therefore, in this work, explicit and non-power series exact analytical solutions for the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation are explored via He’s variational direct scheme. The basic principles of the hybrid method are given as follows:

Step 1. Suppose a nonlinear PDE is presented in general format as follows:

(2.1) W ( x , y , t , Ψ , Ψ x , Ψ y , Ψ t , Ψ x x , Ψ x y , Ψ t t , … ) = 0 .

We commence the following changing shape:

(2.2) Ψ ( x , y , t ) = Z ( ξ ) , ξ = k 1 x + k 2 y − c t ,

where k 1 , k 2 , and c are amounts that will be discovered.

Step 2. By substituting (2.2) into Eq. (2.1), we obtain the following relation:

(2.3) H ( Z , Z ′ , Z ″ , Z ‴ , Z ″ ″ , … ) = 0 ,

where Z ′ = d Z d ξ , Z ″ = d 2 Z d ξ 2 , Z ‴ = d 3 Z d ξ 3 , Z ″ ″ = d 4 Z d ξ 4 , and so on.

Step 3. By employing the variational method for Eq. (2.3), we obtain the following outcome:

(2.4) J ( Z ) = ∫ L ( Z , Z ′ , Z ″ , Z ‴ , Z ″ ″ , … ) d ξ .

Step 4. The solutions of Eq. (2.3) are presented in different behaviors:

Set I : Z ( ξ ) = δ sech ( μ ξ ) .

Set II : Z ( ξ ) = δ sech 2 ( μ ξ ) .

Set III : Z ( ξ ) = δ cos ( μ ξ ) .

Set IV : Z ( ξ ) = δ sinh ( μ ξ ) cosh ( μ ξ ) 3 2 .

Set V : Z ( ξ ) = δ tanh ( μ ξ ) .

Set VI : Z ( ξ ) = δ tanh 2 ( μ ξ ) .

Step 5. Using the following two properties, the exact solutions will be obtained as follows:

(2.5) ∂ J ( Z ) ∂ δ = 0 , ∂ J ( Z ) ∂ μ = 0 .

3 Performance of He’s VDT for a generalized HSI equation

By substituting Eq. (2.3) into Eq. (1.5), the following ODE can be received:

(3.1) A 1 d 4 d x 4 Z ( ξ ) + 2 A 2 d d x Z ( ξ ) d 2 d x 2 Z ( ξ ) + A 3 d 2 d x 2 Z ( ξ ) = 0 ,

with

A 1 = − c k 1 3 , A 2 = − 3 c k 1 2 , A 3 = − c δ 1 k 2 − c k 1 δ 4 + δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 ,

where k 1 , k 2 , and c are free amounts. The functional form is written as follows:

(3.2) J ( Z ) = ∫ A 1 d d ξ Z d 4 d ξ 4 Z + 2 A 2 d d ξ Z 2 d 2 d ξ 2 Z + A 3 d d ξ Z d 2 d ξ 2 Z d ξ .

3.1 Kinky dark solitary wave

Collection I: The first set solution of Eq. (3.1) is taken as follows:

(3.3) Z ( ξ ) = δ sech ( μ ξ ) .

Putting Eq. (3.3) into Eq. (3.2) results in:

(3.4) J ( δ , μ ) = ∫ 0 ∞ { A 1 ( − δ μ sech ( μ ξ ) tanh ( μ ξ ) ) δ μ 4 ( ( cosh ( μ ξ ) ) 4 − 20 ( cosh ( μ ξ ) ) 2 + 24 ) ( cosh ( μ ξ ) ) 5 + 2 A 2 ( − δ μ sech ( μ ξ ) tanh ( μ ξ ) ) 2 × δ μ 2 ( ( cosh ( μ ξ ) ) 2 − 2 ) ( cosh ( μ ξ ) ) 3 + A 3 ( − δ μ sech ( μ ξ ) tanh ( μ ξ ) ) δ μ 2 ( ( cosh ( μ ξ ) ) 2 − 2 ) ( cosh ( μ ξ ) ) 3 d ξ = − δ 2 μ ( 9 A 1 μ 2 sinh ( μ ξ ) ( 7 ( cosh 4 ( μ ξ ) ) + 6 cosh 2 ( μ ξ ) − 8 ) − 4 δ μ A 2 ( 5 cosh 2 ( μ ξ ) − 3 ) − Σ 1 ) 90 cosh 5 ( μ ξ ) ξ = 0 ξ = ∞ ,

with

Σ 1 = 15 A 3 sinh ( μ ξ ) cosh 2 ( μ ξ ) ( cosh 2 ( μ ξ ) − 1 ) ,

(3.5) J ( δ , μ ) = 1 30 δ 2 μ ( 21 c μ 2 k 1 3 + 8 c δ μ k 1 2 − 5 c δ 1 k 2 − 5 c δ 4 k 1 + 5 δ 2 k 1 2 + 5 δ 3 k 1 k 2 + 5 δ 5 k 2 2 ) .

By using the following derivatives, one results:

(3.6) ∂ J ∂ δ = 0 , ∂ J ∂ μ = 0 .

So we have:

(3.7) ∂ J ∂ δ = 1 15 δ μ ( 21 c μ 2 k 1 3 + 8 c δ μ k 1 2 − 5 c δ 1 k 2 − 5 c δ 4 k 1 + 5 δ 2 k 1 2 + 5 δ 3 k 1 k 2 + 5 δ 5 k 2 2 ) + 4 δ 2 μ 2 c k 1 2 15 = 0 ,

(3.8) ∂ J ∂ μ = 21 c δ 2 μ 2 k 1 3 10 + 8 c δ 3 μ k 1 2 15 − 1 6 c δ 2 δ 1 k 2 − 1 6 c δ 2 δ 4 k 1 + 1 6 δ 2 δ 2 k 1 2 + 1 6 δ 2 δ 3 k 1 k 2 + 1 6 δ 2 δ 5 k 2 2 = 0 ,

which leads to:

(3.9) δ = 21 2 c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) ,

μ = − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 21 c k 1 2 .

The final solution is presented as follows:

(3.10) Ψ ( x , y , t ) = 21 2 c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 )

× sech − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 21 c k 1 2 ( k 1 x + k 2 y − c t ) .

3.2 Dark solitary wave

Collection II: The second set solution of Eq. (3.1) is taken as follows:

(3.11) Z ( ξ ) = δ sech 2 ( μ ξ ) .

Plugging Eq. (3.11) into Eq. (3.2), one becomes

(3.12) J ( δ , μ ) = 2 δ 2 μ ( 120 c μ 2 k 1 3 + 35 c δ μ k 1 2 − 14 c δ 1 k 2 − 14 c δ 4 k 1 + 14 δ 2 k 1 2 + 14 δ 3 k 1 k 2 + 14 δ 5 k 2 2 ) 105 .

The following derivatives are given as follows:

(3.13) ∂ J ∂ δ = 4 δ μ ( 120 c μ 2 k 1 3 + 35 c δ μ k 1 2 − 14 c δ 1 k 2 − 14 c δ 4 k 1 + 14 δ 2 k 1 2 + 14 δ 3 k 1 k 2 + 14 δ 5 k 2 2 ) 105 + 2 ∕ 3 δ 2 μ 2 c k 1 2 ,

∂ J ∂ μ = 48 c δ 2 μ 2 k 1 3 7 + 4 ∕ 3 c δ 3 μ k 1 2 − 4 c δ 2 δ 1 k 2 15 − 4 c δ 2 δ 4 k 1 15 + 4 δ 2 δ 2 k 1 2 15 + 4 δ 2 δ 3 k 1 k 2 15 + 4 δ 2 δ 5 k 2 2 15 .

From which, we have:

(3.14) δ = 48 c δ 1 k 2 + 48 c δ 4 k 1 − 48 δ 2 k 1 2 − 48 δ 3 k 1 k 2 − 48 δ 5 k 2 2 5 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) ,

μ = 1 30 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) c k 1 2 .

The last solution of Eq. (1.5) is created as follows:

(3.15) Ψ ( x , y , t ) = 48 c δ 1 k 2 + 48 c δ 4 k 1 − 48 δ 2 k 1 2 − 48 δ 3 k 1 k 2 − 48 δ 5 k 2 2 5 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 )

× sech 2 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 30 c k 1 2 × ( k 1 x + k 2 y − c t ) ) .

3.3 Periodic wave solution

Collection III: The third set solution of Eq. (3.1) is taken as follows:

(3.16) Z ( ξ ) = δ cos ( μ ξ ) .

Inserting Eq. (3.16) into Eq. (3.2) concludes in:

(3.17) J ( δ , μ ) = ∫ 0 T ∕ 4 − 2 3 δ 3 ( sin ( μ ξ ) ) 3 μ 3 A 2 + 1 2 δ 2 ( sin ( μ ξ ) ) 2 μ 2 A 3 − 1 2 δ 2 ( cos ( μ ξ ) ) 2 μ 4 A 1 − δ 2 ( sin ( μ ξ ) ) 2 μ 4 A 1 d ξ

= − 1 12 δ 2 μ 1 2 ( − 9 c μ 3 k 1 3 + 3 c δ 1 k 2 μ + 3 c k 1 δ 4 μ − 3 δ 2 k 1 2 μ − 3 δ 3 k 1 k 2 μ − 3 δ 5 k 2 2 μ ) π − 16 c δ μ k 1 2 .

The following properties are given as follows:

(3.18) ∂ J ∂ δ = 0 , ∂ J ∂ μ = 0 .

These leads to:

(3.19) ∂ J ∂ δ = 1 4 δ μ 2 ( 3 π c μ 2 k 1 3 − π c δ 1 k 2 − π c δ 4 k 1 + π δ 2 k 1 2 + π δ 3 k 1 k 2 + π δ 5 k 2 2 + 16 c δ k 1 2 ) = 0 ,

(3.20) ∂ J ∂ μ = − 1 12 δ 2 μ ( − 18 π c μ 2 k 1 3 + 3 π c δ 1 k 2 + 3 π c δ 4 k 1 − 3 π δ 2 k 1 2 − 3 π δ 3 k 1 k 2 − 3 π δ 5 k 2 2 − 32 c δ k 1 2 ) = 0 ,

which leads to:

(3.21) δ = 3 π ( c δ 1 k 2 + c k 1 δ 4 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 64 c k 1 2 , μ = 3 c k 1 ( c δ 1 k 2 + c k 1 δ 4 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 6 c k 1 2 .

The last solution of Eq. (1.5) is written as follows:

(3.22) Ψ ( x , y , t ) = 3 π ( c δ 1 k 2 + c k 1 δ 4 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 64 c k 1 2

× cos 3 c k 1 ( c δ 1 k 2 + c k 1 δ 4 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 6 c k 1 2 ( k 1 x + k 2 y − c t ) .

3.4 Bright solitary wave

Collection IV: The fourth set solution of Eq. (3.1) is taken as follows:

(3.23) Z ( ξ ) = δ sinh ( μ ξ ) cosh ( μ ξ ) 3 2 .

By putting the related Eq. (3.23) into Eq. (3.2), one obtains

(3.24) J ( δ , μ ) = ∫ 0 ∞ 25 δ 2 μ 2 ( 3 ( cosh ( μ ξ ) ) 3 ∕ 2 μ 2 A 1 ( 25 ( cosh ( μ ξ ) ) 4 − 106 ( cosh ( μ ξ ) ) 2 + 77 ) + Σ 1 ) 96 ( cosh ( μ ξ ) ) 21 ∕ 2 d ξ

= 25 δ 2 μ 233,472 20,117,295 π c μ 2 k 1 3 2 + 16 k 1 2 ( 512 c δ μ + 58,311 π δ 2 ) + 912 π δ 1 k 2 ( c − 1,024 ) + 932,976 π ( − c δ 4 k 1 + δ 3 k 1 k 2 + δ 5 k 2 2 ) ,

Σ 1 = − 40 δ μ A 2 sinh ( μ ξ ) ( ( cosh ( μ ξ ) ) 2 − 1 ) + 12 ( cosh ( μ ξ ) ) 7 ∕ 2 A 3 ( ( cosh ( μ ξ ) ) 2 + 1 ) .

With the following properties, we have:

(3.25) ∂ J ∂ δ = 0 , ∂ J ∂ μ = 0 .

These leads to:

(3.26) ∂ J ∂ δ = 25 δ μ ( 6,705,765 π c μ 2 k 1 3 + 32 k 1 2 ( 256 c δ μ + 19,437 π δ 2 ) + 608 π δ 1 k 2 ( c − 1024 ) + 621,984 π ( − c δ 4 k 1 + δ 3 k 1 k 2 + δ 5 k 2 2 ) ) 77,824 ,

(3.27) ∂ J ∂ μ = 26,470,125 δ 2 π c μ 2 k 1 3 8,192 + 25 δ 2 k 1 2 ( 1,024 c δ μ + 58,311 π δ 2 ) 14,592 + 25 δ 2 π δ 1 k 2 ( c − 1,024 ) 256 + 25,575 δ 2 π ( − c δ 4 k 1 + δ 3 k 1 k 2 + δ 5 k 2 2 ) 256 ,

by solving equations (3.25) leads to:

(3.28) δ = − 171 π ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) 15,686 1,024 c k 1 ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) ,

μ = 4 15,686 c k 1 ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) 117,645 c k 1 2 .

The last solution of Eq. (1.5) is concluded as follows:

(3.29) Ψ ( x , y , t ) = − 171 π ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) 15,686 1,024 c k 1 ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 )

× δ sinh 4 15,686 c k 1 ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) 117,645 c k 1 2 ( k 1 x + k 2 y − c t ) cosh 4 15,686 c k 1 ( c δ 1 k 2 − 1,023 c δ 4 k 1 + 1,023 k 1 2 δ 2 + 1,023 δ 3 k 1 k 2 + 1,023 δ 5 k 2 2 − 1,024 δ 1 k 2 ) 117,645 c k 1 2 ( k 1 x + k 2 y − c t ) 3 2 .

3.5 Dark soliton wave

Collection V: The fifth set solution of Eq. (3.1) is taken as follows:

(3.30) Z ( ξ ) = δ tanh ( μ ξ ) .

Inserting Eq. (3.30) into Eq. (3.2) reached in:

(3.31) J ( δ , μ ) = ∫ 0 ∞ 1 6 ( ( sinh ( μ ξ ) ) 2 − ( cosh ( μ ξ ) ) 2 ) 2 ( 12 ( cosh ( μ ξ ) ) 2 μ 2 A 1 + 3 A 3 ( cosh ( μ ξ ) ) 2 + 4 δ μ A 2 − 24 μ 2 A 1 ) δ 2 μ 2 ( cosh ( μ ξ ) ) 6 d ξ ,

(3.32) J ( δ , μ ) = − 1 30 δ 2 μ ( − 24 c μ 2 k 1 3 + 32 c δ μ k 1 2 + 10 c δ 1 k 2 + 10 c δ 4 k 1 − 10 δ 2 k 1 2 − 10 δ 3 k 1 k 2 − 10 δ 5 k 2 2 ) .

With the following results, we have:

(3.33) ∂ J ∂ δ = 0 , ∂ J ∂ μ = 0 .

So we have:

(3.34) ∂ J ∂ δ = 8 5 c δ μ 3 k 1 3 − 16 k 1 2 δ 2 μ 2 c 5 − 2 3 c δ μ δ 1 k 2 − 2 3 c δ μ δ 4 k 1 + 2 3 δ μ δ 2 k 1 2 + 2 3 δ μ δ 3 k 1 k 2 + 2 3 δ μ δ 5 k 2 2 = 0 ,

(3.35) ∂ J ∂ μ = − 1 15 δ 2 ( − 36 c μ 2 k 1 3 + 32 c δ μ k 1 2 + 5 c δ 1 k 2 + 5 c δ 4 k 1 − 5 δ 2 k 1 2 − 5 δ 3 k 1 k 2 − 5 δ 5 k 2 2 ) = 0 ,

which leads to:

(3.36) δ = − 3 2 c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 − 3 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) ,

μ = − 3 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 6 c k 1 2 .

The last solution of Eq. (1.5) is concluded as follows:

(3.37) Ψ ( x , y , t ) = − 3 2 c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 − 3 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 )

× tanh − 3 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) 6 c k 1 2 ( k 1 x + k 2 y − c t ) .

3.6 Another type dark soliton wave

Collection VI: The sixth set solution of Eq. (3.1) is considered as follows:

(3.38) Z ( ξ ) = δ tanh 2 ( μ ξ ) .

Inserting Eq. (3.38) into Eq. (3.2) results in:

(3.39) J ( δ , μ ) = ∫ 0 ∞ 2 3 ( 3 μ 2 A 1 cosh ( μ ξ ) ( 4 ( cosh ( μ ξ ) ) 4 − 20 ( cosh ( μ ξ ) ) 2 + 15 ) + ( ( cosh ( μ ξ ) ) 2 − 1 ) Σ 1 ) δ 2 μ 2 ( cosh ( μ ξ ) ) 9 d ξ ,

with

Σ 1 = ( 8 sinh ( μ ξ ) δ μ A 2 + 3 A 3 ( cosh ( μ ξ ) ) 3 ) ,

(3.40) J ( δ , μ ) = − 2 δ 2 μ ( − 120 c μ 2 k 1 3 + 35 c δ μ k 1 2 + 14 c δ 1 k 2 + 14 c δ 4 k 1 − 14 δ 2 k 1 2 − 14 δ 3 k 1 k 2 − 14 δ 5 k 2 2 ) 105 .

With the following results:

(3.41) ∂ J ∂ δ = 0 , ∂ J ∂ μ = 0 .

So we have:

(3.42) ∂ J ∂ δ = − 2 δ μ ( − 240 c μ 2 k 1 3 + 105 c δ μ k 1 2 + 28 c δ 1 k 2 + 28 c δ 4 k 1 − 28 δ 2 k 1 2 − 28 δ 3 k 1 k 2 − 28 δ 5 k 2 2 ) 105 = 0 ,

(3.43) ∂ J ∂ μ = − 4 δ 2 ( − 180 c μ 2 k 1 3 + 35 c δ μ k 1 2 + 7 c δ 1 k 2 + 7 c δ 4 k 1 − 7 δ 2 k 1 2 − 7 δ 3 k 1 k 2 − 7 δ 5 k 2 2 ) 105 = 0 ,

which leads to:

(3.44) δ = − 48 c δ 1 k 2 + 48 c δ 4 k 1 − 48 δ 2 k 1 2 − 48 δ 3 k 1 k 2 − 48 δ 5 k 2 2 5 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) ,

μ = 1 30 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) c k 1 2 .

The last solution of Eq. (1.5) is concluded as follows:

(3.45) Ψ ( x , y , t ) = − 48 c δ 1 k 2 + 48 c δ 4 k 1 − 48 δ 2 k 1 2 − 48 δ 3 k 1 k 2 − 48 δ 5 k 2 2 5 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 )

× tanh 2 1 30 − 21 c k 1 ( c δ 1 k 2 + c δ 4 k 1 − δ 2 k 1 2 − δ 3 k 1 k 2 − δ 5 k 2 2 ) c k 1 2 ( k 1 x + k 2 y − c t ) .

Remark: We will utilize one test to verify the got solutions in Section 3. We obtain the values δ 1 = 1 , δ 2 = 2 , δ 3 = 3 , δ 4 = 4 , δ 5 = 5 , then the manner of acting of the kinky dark solution in Eq. (3.10) is designed in Figure 1 by the 3-D and 2-D graphs. The manner of acting of dark solitary in Eq. (3.15) is designed in Figure 2 in the form of 3-D and 2-D graphs. The periodic manner of acting of Eq. (3.22) is investigated in Figure 3. Moreover, the bright soliton behavior of Eq. (3.29), the soliton manner of acting of Eq. (3.36), and the another type soliton wave behavior of Eq. (3.44) have been designed, respectively, in Figures 4, 5, and 6 via the 3-D and 2-D plots. The found computations have been done in Figures 1, 2, 3, 4, 5, and 6 for showing the nature of the received solutions by Eqs (3.10), (3.15), (3.22), (3.29), (3.36), and (3.44), respectively. Obviously, Figures 1–5 show “the kinky dark,” “dark solitary,” “periodic,” “bright solitary,” and “soliton” wave solutions, respectively. Figure 6 shows the perfect another type soliton wave solution, which all can well present the physical interpretation of the obtained solutions for Eq. (1.5).

$Figure 1 Graphs of Ψ \Psi (3.10) with c = 1 c=1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 1

Graphs of Ψ (3.10) with c = 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

$Figure 2 Graphs of Ψ \Psi (3.15) with c = 1 c=1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 2

Graphs of Ψ (3.15) with c = 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

$Figure 3 Graphs of Ψ \Psi (3.22) with c = ‒ 1 c=‒1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 3

Graphs of Ψ (3.22) with c = ‒ 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

$Figure 4 Graphs of Ψ \Psi (3.29) with c = 1 c=1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 4

Graphs of Ψ (3.29) with c = 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

$Figure 5 Graphs of Ψ \Psi (3.36) with c = 1 c=1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 5

Graphs of Ψ (3.36) with c = 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

$Figure 6 Graphs of Ψ \Psi (3.44) with c = 1 c=1 , k 1 = 2 {k}_{1}=2 , k 2 = 3 {k}_{2}=3 , t = 1 t=1 to (I) 3D graph (II) 2D graph.$

Figure 6

Graphs of Ψ (3.44) with c = 1 , k 1 = 2 , k 2 = 3 , t = 1 to (I) 3D graph (II) 2D graph.

4 Stability analysis of generalized HSI equation

The perturbed solution of the Eq. (1.5) is considered as follows:

(4.1) Ψ ( x , y , t ) = ζ + θ Σ ( x , y , t ) ,

where ζ is a free value. Substituting (4.1) into Eq. (1.5), one results

(4.2) θ d 4 d t d x 3 Σ + 3 θ 2 d 2 d x 2 Σ d d t Σ + 3 θ 2 d d x Σ × d 2 d t d x Σ + δ 1 θ d 2 d t d y Σ +

δ 2 θ d 2 d x 2 Σ + δ 3 θ d 2 d x d y Σ + δ 4 θ d 2 d t d x Σ + δ 5 θ d 2 d y 2 Σ = 0 ,

where Σ = Σ ( x , y , t ) and by linerization Eq. (4.2), we obtain

(4.3) θ d 4 d t d x 3 Σ + δ 1 θ d 2 d t d y Σ + δ 2 θ d 2 d x 2 Σ + δ 3 θ d 2 d x d y Σ + δ 4 θ d 2 d t d x Σ + δ 5 θ d 2 d y 2 Σ = 0 .

Theorem 4.1

Presume that the solution of Eq. (4.3) has the following model:

(4.4) Σ ( x , y , t ) = ρ 1 e i ( λ x + μ y + σ t ) ,

where λ and μ are the normalized wave numbers, by inserting (4.4) into Eq. (4.3), then by solving for σ , we can obtain the following relation:

(4.5) σ ( λ , μ ) = λ 2 δ 2 + λ μ δ 3 + μ 2 δ 5 λ 3 − λ δ 4 − μ δ 1 .

Proof

By appending the relation (4.4) in the linear PDE (4.3), one obtain

(4.6) θ d 4 d t d x 3 Σ + δ 1 θ d 2 d t d y Σ + δ 2 θ d 2 d x 2 Σ + δ 3 θ d 2 d x d y Σ + δ 4 θ d 2 d t d x Σ + δ 5 θ d 2 d y 2 Σ

= e i ( λ x + μ y + σ t ) θ ρ 1 ( λ 3 σ − λ 2 δ 2 − λ μ δ 3 − λ σ δ 4 − μ 2 δ 5 − μ σ δ 1 ) = 0 .

Via solving the aforementioned equation, it becomes

(4.7) σ ( λ , μ ) = λ 2 δ 2 + λ μ δ 3 + μ 2 δ 5 λ 3 − λ δ 4 − μ δ 1 .

The theorem is complete.□

In Figure 7, the sign of σ ( λ , μ ) for value of λ for cases ( μ = 1 , 2 and μ = 3 ) and ( μ = − 1 , − 2 and μ = − 3 ) is studied. On the other hand, in Figure 8, the sign σ ( λ , μ ) for the value of λ is stable for cases ( μ = − 2 , 0 and μ = 2 ) and ( μ = 0.01 , 0.2 and μ = 0.8 ). Finally, in Figure 9, the sign of σ ( λ , μ ) for the value of λ is stable for cases ( μ = − 0.01 , − 0.2 and μ = − 0.8 ) and ( μ = − 10 , 0 and μ = 10 ). If the maximum σ ( λ , μ ) of the is exactly 0, the dispersion is named marginally stable.

$Figure 7 Plot σ ( λ , μ ) \sigma \left(\lambda ,\mu ) and wave number λ \lambda under different data δ 1 = 1 {\delta }_{1}=1 , δ 2 = 1.5 {\delta }_{2}=1.5 , δ 3 = 2 {\delta }_{3}=2 , δ 4 = 2.2 {\delta }_{4}=2.2 , δ 5 = ‒ 2 {\delta }_{5}=‒2 .$

Figure 7

Plot σ ( λ , μ ) and wave number λ under different data δ 1 = 1 , δ 2 = 1.5 , δ 3 = 2 , δ 4 = 2.2 , δ 5 = ‒ 2 .

$Figure 8 Plot σ ( λ , μ ) \sigma \left(\lambda ,\mu ) and wave number λ \lambda under different data δ 1 = 1 {\delta }_{1}=1 , δ 2 = 1.5 {\delta }_{2}=1.5 , δ 3 = 2 {\delta }_{3}=2 , δ 4 = 2.2 {\delta }_{4}=2.2 , δ 5 = ‒ 2 {\delta }_{5}=‒2 .$

Figure 8

Plot σ ( λ , μ ) and wave number λ under different data δ 1 = 1 , δ 2 = 1.5 , δ 3 = 2 , δ 4 = 2.2 , δ 5 = ‒ 2 .

$Figure 9 Plot σ ( λ , μ ) \sigma \left(\lambda ,\mu ) and wave number λ \lambda under different data δ 1 = 1 {\delta }_{1}=1 , δ 2 = 1.5 {\delta }_{2}=1.5 , δ 3 = 2 {\delta }_{3}=2 , δ 4 = 2.2 {\delta }_{4}=2.2 , δ 5 = ‒ 2 {\delta }_{5}=‒2 .$

Figure 9

Plot σ ( λ , μ ) and wave number λ under different data δ 1 = 1 , δ 2 = 1.5 , δ 3 = 2 , δ 4 = 2.2 , δ 5 = ‒ 2 .

5 Description of the RTET

The application of the rational tan ( Π ( ξ ) ∕ 2 ) -expansion technique will be studied where for first are given here. We first discuss the mathematical analysis of nonlinear partial differential equations (NPDEs). Hence, we consider the NPDEs of in the following way:

Step 1. Assume a NPDE is produced in a general form in the following:

(5.1) ℒ ( Π , Π x , Π y , Π t , Π x x , Π y y , Π t t , … ) = 0 .

After simple algebraic operations, we obtain

(5.2) Π ( x , y , t ) = Z ( ξ ) , ξ = k 1 x + k 2 y − c t .

Nonlinear ODE

(5.3) S ( Z , k 1 Z ′ , k 2 Z ′ , − c Z ′ , k 1 2 Z ″ , k 2 2 Z ″ , c 2 Z ″ , … ) = 0 .

Step 2. Then, the exact solution of Eq. (5.3) is as follows:

(5.4) Z ( ξ ) = ∑ j = 0 θ η j [ tan ( Π ( ξ ) ∕ 2 ) ] j ∑ j = 0 θ ρ j [ tan ( Π ( ξ ) ∕ 2 ) ] j ,

where η j ( 0 ≤ j ≤ θ ) , ρ j ( 0 ≤ j ≤ θ ) , and η θ , ρ θ ≠ 0 , and, Π = Π ( ξ ) satisfies in the following NODE:

(5.5) Z ′ = λ 1 sin ( Z ) + λ 2 cos ( Z ) + λ 3 .

The particular solutions of Eq. (5.5) will be read as follows:

Family 1: If λ 1 2 + λ 2 2 − λ 3 2 < 0 and λ 2 − λ 3 ≠ 0 , then Z ( ξ ) 2 = tan − 1 λ 1 λ 2 − λ 3 − λ 3 2 − λ 1 2 − λ 2 2 λ 2 − λ 3 tan λ 3 2 − λ 1 2 − λ 2 2 2 ( ξ + ϕ 0 ) .

Family 2: If λ 1 2 + λ 2 2 − λ 3 2 > 0 and λ 2 − λ 3 ≠ 0 , then Z ( ξ ) 2 = tan − 1 λ 1 λ 2 − λ 3 + λ 1 2 + λ 2 2 − λ 3 2 λ 2 − λ 3 tanh λ 1 2 + λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) .

Family 3: If λ 1 2 + λ 2 2 − λ 3 2 > 0 , λ 2 ≠ 0 and λ 3 = 0 , then Z ( ξ ) 2 = tan − 1 λ 1 λ 2 + λ 1 2 + λ 2 2 λ 2 tanh λ 1 2 + λ 2 2 2 ( ξ + ϕ 0 ) .

Family 4: If λ 1 2 + λ 2 2 − λ 3 2 < 0 , λ 3 ≠ 0 and λ 2 = 0 , then Z ( ξ ) 2 = tan − 1 − λ 1 λ 3 + λ 3 2 − λ 1 2 λ 3 tan λ 3 2 − λ 1 2 2 ( ξ + ϕ 0 ) .

Family 5: If λ 2 − λ 3 ≠ 0 and λ 1 = 0 , then Z ( ξ ) 2 = tan − 1 λ 2 + λ 3 λ 2 − λ 3 tanh λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) .

Family 6: If λ 1 = 0 and λ 3 = 0 , then Z ( ξ ) = tan − 1 e 2 λ 2 ( ξ + ϕ 0 ) − 1 e 2 λ 2 ( ξ + ϕ 0 ) + 1 2 e λ 2 ( ξ + ϕ 0 ) e 2 λ 2 ( ξ + ϕ 0 ) + 1 .

Family 7: If λ 2 = 0 and λ 3 = 0 , then Z ( ξ ) = tan − 1 2 e λ 1 ( ξ + ϕ 0 ) e 2 λ 1 ( ξ + ϕ 0 ) + 1 , e 2 λ 1 ( ρ + ϕ 0 ) − 1 e 2 λ 1 ( ξ + ϕ 0 ) + 1 .

Family 8: If λ 1 2 + λ 2 2 = λ 3 2 , then Z ( ξ ) 2 = tan − 1 λ 1 ( ξ + ϕ 0 ) + 2 ( λ 2 − λ 3 ) ( ξ + ϕ 0 ) .

Family 9: If λ 1 = λ 2 = λ 3 = k λ 1 , then Z ( ξ ) 2 = tan − 1 [ e k λ 1 ( ξ + ϕ 0 ) − 1 ] .

Family 10: If λ 1 = λ 3 = k λ 1 and λ 2 = − k λ 1 , then Z ( ξ ) 2 = − tan − 1 e k λ 1 ( ξ + ϕ 0 ) − 1 + e k λ 1 ( ξ + ϕ 0 ) .

Family 11: If λ 3 = λ 1 , then Z ( ξ ) 2 = − tan − 1 ( λ 1 + λ 2 ) e λ 2 ( ξ + ϕ 0 ) − 1 ( λ 1 − λ 2 ) e λ 2 ( ξ + ϕ 0 ) − 1 .

Family 12: If λ 1 = λ 3 , then Z ( ξ ) 2 = tan − 1 ( λ 2 + λ 3 ) e λ 2 ( ξ + ϕ 0 ) + 1 ( λ 2 − λ 3 ) e λ 2 ( ξ + ϕ 0 ) − 1 .

Family 13: If λ 3 = − λ 1 , then Z ( ξ ) 2 = tan − 1 e λ 2 ( ξ + ϕ 0 ) + λ 2 − λ 1 e λ 2 ( ξ + ϕ 0 ) − λ 2 − λ 1 .

Family 14: If λ 2 = − λ 3 , then Z ( ξ ) 2 = tan − 1 λ 1 e λ 1 ( ξ + ϕ 0 ) 1 − λ 3 e λ 1 ( ξ + ϕ 0 ) .

Family 15: If λ 2 = 0 and λ 1 = λ 3 , then Z ( ξ ) 2 = − tan − 1 λ 3 ( ξ + ϕ 0 ) + 2 λ 3 ( ξ + ϕ 0 ) ,

Family 16: If λ 1 = 0 and λ 2 = λ 3 , then Z ( ξ ) 2 = tan − 1 [ λ 3 ( ξ + ϕ 0 ) ] .

Family 17: If λ 1 = 0 and λ 2 = − λ 3 , then Z ( ξ ) 2 = − 2 tan − 1 1 λ 3 ( ξ + ϕ 0 ) .

Family 18: If λ 1 = 0 and λ 2 = 0 , then Z ( ξ ) = λ 3 ( ξ + ϕ 0 ) + C .

Family 19: If λ 2 = λ 3 then Z ( ξ ) 2 = tan − 1 e λ 1 ( ξ + ϕ 0 ) − λ 3 λ 1 , where ϕ 0 is a integration value. Also, steps 3–5 were given by Feng et al. [81].

6 Implementation of rational tan ( Z ( ξ ) ) scheme on the generalized HSI equation

By utilizing the following changing shape, we obtain

(6.1) ξ = k 1 x + k 2 y − c t .

By using the nonlinear model, we obtain

(6.2) A 1 d 4 d x 4 Z ( ξ ) + 2 A 2 d d x Z ( ξ ) d 2 d x 2 Z ( ξ ) + A 3 d 2 d x 2 Z ( ξ ) = 0 ,

with

A 1 = − c k 1 3 , A 2 = − 3 c k 1 2 , A 3 = − c δ 1 k 2 − c k 1 δ 4 + δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 ,

where k 1 , k 2 , and c are not detailed values. The balance amount will be obtain θ = 1 by utilizing the balance technique. Then, the solution is reached as follows:

(6.3) Ψ ( ξ ) = η 0 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 .

First, we substituted the expressions of Z ( ξ ) in (6.3) into (6.2) and collect all terms with the same order of Y ( q ) . Then by equating the coefficient of each polynomial to zero including a system of eleven nonlinear equations and solving them, we obtain the following cases:

First:

(6.4) c = c , η 0 = η 0 , λ 1 = 0 , λ 2 = λ 2 , λ 3 = λ 3 , ρ 0 = ρ 0 , ρ 1 = 0 , η 1 = k 1 ρ 0 ( λ 2 − λ 3 ) ,

k 2 = − k 1 ( k 1 2 λ 2 2 − k 1 2 λ 3 2 + δ 4 ) δ 1 , k 1 = ± 1 2 2 δ 5 ( λ 2 2 − λ 3 2 ) ( δ 1 δ 3 − 2 δ 4 δ 5 + − 4 δ 1 2 δ 2 δ 5 + δ 1 2 δ 3 2 ) δ 5 ( λ 2 2 − λ 3 2 ) .

According to Family 5, using (6.3) becomes

(6.5) Ψ 1 ( ξ ) = η 0 ρ 0 + k 1 ( λ 2 − λ 3 ) tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 2 + λ 3 λ 2 − λ 3 tanh λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) ,

ξ = ± 1 2 2 δ 5 ( λ 2 2 − λ 3 2 ) ( δ 1 δ 3 − 2 δ 4 δ 5 + − 4 δ 1 2 δ 2 δ 5 + δ 1 2 δ 3 2 ) δ 5 ( λ 2 2 − λ 3 2 ) x − k 1 ( k 1 2 λ 2 2 − k 1 2 λ 3 2 + δ 4 ) δ 1 y − c t ,

so that δ 5 ( λ 2 2 − λ 3 2 ) ( δ 1 δ 3 − 2 δ 4 δ 5 + − 4 δ 1 2 δ 2 δ 5 + δ 1 2 δ 3 2 ) > 0 and δ 1 2 δ 3 2 − 4 δ 1 2 δ 2 δ 5 > 0 .

According to Family 6, using (6.3), we obtain

(6.6) Ψ 2 ( ξ ) = η 0 ρ 0 ± 1 2 ( λ 2 − λ 3 ) 2 δ 5 ( δ 1 δ 3 − 2 δ 4 δ 5 + δ 1 2 δ 3 2 − 4 δ 1 2 δ 2 δ 5 ) δ 5 λ 2 tan Z ( ξ ) 2 ,

tan 1 2 Z ( ξ ) = e 2 λ 2 ( ξ + ϕ 0 ) − 1 e 2 λ 2 ( ξ + ϕ 0 ) + 1 , 2 e λ 2 ( ξ + ϕ 0 ) e 2 λ 2 ( ξ + ϕ 0 ) + 1 ,

ξ = ± 1 2 2 δ 5 ( δ 1 δ 3 − 2 δ 4 δ 5 + − 4 δ 1 2 δ 2 δ 5 + δ 1 2 δ 3 2 ) δ 5 λ 2 x − k 1 ( k 1 2 λ 2 2 + δ 4 ) δ 1 y − c t ,

so that δ 5 ( δ 1 δ 3 − 2 δ 4 δ 5 + − 4 δ 1 2 δ 2 δ 5 + δ 1 2 δ 3 2 ) > 0 , and δ 1 2 δ 3 2 − 4 δ 1 2 δ 2 δ 5 > 0 .

Second:

(6.7) c = δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 , η 0 = η 0 , λ 1 = λ 1 , λ 2 = λ 2 , λ 3 = λ 3 , ρ 0 = ρ 0 , ρ 1 = 0 , η 1 = k 1 ρ 0 ( λ 2 − λ 3 ) .

According to Family 1, (6.3) becomes

(6.8) Ψ 1 ( ξ ) = η 0 ρ 0 + k 1 ( λ 2 − λ 3 ) tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 − λ 3 2 − λ 1 2 − λ 2 2 λ 2 − λ 3 tan × λ 3 2 − λ 1 2 − λ 2 2 2 ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 t , λ 1 2 + λ 2 2 − λ 3 2 < 0 , and λ 2 − λ 3 ≠ 0 . According to Family 2, (6.3) becomes

(6.9) Ψ 1 ( ξ ) = η 0 ρ 0 + k 1 ( λ 2 − λ 3 ) tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 + λ 1 2 + λ 2 2 − λ 3 2 λ 2 − λ 3 tanh × λ 1 2 + λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 t , λ 1 2 + λ 2 2 − λ 3 2 > 0 , and λ 2 − λ 3 ≠ 0 . According to Family 6, (6.3) becomes

(6.10) Ψ 3 ( ξ ) = η 0 ρ 0 + k 1 λ 2 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = e 2 λ 2 ( ξ + ϕ 0 ) − 1 e 2 λ 2 ( ξ + ϕ 0 ) + 1 , 2 e λ 2 ( ξ + ϕ 0 ) e 2 λ 2 ( ξ + ϕ 0 ) + 1 ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 + δ 1 k 2 + δ 4 k 1 t . According to Family 8, (6.3) becomes

(6.11) Ψ 4 ( ξ ) = η 0 ρ 0 + k 1 ( λ 2 − λ 3 ) tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 ( ξ + ϕ 0 ) + 2 ( λ 2 − λ 3 ) ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 δ 1 k 2 + δ 4 k 1 t .

Third:

(6.12) c = δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 , η 0 = − 2 k 1 λ 1 ρ 0 ρ 1 + k 1 ρ 1 2 ( λ 2 + λ 3 ) − k 1 ρ 0 2 ( λ 2 − λ 3 ) + η 1 ρ 0 ρ 1 ,

λ 1 = λ 1 , λ 2 = λ 2 , λ 3 = λ 3 , ρ 0 = ρ 0 , ρ 1 = ρ 1 , η 1 = η 1 .

According to Family 1, (6.3) becomes

(6.13) Ψ 1 ( ξ ) = − 2 k 1 λ 1 ρ 0 ρ 1 + k 1 ρ 1 2 ( λ 2 + λ 3 ) − k 1 ρ 0 2 ( λ 2 − λ 3 ) + η 1 ρ 0 ρ 1 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 ,

tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 − λ 3 2 − λ 1 2 − λ 2 2 λ 2 − λ 3 tan λ 3 2 − λ 1 2 − λ 2 2 2 ( ξ + ϕ 0 ) ,

(6.14) Ψ 2 ( ξ ) = − 2 k 1 λ 1 ρ 0 ρ 1 + k 1 ρ 1 2 ( λ 2 + λ 3 ) − k 1 ρ 0 2 ( λ 2 − λ 3 ) + η 1 ρ 0 ρ 1 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 ,

tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 + λ 1 2 + λ 2 2 − λ 3 2 λ 2 − λ 3 tanh λ 1 2 + λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) ,

(6.15) Ψ 3 ( ξ ) = k 1 λ 2 ( − ρ 0 2 + ρ 1 2 ) + η 1 ρ 0 ρ 1 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = e 2 λ 2 ( ξ + ϕ 0 ) − 1 e 2 λ 2 ( ξ + ϕ 0 ) + 1 , 2 e λ 2 ( ξ + ϕ 0 ) e 2 λ 2 ( ξ + ϕ 0 ) + 1 ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 + δ 1 k 2 + δ 4 k 1 t . According to Family VIII, (6.3) becomes

(6.16) Ψ 4 ( ξ ) = − 2 k 1 λ 1 ρ 0 ρ 1 + k 1 ρ 1 2 ( λ 2 + λ 3 ) − k 1 ρ 0 2 ( λ 2 − λ 3 ) + η 1 ρ 0 ρ 1 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 ( ξ + ϕ 0 ) + 2 ( λ 2 − λ 3 ) ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 δ 1 k 2 + δ 4 k 1 t and λ 1 2 + λ 2 2 − λ 3 2 = 0 .

Fourth:

(6.17) c = δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 − k 1 3 λ 3 2 + δ 1 k 2 + δ 4 k 1 , η 0 = k 1 ρ 1 ( λ 2 + λ 3 ) , η 1 = η 1 , k 1 = k 1 , k 2 = k 2 ,

λ 1 = 0 , λ 2 = λ 2 , λ 3 = λ 3 , ρ 0 = 0 , ρ 1 = ρ 1 .

According to Family 1, (6.3) becomes

(6.18) Ψ 1 ( ξ ) = k 1 ( λ 2 + λ 3 ) cot Z ( ξ ) 2 + η 1 ρ 1 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 2 + λ 3 λ 2 − λ 3 tanh λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 − k 1 3 λ 3 2 + δ 1 k 2 + δ 4 k 1 t , and λ 2 − λ 3 ≠ 0 . According to Family 6, (6.3) becomes

(6.19) Ψ 2 ( ξ ) = k 1 λ 2 cot Z ( ξ ) 2 + η 1 ρ 1 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = e 2 λ 2 ( ξ + ϕ 0 ) − 1 e 2 λ 2 ( ξ + ϕ 0 ) + 1 , 2 e λ 2 ( ξ + ϕ 0 ) e 2 λ 2 ( ξ + ϕ 0 ) + 1 ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 + δ 1 k 2 + δ 4 k 1 t .

Fifth:

(6.20) c = c , η 0 = η 0 , η 1 = η 1 , k 1 = k 1 , k 2 = k 2 , λ 1 = 2 ρ 0 λ 3 ρ 1 ρ 0 2 + ρ 1 2 , λ 2 = λ 3 ( ρ 0 2 − ρ 1 2 ) ρ 0 2 + ρ 1 2 , λ 3 = λ 3 , ρ 0 = ρ 0 , ρ 1 = ρ 1 .

According to Family 8, (6.3) becomes

(6.21) Ψ 1 ( ξ ) = η 0 + η 1 tan Z ( ξ ) 2 ρ 0 + ρ 1 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 ( ξ + ϕ 0 ) + 2 ( λ 2 − λ 3 ) ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − c t and λ 1 2 + λ 2 2 − λ 3 2 = 0 .

Sixth:

(6.22) c = δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 , η 1 = k 1 ρ 0 ( λ 1 2 + λ 2 2 − λ 2 2 ) ( 8 λ 1 2 − λ 2 2 + λ 3 2 ) + 3 η 0 λ 1 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) 9 λ 1 2 ( λ 2 + λ 3 ) ,

η 0 = η 0 , k 1 = k 1 , k 2 = k 2 , λ 1 = λ 1 , λ 2 = λ 2 , λ 3 = λ 3 , ρ 0 = ρ 0 , ρ 1 = 1 3 ρ 0 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) λ 1 ( λ 2 + λ 3 ) .

According to Family 1, (6.3) becomes

(6.23) Ψ 1 ( ξ ) = η 0 + k 1 ρ 0 ( λ 1 2 + λ 2 2 − λ 2 2 ) ( 8 λ 1 2 − λ 2 2 + λ 3 2 ) + 3 η 0 λ 1 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) 9 λ 1 2 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 ρ 0 + 1 3 ρ 0 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) λ 1 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 ,

tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 − λ 3 2 − λ 1 2 − λ 2 2 λ 2 − λ 3 tan λ 3 2 − λ 1 2 − λ 2 2 2 ( ξ + ϕ 0 ) ,

(6.24) Ψ 2 ( ξ ) = η 0 + k 1 ρ 0 ( λ 1 2 + λ 2 2 − λ 2 2 ) ( 8 λ 1 2 − λ 2 2 + λ 3 2 ) + 3 η 0 λ 1 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) 9 λ 1 2 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 ρ 0 + 1 3 ρ 0 ( 2 λ 1 2 − λ 2 2 + λ 3 2 ) λ 1 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 ,

tan 1 2 Z ( ξ ) = λ 1 λ 2 − λ 3 + λ 1 2 + λ 2 2 − λ 3 2 λ 2 − λ 3 tanh λ 1 2 + λ 2 2 − λ 3 2 2 ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 ( λ 1 2 + λ 2 2 − λ 3 2 ) + δ 1 k 2 + δ 4 k 1 t , λ 1 2 + λ 2 2 − λ 3 2 > 0 , and λ 2 − λ 3 ≠ 0 . According to Family 8, (6.3) becomes

(6.25) Ψ 3 ( ξ ) = η 0 + η 0 λ 1 λ 2 + λ 3 tan Z ( ξ ) 2 ρ 0 + ρ 0 λ 1 λ 2 + λ 3 tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = λ 1 ( ξ + ϕ 0 ) + 2 ( λ 2 − λ 3 ) ( ξ + ϕ 0 ) ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 δ 1 k 2 + δ 4 k 1 t , and λ 1 2 + λ 2 2 − λ 3 2 = 0 . According to Family 12, (6.3) becomes

(6.26) Ψ 4 ( ξ ) = η 0 − 1 9 k 1 λ 2 4 ρ 0 − 9 k 1 λ 2 2 λ 3 2 ρ 0 + 3 η 0 λ 2 2 λ 3 − 9 η 0 λ 3 3 λ 3 2 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 ρ 0 − 1 3 ρ 0 ( λ 2 2 − 3 λ 3 2 ) λ 3 ( λ 2 + λ 3 ) tan Z ( ξ ) 2 , tan 1 2 Z ( ξ ) = ( λ 2 + λ 3 ) e λ 2 ( ξ + ϕ 0 ) + 1 ( λ 2 − λ 3 ) e λ 2 ( ξ + ϕ 0 ) − 1 ,

so that ξ = k 1 x + k 2 y − δ 2 k 1 2 + δ 3 k 1 k 2 + δ 5 k 2 2 k 1 3 λ 2 2 + δ 1 k 2 + δ 4 k 1 t .

7 Conclusion

In this article, the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation was investigated, and the obtained soliton solutions by employing the HVDT were analyzed. In continuing, the modulation instability was applied to discuss the stability of earned solutions. It was quite visible that these novel schemes have plenty of family solutions containing rational exponential, hyperbolic, and periodic functions with choosing uncommon amounts. Also, the semi-inverse variational principle was used for the gHSI equation. Six major cases containing the kinky dark solitary, dark solitary, periodic, bright solitary, dark soliton, and another type dark soliton wave solutions were studied from six different analyses. Also, the rational tan ( Z ( ξ ) ) scheme on the generalized nonlinear wave equation was investigated, and four sets solutions were obtained. Moreover, under certain requirements the asymptotic behavior of the linerization solution was analyzed to prove that the modulation instability is stable for some points. The applicability and effectiveness of the acquired solutions were presented through the numerical results in the form of 3D and 2D graphs. The obtained results show that the proposed HVDT was simple, straightforward and productive, which was expected to bring a light to the study of the traveling wave theory. For the aforementioned results, we use Maple software to obtain the three-dimensional plots and density plots. There is no doubt that the significance of this study, at the same time, nonlinear partial differential equations are also worthy of further research and exploration. After that, we will also devote ourselves to new research work, and sincerely hope that we and other researchers can explore other interesting results. This research sets the foundation for future investigations in physics phenomena and experiments on even more complex systems that do not follow straightforward rules, and how these can be used in real-life situations.

Acknowledgments

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-51).

Funding information: This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-51).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Hirota R, Satsuma J. N-soliton solutions of model equations for shallow water waves. J Phys Soc Japan. 1976;40(2):611–12. 10.1143/JPSJ.40.611Search in Google Scholar

[2] Hietarinta J. Introduction to the Hirota bilinear method, in Integrability of nonlinear systems, Kosmann-Schwarzbach Y, Grammaticos B, and Tamizhmani KM, Eds., Berlin, Heidelberg: Springer; 1997, pp. 95–103. 10.1007/BFb0113694Search in Google Scholar

[3] Ma WX, Li J, Khalique CM. A study on lump solutions to a generalized Hirota–Satsuma–Ito equation in (2+1)-dimensions. Complexity. 2018;2018:9059858. 10.1155/2018/9059858Search in Google Scholar

[4] Dehghan M, Manafian J, Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Num Meth Partial Diff Eq J. 2010;26:448–79. 10.1002/num.20460Search in Google Scholar

[5] Dehghan M, Manafian J. The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method. Z Naturforsch A. 2009;64a:420–30. 10.1515/zna-2009-7-803Search in Google Scholar

[6] Manafian J, Lakestani M. Application of tan(ϕ∕2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity. Optik. 2016;127:2040–54. 10.1016/j.ijleo.2015.11.078Search in Google Scholar

[7] Fei R, Guo Y, Li J, Hu B, Yang L. An improved BPNN method based on probability density for indoor location. IEICE Trans Inform Sys. 2023;E106.D(5):773–85. 10.1587/transinf.2022DLP0073Search in Google Scholar

[8] Chen C, Han D, Chang CC. MPCCT: Multimodal vision-language learning paradigm with context-based compact transformer. Pattern Recognition. 2024;147:110084. 10.1016/j.patcog.2023.110084Search in Google Scholar

[9] Chen C, Han D, Shen X. CLVIN: Complete language-vision interaction network for visual question answering. Knowledge-Based Sys. 2023;275:110706. 10.1016/j.knosys.2023.110706Search in Google Scholar

[10] Shi S, Han D, Cui M. A multimodal hybrid parallel network intrusion detection model. Connection Sci. 2023;35:2227780. 10.1080/09540091.2023.2227780Search in Google Scholar

[11] Wanh H, Han D, Cui M, Chen C. NAS-YOLOX: a SAR ship detection using neural architecture search and multi-scale attention. Connection Sci. 2023;35:1–32. 10.1080/09540091.2023.2257399Search in Google Scholar

[12] Han D, Zhou HX, Weng TH, Wu Z, Han B, Li KC, Pathan ASK. LMCA: a lightweight anomaly network traffic detection model integrating adjusted mobilenet and coordinate attention mechanism for IoT. Telecommun Sys. 2023;84:549–64. 10.1007/s11235-023-01059-5Search in Google Scholar

[13] Dou J, Liu J, Wang Y, Zhi L, Shen J, Wang G. Surface activity, wetting, and aggregation of a perfluoropolyether quaternary ammonium salt surfactant with a hydroxyethyl group. Molecules. 2023;28(20):7151. 10.3390/molecules28207151Search in Google Scholar PubMed PubMed Central

[14] Chen B, Zhang X, Liu Y, Ma X, Wang X, Cao X, et al. Magnetic porous carbons derived from iron-based metal-organic framework loaded with glucose for effective extraction of synthetic organic dyes in drinks. J Chromatography A. 2022;1661:462716. 10.1016/j.chroma.2021.462716Search in Google Scholar PubMed

[15] Chen X, Yang P, Wang M, Hu F, Xu J. Output voltage drop and input current ripple suppression for the pulse load power supply using virtual multiple quasi-Notch-filters impedance. IEEE Trans Power Elec. 2023;38(8):9552–65. 10.1109/TPEL.2023.3275304Search in Google Scholar

[16] Shi XL, Du M, Sun B, Liu S, Jiang L, Hu Q, et al. A novel fiber-supported super base catalyst in the spinning basketreactor for cleaner chemical fixation of CO_2 with 2-aminobenzonitriles in water. Chem Eng J. 2022;430(3):1332045. 10.1016/j.cej.2021.133204Search in Google Scholar

[17] Zheng Y, Wang Y, Liu J. Research on structure optimization and motion characteristics of wearable medical robotics based on improved particle swarm optimization algorithm. Future Gener Comput Sys. 2022;129:187–98. 10.1016/j.future.2021.11.021Search in Google Scholar

[18] Zhang G, Li W, Yu M, Huang H, Wang Y, Han Z. Electric-field-driven printed 3D highly ordered microstructure with cell feature size promotes the maturation of engineered cardiact issues. Adv Sci. 2023;10(11):2206264. 10.1002/advs.202206264Search in Google Scholar PubMed PubMed Central

[19] Li W, Bu J, Li X, Peng H, Niu Y, Zhang Y. A survey of DeFi security: challenges and opportunities. J KingSaud Univ Comput Inform Sci. 2022;34(10):10378–404. 10.1016/j.jksuci.2022.10.028Search in Google Scholar

[20] Hong J, Gui L, Cao J. Analysis and experimental verification of the tangential force effect on electromagnetic vibration of PM motor. IEEE Trans Energy Conver. 2023;38(3):1893–1902. 10.1109/TEC.2023.3241082Search in Google Scholar

[21] Liao L, Guo Z, Gao Q, Wang Y, Yu F, Zhao Q, et al. Color image recovery using generalized matrix completion over higher-order finite dimensional algebra. Axioms. 2023;12(10):954. 10.3390/axioms12100954Search in Google Scholar

[22] Zhang X, Feng Z, Zhang X. On reachable set problem for impulse switched singular systems with mixed delays. IET Control Theo Appl. 2023;17(5):628–38. 10.1049/cth2.12390Search in Google Scholar

[23] He X, Xiong Z, Lei C, Shen Z, Ni A, Xie Y, et al. Excellent microwave absorption performance of LaFeO_3/Fe_3O_4/C perovskite composites with optimized structure and impedance matching. Carbon. 2023;213:118200. 10.1016/j.carbon.2023.118200Search in Google Scholar

[24] Meng S, Meng F, Chi H, Chen H, Pang A. A robust observer based on the nonlinear descriptor systems application to estimate the state of charge of lithium-ion batteries. J Frankl Inst. 2023;360(16):11397–413. 10.1016/j.jfranklin.2023.08.037Search in Google Scholar

[25] Chen D, Zhao J, Qin SR. SVM strategy and analysis of a three-phase quasi-Z-source inverter with high voltage transmission ratio. Sci China Tech Sci. 2023;66:2996–3010. 10.1007/s11431-022-2394-4Search in Google Scholar

[26] Zhang G, Song D, Jiang J, Li W, Huang H, Yu Z, et al. Electrically assisted continuous vat photo polymerization 3D printing for fabricating high-performance ordered graphene/polymer composites. Composites Part B: Eng. 2023;250:110449. 10.1016/j.compositesb.2022.110449Search in Google Scholar

[27] Wang J, Xu Z, Zheng X, Liu Z. A fuzzy logic path planning algorithm based on geometric landmarks and kinetic constraints. Inform Tech Contorol 2022;51(3):499–514. 10.5755/j01.itc.51.3.30016. Search in Google Scholar

[28] Seadawy AR, Manafian J. New soliton solution to the longitudinal wave equationin a magneto-electro-elastic circular rod. Results Phys. 2018;8:1158–67. 10.1016/j.rinp.2018.01.062Search in Google Scholar

[29] Feng F, Manafian J, Ilhan OA, Rao AM, Agadi AH. Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP-BBM equation. Int J Modern Phys B. 2021;35(27):2150275. Search in Google Scholar

[30] Liu W, Bai X, Yang H, Bao R, Liu J. Tendon driven bistable origami flexible gripper for high-speed adaptive grasping. IEEE Rob Autom Let. 2024;9(6):5417–24.10.1109/LRA.2024.3389413Search in Google Scholar

[31] Zhou X, Ilhan OA, Zhou F, Sutarto S, Manafian J. Lump and interaction solutions to the (3+1)-dimensional variable-coefficient nonlinear wave equation with multidimensional binary bell polynomials. J Function Spaces. 2021;2021:4550582. 10.1155/2021/4550582Search in Google Scholar

[32] Li G, Manafian J, Seyyedi M, Sivaraman R, Zeynalli SM. Periodic, Cross-Kink, and interaction between stripe and periodic wave solutions for generalized Hietarinta equation: Prospects for applications in environmental engineering. Adv Math Phys. 2022;2022:6445482. 10.1155/2022/6445482Search in Google Scholar

[33] Li J, Manafian J, Wardhana A, Othman AJ, Husein I, Al-Thamir M, et al. N-lump to the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation. Complexity. 2022;2022:4383100. 10.1155/2022/4383100Search in Google Scholar

[34] Olalekan OS, Abbyssinia M. Cash transfers and quality of life among households in the Eastern Cape, South Africa: Application of a generalized linear model. Int J Innovative Research Sci Stud. 2023;6(4):804–12. 10.53894/ijirss.v6i4.2006Search in Google Scholar

[35] Bouchaala F, Ali MY, Matsushima J. Estimation of seismic attenuation in carbonaterocks using three different methods: Application on VSP data from Abu Dhabi oil field. J Appl Geophys. 2016;129:79–91. 10.1016/j.jappgeo.2016.03.014Search in Google Scholar

[36] Ali F, Zahid M, Hou Y, Manafian J, Rana MA, Hajar A. A theoretical study of reverse roll coating for a non-isothermal third-grade fluid under lubrication approximation theory. Math Prob Eng. 2022;2022:5029132. 10.1155/2022/5029132Search in Google Scholar

[37] Mallampalli P, Janardhanan S, Karottu KV, Ommi G. Numerical study onmaneuvering a container ship in shallow water waves. Int JInnovative Research Sci Stud. 2023;6(4):913–26. 10.53894/ijirss.v6i4.2094Search in Google Scholar

[38] Krishna G, Singh R, Gehlot A, Singh P, Rana S, Akram SV, et al. An imperative role of studying existing battery datasets and algorithms for battery management system. Rev Comput Eng Research. 2023;10(2):28–39. 10.18488/76.v10i2.3413Search in Google Scholar

[39] Bouchaala F, Guennou C. Estimation of viscoelastic attenuation of real seismic databy use of ray tracing software: Application to the detection of gas hydrates and free gas. Comptes Rendus Giosci. 2012;344(2):57–662012. 10.1016/j.crte.2011.12.006Search in Google Scholar

[40] Ghiasi M, Niknam T, Wang Z, Mehrandezh M, Dehghani M, Ghadimi N. A comprehensive review of cyber-attacks and defense mechanisms for improving security in smart grid energy systems: Past, present and future. Elec Power Sys Res. 2023;215:108975. 10.1016/j.epsr.2022.108975Search in Google Scholar

[41] Chang L, Wu Z, Ghadimi N. A new biomass-based hybrid energy system integrated with a flue gas condensation process and energy storage option: an effort to mitigate environmental hazards. Process Safety Environ Protect. 2023;177:959–75. 10.1016/j.psep.2023.07.045Search in Google Scholar

[42] Huang H, Zhang G, Li W, Yu Z, Peng Z, Wang F, et al. The theoretical model and verifcation of electric-feld-driven jet 3D printing for large-height and conformal micro/nano-scale parts. Virtual Phys Prototyp. 2023;18(1):e2140440. 10.1080/17452759.2022.2140440Search in Google Scholar

[43] Jiang L. A fast and accurate circle detection algorithm based on random sampling, Future Gener Comput Syst. 2021;123:245256. 10.1016/j.future.2021.05.010Search in Google Scholar

[44] Wang J, Xu Z, Zengh X, Liu Z. Fuzzy event-triggered sliding mode depth control of unmanned under-water vehicles. Ocean Eng. 2022;266(1):112725. 10.1016/j.oceaneng.2022.112725Search in Google Scholar

[45] Zhao Y, Wang J, Cao G, Yuan Y, Yao X, Qi L. Intelligent control of multilegged robot smooth motion: a review. IEEE Access 2022;11:86645–85. 10.1109/ACCESS.2023.3304992Search in Google Scholar

[46] Lin R, Liu Z, Han P, Lin R, Lu Y, Cao H, et al. A machine learning study on super lattice electron blocking layer design for AlGaN deep ultraviolet light-emitting diodes using the stacked XGBoost/Light GBM algorithm. J Mater Chem C. 2022;10:17602–10. 10.1039/D2TC02335KSearch in Google Scholar

[47] Lin R, Han P, Wang Y, Lin R, Lu Y, Liu Z, et al. Low resistance asymmetric III-nitride tunnel junctions designed by machine learning. Nanomaterials. 2021;11(10):2466. 10.3390/nano11102466Search in Google Scholar PubMed PubMed Central

[48] Lin R, Han P, Wang Y, Lin R, Lu Y, Liu Z, et al. BAlN alloy for enhanced two-dimensional electron gas characteristics of GaN/AlGaN hetero structures. J Phys D Appl Phys. 2020;53:48LT01. 10.1088/1361-6463/aba4d5Search in Google Scholar

[49] Liu Z, Lu Y, Wang Y, Lin R, Xiong C, Li X. Polarization modulation at last quantum barrier for high efficiency AlGaN-based UV LED. IEEE Photonics J. 2022;14(1):8210208. 10.1109/JPHOT.2021.3139265Search in Google Scholar

[50] Tang X, Lu Y, Lin R, Liao CH, Zhao Y, Li KH, et al. Flexible self-powered DUV photodetectors with high responsivity utilizing Ga2O3/NiO heterostructure on buffered Hastelloy substrates. Appl Phys Lett. 2023;122:121101. 10.1063/5.0146030Search in Google Scholar

[51] Chen D, Zhao T, Han L, Feng Z. Single-stage multi-input Buck type high-frequency linkas inverters with series and simultaneous power supply. IEEE Trans Power Elec. 2022;37(6):7411–21. 10.1109/TPEL.2021.3139646Search in Google Scholar

[52] Chen D, Zhao T, Xu S. Single-stage multi-input buck type high-frequency linkas inverters with multiwinding and time-sharing power supply. IEEE Trans Power Elec. 2022;37(10):12763–73. 10.1109/TPEL.2022.3176377Search in Google Scholar

[53] Mehrpooya M, Ghadimi N, Marefati M, Ghorbanian SA. Numerical investigation of a new combined energy system includes parabolic dish solar collector, stirling engine and thermoelectric device. Int J Energy Res. 2021;45(11):16436–55. 10.1002/er.6891Search in Google Scholar

[54] Jiang W, Wang X, Huang H, Zhang D, Ghadimi N. Optimal economic scheduling of microgrids considering renewable energy sources basedon energy hub model using demand response and improved water wave optimization algorithm. J Energy Storage. 2022;55(1):105311. 10.1016/j.est.2022.105311Search in Google Scholar

[55] Han E, Ghadimi N. Model identification of proton-exchange membrane fuel cells based on a hybrid convolutional neural network and extreme learning machine optimized by improved honey badger algorithm. Sustain Energy Tech Asses. 2022;52:102005. 10.1016/j.seta.2022.102005Search in Google Scholar

[56] Zhang L, Zhang J, Gao W, Bai F, Li N, Ghadimi N. A deep learning outline aimed at prompt skin cancer detection utilizing gated recurrent unit networks and improved orca predation algorithm. Biomed Signal Proces Control. 2024;90:105858. 10.1016/j.bspc.2023.105858Search in Google Scholar

[57] Haozhi L, Ghadimi N. Hybrid convolutional neural network and flexible dwarf mongoose optimization algorithm for strong kidney stone diagnosis. Biomed Signal Proces Control. 2024;91:106024. 10.1016/j.bspc.2024.106024Search in Google Scholar

[58] Yu D, Zhang T, He G, Nojavan S, Jermsittiparsert K, Ghadimi N. Energy management of wind-PV-storage-grid based large electricity consumer using robust optimization technique. J Energy Storage. 2020;27:101054. 10.1016/j.est.2019.101054Search in Google Scholar

[59] Zhang M, Lyu H, Bian H, Ghadimi N. Improved chaos grasshopper optimizer and its application to HRES techno-economic evaluation. Heliyon. 2024;10(2):e24315. 10.1016/j.heliyon.2024.e24315Search in Google Scholar PubMed PubMed Central

[60] Yuan Z, Wang W, Wang H, Ghadimi N. Probabilistic decomposition-based security constrained transmission expansion planning incorporating distributed series reactor. IET Gener Trans Distrib. 2020;14(17):3478–87. 10.1049/iet-gtd.2019.1625Search in Google Scholar

[61] Mir M, Shafieezadeh M, Heidari MA, Ghadimi N. Application of hybrid forecast engine based intelligent algorithm and feature selection for wind signal prediction. Evolv Sys. 2020;11(4):559–73. 10.1007/s12530-019-09271-ySearch in Google Scholar

[62] Zhang J, Khayatnezhad M, Ghadimi N. Optimal model evaluation ofthe proton exchange membrane fuel cells based on deep learning and modified african vulture optimization algorithm. Energy Sources A. 2022;44:287–305. 10.1080/15567036.2022.2043956Search in Google Scholar

[63] Kai Y, Chen S, Zhang K, Yin Z. Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation. Waves Random Complex. Media. 2022. 10.1080/17455030.2022.2044541. Search in Google Scholar

[64] Wang L, Zou T, Cai K, Liu Y. Rolling bearing fault diagnosis method based on improved residual shrinkage network. J Brazilian Soc Mech Sci Eng. 2024;46(3):172.10.1007/s40430-024-04729-wSearch in Google Scholar

[65] Kai Y, Yin Z. On the Gaussian traveling wave solution to a special kind of Schrödinger equationwith logarithmic nonlinearity. Modern Phys Let B. 2022;36(02):2150543. 10.1142/S0217984921505436Search in Google Scholar

[66] Zhang X, Hu Z, Liu Y. Fast generation of GHZ-like states using collective-spin XYZ model. Phys Rev Let. 2024;132(11):113402.10.1103/PhysRevLett.132.113402Search in Google Scholar PubMed

[67] Zhu C, Al-Dossari M, Rezapour S, Shateyi S, Gunay B. Analytical optical solutions to the nonlinear Zakharov system via logarithmic transformation. Results Phys. 2024;56:107298.10.1016/j.rinp.2023.107298Search in Google Scholar

[68] Zhu C, Al-Dossari M, Rezapour S, Gunay B. On the exact soliton solutions and different wave structures to the (2+1) dimensional Chaffee–Infante equation. Results in Phys. 2024;57:107431.10.1016/j.rinp.2024.107431Search in Google Scholar

[69] Zhu C, Al-Dossari M, Rezapour S, Alsallami SAM, Gunay B. Bifurcations, chaotic behavior, and optical solutions for the complex Ginzburg–Landau equation. Results Phys. 2024;59:107601.10.1016/j.rinp.2024.107601Search in Google Scholar

[70] Zhu H, Lu Y, Cai L. Wavelength-shift-free racetrack resonator hybrided with phase change material for photonic in-memory computing. Opt Express. 2023;31(12):18840–50. 10.1364/OE.489525Search in Google Scholar PubMed

[71] Gao N, Liu J, Deng J, Chen D, Huang Q, Pan G. Design and performance of ultra-broadband composite meta-absorber in the 200Hz–20kHz range. J Sound Vibration. 2024;574:118229. 10.1016/j.jsv.2023.118229Search in Google Scholar

[72] Kumar S, Niwas M, Osman MS, Abdou MA. Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations. Communicat Theo Phys. 2021;73(10):105007. 10.1088/1572-9494/ac11eeSearch in Google Scholar

[73] Niwas M, Kumar S. Multi-peakons, lumps, and other solitons solutions for the (2+1)-dimensional generalized Benjamin? Ono equation: an inverse (G’/G)-expansion method and real-world applications. Nonlinear Dyn. 2023;111(24):22499–512. 10.1007/s11071-023-09023-3Search in Google Scholar

[74] Kumar S, Hamid I, Abdou MA. Dynamic frameworks of optical soliton solutions and soliton-like formations to Schrödinger–Hirota equation with parabolic law non-linearity using a highly efficient approach. Opt Quant Elec. 2023;55(14):1261. 10.1007/s11082-023-05461-wSearch in Google Scholar

[75] Gu Y, Aminakbari N. Bernoulli (G’/G)-expansion method fornonlinear Schrödinger equation with third-order dispersion. Modern Phys Let B. 2022;36(11):2250028. 10.1142/S0217984922500282Search in Google Scholar

[76] Gu Y, Chen B, Ye F, Aminakbari N. Soliton solutions of nonlinear Schrödinger equation with the variable coefficients under the influence of Woods–Saxon potential. Results Phys. 2022;42:105979. 10.1016/j.rinp.2022.105979Search in Google Scholar

[77] Gu Y, Aminakbari N. New optical soliton solutions for the variable coefcients nonlinear Schrödinger equation. Opt Quant Elec. 2022;54:255. 10.1007/s11082-022-03645-4Search in Google Scholar

[78] Chen-Wang, Dai HP, Wei-Tan, Feng YX. Some new nonlinear wave solutions and dynamical behavior of the (3+1)-dimensional Ito equation. Results Phys. 2024;56:107250. 10.1016/j.rinp.2023.107250Search in Google Scholar

[79] Inc M, Az-Zóbi EA, Jhangeer A, Rezazadeh H, Ali MN, Kaabar MKA. New soliton solutions for the higher-dimensional non-local Ito equation. Nonlinear Eng. 2021;10:374?84. 10.1515/nleng-2021-0029Search in Google Scholar

[80] Az-Zóbi EA, Alzoubi WA, Akinyemi L, Senol M, Masaedeh BS. A variety of wave amplitudes for the conformable fractional (2+1)-dimensional Ito equation. Modern Phys Let B. 2021;35(15):2150254. 10.1142/S0217984921502547Search in Google Scholar

[81] Feng B, Manafian J, Ilhan OA, Rao AM, Agadi AH. Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation. Int J Modern Phys B. 2021;35(27):2150275. 10.1142/S0217979221502751Search in Google Scholar

Received: 2023-12-30

Revised: 2024-04-05

Accepted: 2024-04-09

Published Online: 2024-06-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0027

Keywords for this article

traveling wave solutions; He’s variational direct technique; The rational tan(Π(ξ)) scheme; generalized Hirota–Satsuma–Ito equation; modulation instability

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