Dynamical Behaviors of Lumpoff and Rogue Wave Solutions for Nonlocal Gardner Equation

Manafian, Jalil; Shahabi Takami, Seyedeh Fatemeh; Eslami, Baharak; Malmir, Somaye; Sood, Anju; Sabherwal, Amarpreet Kaur

doi:https://doi.org/10.1155/2022/9360263

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 9360263 | https://doi.org/10.1155/2022/9360263

Dynamical Behaviors of Lumpoff and Rogue Wave Solutions for Nonlocal Gardner Equation

Jalil Manafian,^1,2Seyedeh Fatemeh Shahabi Takami,³Baharak Eslami,⁴Somaye Malmir,⁴Anju Sood,⁵and Amarpreet Kaur Sabherwal^6,7

Academic Editor: Abdullahi Yusuf

Received16 Jun 2022

Accepted21 Sept 2022

Published14 Oct 2022

Abstract

In this paper, we got a novel kind of rogue waves with the predictability of (2 + 1)-dimensional nonlocal Gardner equation with the aid of Maple according to the Hirota bilinear model. We first construct a general quadratic function to derive the general lump solution for the mentioned equation. At the same time, the lumpoff solutions are demonstrated with more free autocephalous parameters, in which the lump solution is localized in all directions in space. Moreover, when the lump solution is cut by twin-solitons, special rogue waves are also introduced. Based on the data available in the literature, the resulting soliton solutions are innovative, developed, distinctive, and significant and can be applied to more complex phenomena, and they are immensely active for nonlinear models of classical and fractional-order type. To examine the dynamic behavior of the waves, contour 3D and 2D plots of several obtained findings are sketched by assigning specific values to the parameters. Furthermore, we obtain new sufficient solutions containing cross-kink, periodic-kink, multi-waves, and solitary wave solutions. It is worth noting that the emerging time and place of the rogue waves depend on the moving path of the lump solution.

1. Introduction

Nonlinear physical structures have been linked with nonlinear equations concerning several disciplines, like thermodynamics, mechanics, fluid dynamics, wave propagation, plasma physics, fluid flow, nonlinear networks, optical fibers, and soil consolidations, to develop vital phenomenon and implementations ([1–4])[5, 6]. The role of nonlinearity in waves is quite significant, mostly throughout non-linear sciences; research development towards exact solutions of partial differential nonlinear equations has always been a major endeavor for the past couple of years. All the while, several powerful, efficient, and reliable methods for attempting to seek exact analytical solutions to traveling waves have been established: for example, the -soliton solution [7], a fractional-order multiple-model type-3 fuzzy control [8], a calculation methodology for geometrical characteristics [9], a combination of group method of data handling and computational fluid dynamics [10], the truss optimization with metaheuristic algorithms [11], the extended generalized Darboux method [12], the Lax pair technique [13], intermolecular interactions to the modern acid-base theory [14], the deep learning for Feynman’s path integral [15], the Darboux–Bäcklund technique [16], the Hirota bilinear technique [17], the multiple exp-function method [18], optimal structure design [19], the experimental study on circular steel [20, 21], an influence of seismic orientation on the statistical distribution [22], effects of actual loading waveforms on the fatigue behaviors [23], and so on [24–27]. Hirota bilinear method is an efficient instrument to construct exact solutions of NLEEs, and there exist a lot of completely integrable equations that are investigated in this procedure, for example, the generalized bilinear equations [28], the Kadomtsev–Petviashvili (KP)–Benjamin–Bona–Mahony equation [29], the optimal Galerkin-homotopy asymptotic method [30], the (2 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation [31], the KP equation [32], the B-type KP equation [33], the optical interactions within magnetic layered structures [34], the (2 + 1)-dimensional breaking soliton equation [35], and the bidirectional Sawada–Kotera equation [36].

As we all know, the (1 + 1)-dimensional Gardner equation is an important equation and is a mixed version of the well-known Korteweg-De Vries (KdV) equation and the modified KdV (mKdV) equation [37, 38], and its properties are widely used in the field of mathematical physics, fluid dynamics, hydrodynamics, quantum mechanics, and nonlinear optics which is presented as follows:the Gardner equation is completely integrable. The diverse kinds of exact solutions of Eq. (1) were studied by several techniques where interested researchers have worked on it, for example, using the inverse spectral transform method and constructing explicitly the exact solutions by the -dressing method [39], utilizing the Casorati and Grammian determinant solutions to the (2 + 1)-dimensional Gardner equation [40], the tan h method with kink solutions [41], and the nonlinearization technique of Lax pairs to find the integrable decompositions for the (2 + 1)-dimensional Gardner [42]. Konopelchenko and Dubrovsky suggested the integro-differential form for the (2 + 1)-dimensional (D) Gardner equation as below:where , and are nonzero arbitrary values, and . Also, the operator . All cases for parameters , and have been investigated in the valuable works such as nonlocal symmetry and exact solutions [43], N-soliton solution and soliton resonances [44], and lie symmetries and invariant solutions [45] of (2 + 1)-dimensional Gardner equation.

Different kinds of probes were studied by proficient scholars in which many of them can be indicated, for instance, the generalized higher-order NLSE [46], the double dispersive equation in Murnaghan’s rod [47], the fractional (2 + 1)-D Boussinesq dynamical model [48], the conformable time-fractional Wu-Zhang system [49], and the time-fractional model of Lassa hemorrhagic fever spreading [50]. In the valuable works, the important periodic and breather solutions to the KP-BBM equation [51] and generalized Bogoyavlensky–Konopelchenko equation [52] with the help of the Hirota technique were obtained. Authors of Reference [53] investigated the exact soliton solutions of a nonlinear Schrödinger equation including Kudryashov’s sextic power-law nonlinearity by two efficient analytical methods. Cinar et al.[54] studied the soliton solutions for the perturbed Fokas–Lenells equation which has a vital role in optics by using Sardar subequation method. In Reference [55], the analytical solutions of simplified modified Camassa–Holm equations with various derivative operators, namely, conformable and M-truncated derivatives were obtained. Also, the analytical solutions of (2 + 1) dimensional Heisenberg ferromagnetic spin equation which describes the nonlinear dynamics of the ferromagnetic materials by using the extended rational sine-cosine and sin h-cos h methods were reached [56]. The dark, singular, combined dark-singular soliton, singular periodic wave, and rational function solutions for the (2 + 1)-dimensional Biswas–Milovic equation for the description of pulse propagation in optical fiber were constructed [57]. Some authors worked on the nonlinear PDEs including the nonlinear directional couplers in nonlinear optics [58]; an improved perturbed Schrödinger equation with Kerr law nonlinearity in nonlinear optics [59]; the fractional analysis of fusion and fission process in plasma physics [60]; traveling, periodic, quasiperiodic, and chaotic structures of perturbed Fokas–Lenells model [61]; and dynamics of longitudinal bud equation among a magneto-electro-elastic round rod [62]. In Reference [63], the (G’/G)-expansion and the Ricatti transformation method were used for solving the (2 + 1)-dimensional Boussinesq equation. The analytical solutions to integer and fractional-order to KdV and Fornberg–Whitham equations were obtained by using the Laplace transforms, Sumudu transforms, and Elzaki transforms [64]. Scholars of Reference [65] studied Yang transform homotopy perturbation method to nonlinear fractional order KdV and Burger equation with exponential-decay kernel and provided formulae for the Yang transform of Caputo–Fabrizio fractional-order derivatives.

The lump, lumpoff, rogue wave, lump-periodic, periodic wave, cross-kink, multiwave, and solitary wave solutions of (2) will be investigated and analyzed. According to the Hirota bilinear model (2), the general periodic wave solution is concluded in Section 2 and also the new periodic solutions which can be arisen with more than five classes. Finally, conclusions are given is Section 10.

2. General Lump Solutions

According to Reference [66], the Hirota bilinear form of the Gardner equation is as follows:under the condition of transformation:

Here, the -operator is defined as follows:

With the help of transformation (4), the general lump solutions of Eq. (3) can be given. We introduce ansatz and as follows:with,in which are the symmetric matrixes and and are the positive constants. Then,or we can write the custom form as below:

Substituting relations (10) and (11) into Eq. (3) and collecting all the coefficients of , we can get twenty determining equations. Solving these resulting equations, we have the following cases:

Case 1. Under the transformation , the corresponding general lump solutions is read as follows:in which,

Case 2. Under the transformation , the corresponding general lump solutions read as follows (see Figure 1):in which

3. Lumpoff Solution

In the section, the lumpoff solutions of the Gardner equation, a solution with the interaction between lump wave and stripe soliton wave, are investigated. On the basis of and of the general lump solutions, the lumpoff solutions can be written as follows:

We can easily detect that it consists of two parts: lump wave part and solitary wave part. It is worth noting that the solitary wave part is dominant when . Otherwise, the lump solution only arises when . To summarize, the existence of the soliton is based on the existence of the lump. Then, substituting (16) and (17) into Eq. (3), the selected constants of and are obtained as follows:

Case 3. Under the transformation , the corresponding general lumpoff solutions read as follows:in which,Figure 2 presents the overtaking interactions between lump wave and stripe soliton wave containing 3D plot, density plot, and 2D plot, respectively, .

4. Rogue Waves with Predictability

In the section, we will explore the rogue waves with predictability for the (2 + 1)-dimensional Gardner equation. The so-called predictability is that the emerging time and place of the rogue waves can be described by some particular expressions. Let us consider a more general ansatz as follows:

Substituting (21) and (22) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following case:

Case 4. Under the transformation , the corresponding special rogue solution is read as follows:in which,Actually, the hyperbolic function cos h part that predominates in and led to the lump waves which are always disappeared. Figure 3 presents the overtaking interactions between lump wave and stripe soliton wave containing 3D plot, density plot, and 2D plot, respectively, .

5. Lump-Periodic Solutions

In the section, we will explore the lump-periodic solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:

Substituting (26) and (27) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following case:

Case 5. Under the transformation , the corresponding special lump-periodic solution can be read as follows:in which,Figure 4 presents the overtaking interactions between lump wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

6. Double Breather Solutions

In the section, we will explore the breather solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:

Substituting (31) and (32) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following case:

Case 6. Under the transformation , the corresponding special breather solution is read as follows:in which,Figure 5 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

Case 7. Under the transformation , the corresponding special breather solution is read as follows:in which,Figure 6 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

Case 8. Under the transformation , the corresponding special breather solution is read as follows:in which,Figure 7 presents the overtaking interactions between stripe wave and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

7. Periodic Cross-Kink Solutions

In the section, we will explore the periodic cross-kink solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:

Substituting (42) and (43) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following cases:

Case 9. Under the transformation , the corresponding special periodic cross-kink solution is read as follows:in which,

Case 10. .in which,

Case 11. in which,

Case 12. in which,

Case 13. in which,

Case 14. in which,Figure 8 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

Case 15. in which,Figure 9 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

8. Periodic Wave Solutions

In the section, we will explore the periodic wave solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:

Substituting (59) and (60) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following cases:

Case 16. Under the transformation , the corresponding special periodic wave solution is read as follows:in which,

Case 17. in which,

Case 18. in which,

Case 19. in which,

Case 20. in which,

Case 21. in which,Figure 10 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

Case 22. in which,Figure 11 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, density plot, and 2D plot, respectively, .

9. Multiwave Solutions

In the section, we will explore the multiwave solutions for the (2 + 1)-dimensional Gardner equation. Let us consider a more general ansatz as follows:

Substituting (76) and (77) into Eq. (3) and collecting all relevant coefficients of , and , a series of equations have been obtained. Solving these equations, we have the following cases:

Case 23. Under the transformation , the corresponding special multiwave solution is read as follows:in which,Figure 12 presents the overtaking interactions between two stripe waves and periodic wave containing 3D plot, contour plot, and 2D plot, respectively, .

Case 24. Under the transformation , the corresponding special multiwave solution is read as follows:in which,Figure 13 presents the overtaking interactions between two stripe waves and periodic wave containing density plot and 2D plot, respectively, .

9.1. Solitary Wave Solutions

Here, we will consider the solitary wave solution by selecting the below function in which (1) has been taken as follows:

Plugging (84) and (85) into Eq. (3) and collecting all relevant coefficients of and , a series of equations have been obtained. Solving these equations, we have the following solutions:

9.1.1. First Solution

Under the transformation , the corresponding special solitary wave solution is read as follows:in which,

9.1.2. Second Solution

Under the transformation , the corresponding special solitary wave solution is read as follows:in which,

10. Conclusion

In this paper, the lump solutions, lumpoff solutions, and rogue waves with predictability of the (2 + 1)-dimensional Gardner equation are investigated with more arbitrary autocephalous parameters. It is not hard to see that the general lump solution is an algebraically localized wave decayed in all space directions and exists in all time. We constructed a general quadratic function because of deriving the general lump solution for this equation. Hence, the lumpoff solutions are shown with more free autocephalous parameters, in which the lump solution localized in all directions in space. Moreover, when the lump solution is cut by twin-solitons, the special rogue waves are also introduced. Furthermore, we obtain new sufficient solutions containing cross-kink, periodic-kink, multiwaves, and solitary wave solutions. From the acquired results, it can be concluded that the procedures followed in this analysis can be implemented in a simple and straightforward manner to create new exact solutions of a lot of other nonlinear partial differential equations in terms of Hirota operator.

Data Availability

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

Conflicts of Interest

The author declares that he has no conflicts of interest.

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Copyright © 2022 Jalil Manafian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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