Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method

1 Introduction

Nonlinear partial differential equations (NLPDEs) are a fundamental mathematical tool used in many disciplines, including physics, science, and engineering, where they are used to describe and understand a wide range of physical processes. Numerous analytical and numerical techniques have been used to study NLPDEs. In this context, the Hirota bilinear method [1–5], the Backlund transformation method [6–8], the Darboux transformation technique [9,10], the exp-function method [11], the inverse scattering method [12], Galerkin’s technique [13], the unified method (UM) and its generalized form (GUM) [14,15], the polynomial function method [16,17], the modified G ′ G -expansion [18], the modified Kudryashov expansion technique [19], the Lie-symmetry method [20], the generalized exponential rational function method [21], the new modified Laplace-decomposition method [22], and other techniques are included in the broad range of approaches to study NLPDEs. However, those approaches are frequently heuristic, but show remarkable potential in producing single wave solutions for a wide variety of NLPDEs [23].

Conversely, coastal ecosystems (which are areas where land and sea meet) are dynamic and delicate places that are constantly influenced by the frequent shifting actions of ocean waves [24,25]. The thorough study of these waves is crucial for the proper management and preservation of coastal environments and infrastructure, as they have a considerable impact on coastal stability, sediment transport, and the overall geomorphology of these locations. Advances in the research of coastal dynamics have made it possible to better understand how these waves interact with coastal landscapes by NLPDEs. In the present study, we investigate the coastal ecosystem by employing the modified regularized long-wave equation (MRLWE) given as [23,26,27]

where R ( x , t ) is the wave function characterizing the water wave behavior along the coast, t is the time, and x is the spatial coordinate, which identifies the spot on a linear coastline where the wave is being observed. The wave dissipation and nonlinearity are influenced by θ (an environmental parameter that affects the conservation and redistribution of wave energy) and γ is an additional environmental element that affects how waves disperse and alter in shape at various wavelengths. We assume that Eq. (1) captures how water waves spread in a coastal setting, especially when taking into account how different variables interact. This study makes use of the Hirota bilinear transformation method, a potent method for solving nonlinear wave equations, to examine and understand these intricate relationships. Thus, an in-depth analysis of wave patterns and their behavior in coastal environments is made possible.

The importance of Eq. (1) transcends boundaries between disciplines, offering understanding into a range of phenomena such as shallow water waves, ion-acoustic waves in plasma, pressure waves in liquids, and gas bubbles and phonon waves in nonlinear crystals [26,27]. As a result, the MRLWE acts as a fundamental model for understanding a range of physical processes. In recent efforts in this field of study, several research works have examined the MRLWE by utilizing a variety of numerical techniques to analyze and solve it. For instance, Hammad and El-Azab [28] employed the Chebyshev–Chebyshev spectral collocation technique (C-C SCM), and Zheng et al. [29] explored the barycentric interpolation collocation method specifically for the MRLWE. Other approaches hinge on spectrum methods, such as the Fourier spectral method employed by Hassan [30] and the application of a spectral method by Ben-Yu and Manoranjan [31]. Furthermore, a number of finite element techniques are based on the least-square principle [32], the septic B-spline collocation approach [33] and the Petrov–Galerkin finite-element method [34]. Other additional techniques are the subdomain finite-element method [35], the Riccati–Bernoulli sub-ODE method [35], and the finite-difference approach for space discretization [36], which makes use of delta-shaped basis functions. Jhangeer et al. [37] investigated fractional MRLWE (both perturbed and unperturbed) using analytical and numerical methods, adding to the wide range of techniques used to analyze and solve the MRLWE.

In the field of soliton research, the Hirota bilinear approach is an essential instrument with broad applications across multiple scientific disciplines. This technique has been essential in revealing intricate wave patterns that originate from a variety of NLPDEs. Several notable examples are Alsallami et al. [38], who used it in solving stochastic-fractional Drinfel’d–Sokolov–Wilson equations, Ceesay et al. [5] applied it in investigating fluid ion wave phenomena using the BBMPB equation, Garcia Guirao et al. [39] employed it in analyzing the recently extended nonlinear (2+1)-dimensional Boussinesq equation with fourth-order terms. Yang and Wei [40] applied Hirota’s bilinear approach to find bilinear equations with ambiguous coefficients for NLPDEs. Likewise, Wazwaz [41] used a combination of Hirota’s methodology and the tanh-coth technique to derive many solutions for the Sawada–Kotera–Kadomtsev–Petviashvili problem. A thorough analysis of bilinear equations was carried out concurrently by Hereman and Zhuang [42]. Additionally, the Hirota bilinear approach has been utilized by scholars in a variety of models to further their research. Using the Hirota bilinear technique, for example, Rizvi et al. [43] investigated KMM in saturated ferromagnetic materials. The study of plasma physics and fluid dynamics was conducted by Wang et al. [44] for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation, Zhao and Wazwaz [45] employed the Hirota bilinear approach to search for multifold soliton solutions, Khan and Wazwaz [46] also obtained a number of unique structures and manifold periodic-soliton solutions for the CBS model using the Hirota bilinear form and ansatz functions.

Conversely by utilizing the bilinear forms of the (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation, Zhao et al. [47] derived lump soliton and mixed lump stripe solutions. Using generalized bilinear operators, Ren et al. [48] investigated an extended (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff-like equation. Yan et al. [49] investigated the (2+1)-dimensional gBS equation using the Hirota bilinear transformation, while Dong et al. [50] studied the (3+1)-dimensional Hirota bilinear equation using the Hirota bilinear approach. Using the Hirota bilinear approach, Feng et al. [51] examined the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. In addition, Rizvi et al. [52] investigated the coupled Higgs equation (CHE) using the Hirota bilinear technique among other transformations. In summary, the broad range of applications of the Hirota bilinear approach is exemplified by its findings, which go beyond standard solitons to include a variety of remarkable wave patterns, including lumps, breathers, kinks, and rogue waves. These solutions find applications in fields such as shallow water wave modeling, optical pulse analysis, fluid dynamics, etc., and they are not limited to theoretical constructs. Altogether, these efforts highlight the lasting importance of the Hirota bilinear method, which cuts across academic borders and enables significant progress in our comprehension of wave phenomena, soliton solutions, and their real-world applications.

It is worth pointing out that this work reports on some important classes of solutions for the MRLWE. The literature provides various results on exact solutions for many partial differential equations in mathematical physics. For example, there are reports on breather, lump, M-shape, and other interaction for the Poisson–Nernst–Planck equation in biological membranes [53]. Also, there are reports on solitary wave structures for a variety of some nonlinear models of surface wave propagation with geometric interpretations [54], N1-soliton solution for Schrödinger’s equation with competing weakly nonlocal and parabolic law nonlinearities [55] and stable optical solitons for the higher-order non-Kerr nonlinear Schrödinger equation via the modified simple equation method [56], solitary wave solutions to the space–time fractional Landau–Ginsburg–Higgs equation via three consistent methods [57], optical solitons solutions of the fractional complex Ginzburg–Landau equation via altered methods [58], exact soliton solutions of the time-fractional clannish random Walker’s parabolic equation via dual techniques [59], and exact solutions for the nonlinear fractional ( 3 + 1 ) -dimensional WBBM equation [60].

The novelty of the article is the use of the Hirota bilinear method to derive new M-shaped soliton solutions for the MRLWE. Solutions in the form of M-shapes are found in various phenomena, but most importantly in coastal environments. Indeed, the propagation of waves on shallow coastal environments (such as shallow beaches) is usually witnessed in the form of this type of shapes. In fact, as far as the authors could verify, this is the first work in the literature in which M-shaped soliton solutions are derived for the MRLE using any method. As shown by the figures in this article, these solutions are M-shaped, and they may have potential applications in various areas of mathematical physics, including the mathematical modeling of the dynamics of coastal environments.

2 Methodology

Although prior studies have investigated the MRLWE using a variety of mathematical techniques, to the best of our knowledge, there is still a clear gap in the literature when it comes to examining the various M-shape symmetry wave structures using the Hirota bilinear approach. The present work explores the effects of different M-shape water wave patterns on coastal environments. Our study attempts to provide a deeper understanding of how these wave patterns affect sediment movement, erosion, and the general stability of coastal regions by utilizing the Hirota bilinear transformation as a basic analytical tool within the framework of the MRLWE. The goal of this research is to improve our knowledge of dynamic coastal processes, which will ultimately lead to more informed and successful coastal management and conservation measures. The knowledge gathered from this study can aid in paving the way for sustainable methods of coastal management and preservation as we continue to negotiate the intricate interactions between water waves and coastlines.

In this section, we outline the procedure used to identify the model’s soliton solutions for this study. We begin by stating the general shape of the NLPDE. To that end, let us assume that the following equation is satisfied:

Here, we let Z be a polynomial function in terms of the function R ( x , t ) and its partial derivatives in both space and time. The following are the primary stages in the method used in this work.

• Stage 1. Apply the standard transformation

  (3) R ( x , t ) = S ( η ) , η = x − c t − b .

  Obviously, the real variables x and t are used here to produce the compound variable η . The constant c is real and it represents a wave speed. Using this transformation, we can convert Eq. (2) into the ODE

  (4) Z ( S , S ′ , S ″ , S ‴ , … ) = 0 ,

  where Z is now a polynomial Z in S ( η ) and its derivatives S ′ ( η ) , S ″ ( η ) , S ‴ ( η ) , …

• Stage 2. We assume that the solution to Eq. (4) can be expressed using logarithms as

  (5) S = ln ( f ( η ) ) η η ,

  where f ( η ) is an unknown function to be determined.

• Stage 3. In this stage, we will determine S , S ′ , S ″ , S ‴ , … in Eq. (5) and substitute their expressions into Eq. (4). This will lead to an equation in terms of f ( η ) and its derivatives up to any possible order.

• Stage 4. We integrate the equation obtained in Stage 3 as many times as possible while keeping the constants of integration at 0 for simplicity to obtain a bilinear form.

• Stage 5. The numerous wave structures under consideration are now inserted into the bilinear equation generated in Stage 4. In each case, we expand, simplify, and collect similar terms, and equate them to 0. Finally, in each situation, we solve the set of equations to obtain suitable solution(s).

• Stage 6. We substitute these classes of solution(s) into the different wave structures obtained in Stage 5. Next, we substitute the resulting function in Eq. (5) to obtain the solutions for S ( η ) .

• Stage 7. Finally, we substitute x − c t − b in Eq. (3) for η in the solution(s) of S ( η ) , which were obtained in Stage 6. In that way, we reach the desired solutions.

In the following sections, we will apply the procedure described above for solving the MRLWE. Various soliton solutions for that equation will be determined as a consequence. To that end, using the traveling wave transformation in Eq. (3), we reduce Eq. (1) to the following ODE:

We assume that the solution to Eq. (6) can be expressed using natural logarithm as

We now determine S ′ , S ″ , and S ‴ of Eq. (7), substitute them into Eq. (6), and integrate equation once to obtain the following simplified bilinear form:

We will substitute various wave structures under consideration into Eq. (8). In each case, we will expand, simplify, collect similar terms, and equate them to 0. In order to obtain the ideal solution(s) in each case, we will solve the resulting set of equations. This task will be performed in the subsequent sections of this article. As a consequence, we will obtain M-shaped rational waves with one of two kinks. We will also study the interaction between rogue and kink M-shaped waves.

Before we close this stage of our work, it is important to point out that one of the main advantages of the bilinear method proposed by Hirota is that it is an algebraic procedure. Conversely, many other approximation methods are analytic rather than algebraic. For example, it is well known that the inverse scattering transform method is powerful in the sense that it is capable of handling arbitrary initial data but, at the same time, it is much more complex. According to the literature, Hirota’s bilinear method is one of the fastest methods to calculate soliton solutions for partial differential equations [61].

3 Results and discussion

3.1 Rational waves with one kink

We produce various kinds of solutions using the following transformation [5,38]:

(9) f = w 1 exp ( d 1 η + d 2 ) + ( η r 1 + r 2 ) 2 + ( η r 3 + r 4 ) 2 + r 5 .

Eq. (9) and its derivatives up to the fourth order are substituted into Eq. (8) and simplified. Then, identical terms are grouped together, and the coefficients of each expression were set to zero. From this system of equations, we obtained the following sets of constant values (called here classes) and their corresponding solutions for the MRLEW

Class 1.

d 1 = − 11 3 c − 1 3 c γ , r 1 = − 5 11 3 c − 1 r 4 6 c γ , r 2 = − r 4 , r 3 = 5 11 3 c − 1 r 4 6 c γ .

Substituting these values into Eq. (9) and substituting the result into (7), we obtain that

(10) S 1 , 1 ( η ) = 198 a ( c − 1 ) e − G 1 ( 25 e G 1 r 4 4 ( 60 33 c − 1 c γ η + 108 c γ + 275 ( c − 1 ) η 2 ) − 54 c γ e d 2 r 5 w 1 ) ( 162 c γ w 1 e d 2 − G 1 + r 4 2 ( 18 c γ + 5 33 c − 1 η ) 2 + 162 c γ r 5 ) 2 − 198 a ( c − 1 ) e − G 1 ( r 4 2 ( e d 2 w 1 ( 360 33 c − 1 c γ η + 2,538 c γ + 275 ( c − 1 ) η 2 ) + 1,350 c γ e G 1 r 5 ) ) ( 162 c γ w 1 e d 2 − G 1 + r 4 2 ( 18 c γ + 5 33 c − 1 η ) 2 + 162 c γ r 5 ) 2 ,

where G 1 = 11 3 c − 1 c η 3 γ . Consequently, the M-shaped rational wave with one kink solution of Eq. (1) in this case is

(11) R 1 , 1 ( x , t ) = N 1 , 1 D 1 , 1 .

Here,

(12) N 1 , 1 = 198 a ( c − 1 ) e G 2 ( 25 e − G 2 r 4 4 ( − 60 c γ G 3 ( b + c t − x ) + 275 ( c − 1 ) ( b + c t − x ) 2 + 108 c γ ) − 54 c γ e d 2 r 5 w 1 ) − 198 a ( c − 1 ) e G 2 ( 25 e − G 2 ( r 4 2 ( e d 2 w 1 ( − 360 c γ G 3 × ( b + c t − x ) + 275 ( c − 1 ) ( b + c t − x ) 2 + 2,538 c γ ) + 1,350 c γ e − G 2 r 5 ) ) )

and

(13) D 1 , 1 = ( r 4 2 ( 5 G 3 ( b + c t − x ) − 18 c γ ) 2 + 162 c γ w 1 e d 2 + G 2 + 162 c γ r 5 ) 2 ,

where G 2 = 11 3 c − 1 c ( b + c t − x ) 3 γ and G 3 = 33 c − 1 . For illustration purposes, Figure 1(a) shows the surface plot of the solution function R 1 , 1 ( x , t ) for some values of the model parameters. Meanwhile, Figure 1(b) provides a contour plot of the solution.

Figure 1

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 1 , 1 ( x , t ) by taking a = 0.09994 , b = 0.0009 , c = 0.799 , γ = 0.996 , d 2 = 0.994 , r 4 = 0.999 , r 5 = 0.99 , w 1 = 0.5 .

Class 2.

d 1 = − c − 1 17 c γ , r 1 = r 3 = 0 , r 5 = − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ + r 2 2 + r 4 2 .

Substituting these constants into Eq. (9) and then the result in Eq. (7), we reach

(14) S 1 , 2 ( η ) = 14 a ( c − 1 ) w 1 ( 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ + 476 c γ ( r 2 2 + r 4 2 ) ) e c − 1 c η 17 γ + d 2 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ e c − 1 c η 17 γ + 238 c γ 2 ( r 2 2 + r 4 2 ) e c − 1 c η 17 γ + e d 2 w 1 2 .

Consequently, the M-shaped rational wave with one kink solution in this case is obtained as

(15) R 1 , 2 ( x , t ) = 14 a ( c − 1 ) w 1 ( 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ + 476 c γ r 2 2 + 476 c γ r 4 2 ) e c − 1 c ( b + ct − x ) 17 γ + d 2 238 c γ w 1 e c − 1 c ( b + ct − x ) 17 γ + d 2 + 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ + 476 c γ r 2 2 + 476 c γ r 4 2 2 .

Figure 2(a) shows the surface plot of the solution function R 1 , 2 ( x , t ) for some values of the model parameters, while Figure 2(b) provides a two-dimensional contour plot of the solution.

Figure 2

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 1 , 2 ( x , t ) by taking a = 0.349 , b = 0.047 , c = 0.079 , γ = 0.11 , d 2 = 1.6 , r 2 = 0.01 , r 4 = 0.1 , w 1 = 0.00045 .

Class 3.

d 1 = − c − 1 17 c γ , r 1 = r 3 , r 4 = − r 2 , r 5 = − 17 c − 1 c γ − 17 c − 1 c γ 17 14 c γ .

Proceeding as in the previous classes, we reach the following equation:

(16) S 1 , 3 ( η ) = a ( c − 1 ) w 1 e d 2 − c − 1 c η 17 γ 17 c γ + 4 r 3 2 − ( c − 1 ) 3 ∕ 2 c γ 3 ∕ 2 14 1 7 3 ∕ 4 ( c − 1 ) 3 ∕ 2 + w 1 e d 2 − c − 1 c η 17 γ + 2 η 2 r 3 2 + 2 r 2 2 − a c − 1 c w 1 e d 2 − c − 1 c η 17 γ 17 γ − 4 η r 3 2 2 − ( c − 1 ) 3 ∕ 2 c γ 3 ∕ 2 14 1 7 3 ∕ 4 ( c − 1 ) 3 ∕ 2 + w 1 e d 2 − c − 1 c η 17 γ + 2 η 2 r 3 2 + 2 r 2 2 2 .

It follows that the M-shaped rational wave with one kink solution of Eq. (1) is given by

(17) R 1 , 3 ( x , t ) = a ( c − 1 ) w 1 e c − 1 c ( b + c t − x ) 17 γ + d 2 17 c γ + 4 r 3 2 w 1 e c − 1 c ( b + c t − x ) 17 γ + d 2 + 2 r 3 2 ( b + c t − x ) 2 + − ( c − 1 ) 3 ∕ 2 c γ 3 ∕ 2 14 1 7 3 ∕ 4 ( c − 1 ) 3 ∕ 2 + 2 r 2 2 − a c − 1 c w 1 e c − 1 c ( b + c t − x ) 17 γ + d 2 17 γ + 4 r 3 2 ( b + c t − x ) 2 w 1 e c − 1 c ( b + c t − x ) 17 γ + d 2 + 2 r 3 2 ( b + c t − x ) 2 + − ( c − 1 ) 3 ∕ 2 c γ 3 ∕ 2 14 1 7 3 ∕ 4 ( c − 1 ) 3 ∕ 2 + 2 r 2 2 2 .

Figure 3(a) shows the surface plot of the solution function R 1 , 3 ( x , t ) for some values of the model parameters. Meanwhile, Figure 3(b) provides a contour plot of the solution.

Figure 3

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 1 , 3 ( x , t ) by taking a = 0.0689 , b = 0.049 , c = 0.079 , γ = 0.11 , d 2 = 0.0054 , r 2 = 0.01 , r 3 = 0.0001 , and w 1 = 0.00055 .

Class 4.

d 1 = c − 1 17 c γ , r 1 = r 2 = 0 , r 3 → 0 , r 5 = − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ − r 2 2 − r 4 2 .

Proceeding as before, we obtain

(18) S 1 , 4 ( η ) = − a c − 1 c w 1 e c − 1 c η 17 γ + d 2 17 γ + 2 r 3 ( η r 3 + r 4 ) 2 − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ + w 1 e c − 1 c η 17 γ + d 2 + η r 3 ( η r 3 + 2 r 4 ) 2 + a ( c − 1 ) w 1 e c − 1 c η 17 γ + d 2 + 34 c γ r 3 2 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ + 238 c γ w 1 e c − 1 c η 17 γ + d 2 + η r 3 ( η r 3 + 2 r 4 ) ( 4,046 c 2 γ 2 ) − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ + w 1 e c − 1 c η 17 γ + d 2 + η r 3 ( η r 3 + 2 r 4 ) 2 .

It follows that the M-shaped rational wave with one kink solution for this case is given by

(19) R 1 , 4 ( x , t ) = a − 2 r 3 ( r 4 − r 3 ( b + c t − x ) ) + c − 1 c e − H 1 w 1 17 γ r 3 ( − b − c t + x ) ( 2 r 4 − r 3 ( b + c t − x ) ) + − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ + e − H 1 w 1 2 + a ( ( ( c − 1 ) e − H 1 w 1 + 34 c γ r 3 2 ) ( 238 c γ ( r 3 2 ( b + c t − x ) 2 − 2 r 4 r 3 ( b + c t − x ) + e − H 1 w 1 ) + 17 4 14 − ( c − 1 ) 3 ∕ 2 c γ ) ) ( 4,046 c 2 γ 2 ) r 3 ( − b − c t + x ) ( 2 r 4 − r 3 ( b + c t − x ) ) + − ( c − 1 ) 3 ∕ 2 c γ 14 1 7 3 ∕ 4 c γ + e − H 1 w 1 2 ,

where H 1 = c − 1 c ( b + c t − x ) 17 γ + d 2 . For the sake of illustration, Figure 4(a) shows the surface plot of the solution function R 1 , 4 ( x , t ) for some values of the model parameters. Meanwhile, Figure 4(b) is a contour plot of the solution.

Figure 4

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 1 , 4 ( x , t ) by taking a = 0.0689 , b = 0.049 , c = 0.000169 , γ = 0.11 , d 2 = 0.0054 , r 3 = 0.01 , r 4 = 0.0001 , w 1 = 0.055 .

Class 5.

d 1 = c − 1 17 c γ , r 1 = − r 3 , r 2 = r 4 = 0 , r 5 = c − 1 c 17 γ − c − 1 17 c γ 238 c γ .

Substituting these parameter values into Eq. (9) and then the result into Eq. (7), we have

(20) S 1 , 5 ( η ) = − a c − 1 w 1 e c − 1 η 17 c γ + d 2 17 c γ + 4 η r 3 2 2 c − 1 c 17 γ − c − 1 17 c γ 238 c γ + w 1 e c − 1 η 17 c γ + d 2 + 2 η 2 r 3 2 2 + a c − 1 c 17 γ − c − 1 17 c γ 238 c γ + w 1 e c − 1 η 17 c γ + d 2 + 2 η 2 r 3 2 ( c − 1 ) w 1 e c − 1 η 17 c γ + d 2 17 c γ + 4 r 3 2 c − 1 c 17 γ − c − 1 17 c γ 238 c γ + w 1 e c − 1 η 17 c γ + d 2 + 2 η 2 r 3 2 2 .

The solution in this case is provided by the expression

(21) R 1 , 5 ( x , t ) = − 56,644 a c γ c − 1 c e − H 2 w 1 17 γ − 4 r 3 2 ( b + c t − x ) 2 ( 476 c γ r 3 2 ( b + c t − x ) 2 + 17 4 14 ( c − 1 ) 3 ∕ 2 c γ + 238 c γ e − H 2 w 1 ) 2 + 56,644 a c γ ( c − 1 ) e − H 2 w 1 17 c γ + 4 r 3 2 2 r 3 2 ( b + c t − x ) 2 + ( c − 1 ) 3 ∕ 2 c γ 3 ∕ 2 14 1 7 3 ∕ 4 ( c − 1 ) 3 ∕ 2 + e − H 2 w 1 ( 476 c γ r 3 2 ( b + c t − x ) 2 + 17 4 14 ( c − 1 ) 3 ∕ 2 c γ + 238 c γ e − H 2 w 1 ) 2 ,

where H 2 = c − 1 c ( b + c t − x ) 17 γ + d 2 . Figure 5(a) provides the surface plot of the solution function R 1 , 5 ( x , t ) for some values of the model parameters. Meanwhile, Figure 5(b) provides a contour plot of the solution.

Figure 5

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 1 , 5 ( x , t ) by taking a = 0.069 , b = 0.049 , c = 0.0009 , γ = 0.41 , d 2 = 0.54 , r 2 = 0.01 , r 3 = 0.1 , w 1 = 0.055 .

The M-shaped solutions for the MRLWE derived in this section physically correspond to trains of waves traveling on the surface of a fluid. We may think of the fluid in this case as the surface of ocean water on a shallow zone, similar to a coastal environment. In this case, the solutions present some perturbations which are natural in the propagation of water waves on a beach. These solutions could be used to describe water fronts on the surface of an ocean, which are entirely due to the effect of the wind.

3.2 M-shaped rational wave

The purpose of the present section is to derive various solutions for the mathematical model under investigation in this work, using the following transformation [5,38]:

(22) f = ( η r 1 + r 2 ) 2 + ( η r 3 + r 4 ) 2 + r 5 .

Eq. (22) and its derivatives up to the fourth order will be substituted into Eq. (8). We will simplify algebraically and group similar terms using η with different values. Afterward, the coefficients of each expression will be set equal to zero. The following classes of solutions are derived in that way.

Class 1.

r 5 = − 15 r 1 2 r 2 2 − r 3 2 r 2 2 − 28 r 1 r 3 r 4 r 2 − r 1 2 r 4 2 − 15 r 3 2 r 4 2 r 1 2 + r 3 2 .

Substituting this value into Eq. (22), and substituting then the resulting expression into Eq. (7), we have

(23) S 2 , 1 ( η ) = − 2 a ( r 1 2 + r 3 2 ) 2 ( η 2 r 1 4 + r 3 2 ( η 2 r 3 2 + 2 η r 4 r 3 + 16 r 4 2 ) + 2 η r 2 r 1 3 + 2 r 1 2 ( η r 3 ( η r 3 + r 4 ) + 8 r 2 2 ) + 2 r 2 r 3 r 1 ( η r 3 + 16 r 4 ) ) ( η 2 r 1 4 + r 3 2 ( η 2 r 3 2 + 2 η r 4 r 3 − 14 r 4 2 ) + 2 η r 2 r 1 3 + 2 r 1 2 ( η r 3 ( η r 3 + r 4 ) − 7 r 2 2 ) + 2 r 2 r 3 r 1 ( η r 3 − 14 r 4 ) ) 2 .

So, the M-shaped rational wave solution of Eq. (1) in this case is obtained as

(24) R 2 , 1 ( x , t ) = − N 2 , 1 D 2 , 1 ,

Here,

(25) N 2 , 1 = 2 a ( r 1 2 + r 3 2 ) 2 ( r 1 4 ( b + c t − x ) 2 − 2 r 2 r 1 3 ( b + c t − x ) + 2 r 1 2 ( r 3 ( − b − c t + x ) ( r 4 − r 3 ( b + c t − x ) ) + 8 r 2 2 ) + 2 r 2 r 3 r 1 ( 16 r 4 − r 3 ( b + c t − x ) ) + J 1 r 3 2 ) ,

(26) D 2 , 1 = ( r 1 4 ( b + c t − x ) 2 − 2 r 2 r 1 3 ( b + c t − x ) + 2 r 1 2 ( r 3 ( − b − c t + x ) ( r 4 − r 3 ( b + c t − x ) ) − 7 r 2 2 ) + 2 r 2 r 3 r 1 ( r 3 ( − ( b + c t − x ) ) − 14 r 4 ) + J 2 r 3 2 ) 2 .

Also, J 1 = r 3 2 ( b + c t − x ) 2 − 2 r 4 r 3 ( b + c t − x ) + 16 r 4 2 and J 2 = r 3 2 ( b + c t − x ) 2 − 2 r 4 r 3 ( b + c t − x ) − 14 r 4 2 . For illustration purposes, Figure 6(a) shows the surface plot of the solution function R 2 , 1 ( x , t ) for some values of the model parameters. Meanwhile, Figure 6(b) provides a contour plot of the solution.

Figure 6

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 2 , 1 ( x , t ) by taking a = 2.009 , b = 0.97 , c = 8.999 , r 1 = 0.953 , r 2 = 1.34 , r 3 = 0.00021 , and r 4 = 2.91 .

Class 2.

r 5 = − 2 26 ( r 1 2 + r 3 2 ) 2 ( r 1 r 2 + r 3 r 4 ) 4 + ( r 1 2 + r 3 2 ) ( ( 13 r 2 2 + r 4 2 ) r 1 2 + 24 r 2 r 3 r 4 r 1 + r 3 2 ( r 2 2 + 13 r 4 2 ) ) ( r 1 2 + r 3 2 ) 2 .

Proceeding as before, we obtain

(27) S 2 , 2 ( η ) = − N 2 , 2 D 2 , 2 ,

where

(28) N 2 , 2 = 2 a ( r 1 2 + r 3 2 ) 3 ( J 3 + ( r 1 2 + r 3 2 ) ( η 2 r 1 4 + r 3 2 ( η 2 r 3 2 + 2 η r 4 r 3 + 14 r 4 2 ) + 2 η r 2 r 1 3 + 2 r 1 2 ( η r 3 ( η r 3 + r 4 ) + 7 r 2 2 ) + 2 r 2 r 3 r 1 ( η r 3 + 14 r 4 ) ) ) ,

(29) D 2 , 2 = ( ( r 1 2 + r 3 2 ) ( η 2 r 1 4 + r 3 2 ( η 2 r 3 2 + 2 η r 4 r 3 − 12 r 4 2 ) + 2 η r 2 r 1 3 + 2 r 1 2 ( η r 3 ( η r 3 + r 4 ) − 6 r 2 2 ) + 2 r 2 r 3 r 1 ( η r 3 − 12 r 4 ) ) − J 3 ) 2 .

In this case, J 3 = 2 26 ( r 1 2 + r 3 2 ) 2 ( r 1 r 2 + r 3 r 4 ) 4 . It follows that the M-shaped rational wave solution for this case is given as

(30) R 2 , 2 ( x , t ) = − N 2 , 2 ′ D 2 , 2 ′ ,

where

(31) N 2 , 2 ′ = 2 a ( r 1 2 + r 3 2 ) 3 ( ( r 1 2 + r 3 2 ) ( r 1 4 ( b + c t − x ) 2 − 2 r 2 r 1 3 ( b + c t − x ) + 2 r 1 2 ( r 3 ( − b − c t + x ) ( r 4 − r 3 ( b + c t − x ) ) + 7 r 2 2 ) + 2 r 2 r 3 r 1 ( 14 r 4 − r 3 ( b + c t − x ) ) + r 3 2 J 4 ) + J 3 ) ,

(32) D 2 , 2 ′ = ( ( r 1 2 + r 3 2 ) ( r 1 4 ( b + c t − x ) 2 − 2 r 2 r 1 3 ( b + c t − x ) + 2 r 1 2 ( r 3 ( − b − c t + x ) ( r 4 − r 3 ( b + c t − x ) ) − 6 r 2 2 ) + 2 r 2 r 3 r 1 ( r 3 ( − ( b + c t − x ) ) − 12 r 4 ) + J 5 r 3 2 ) − J 3 ) 2 .

In this case, J 4 = r 3 2 ( b + c t − x ) 2 − 2 r 4 r 3 ( b + c t − x ) + 14 r 4 2 and J 5 = r 3 2 ( b + c t − x ) 2 − 2 r 4 r 3 ( b + c t − x ) − 12 r 4 2 . Figure 7(a) shows the surface plot of the solution function R 2 , 2 ( x , t ) for some values of the model parameters. Meanwhile, Figure 7(b) provides a contour plot of the solution.

Figure 7

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 2 , 2 ( x , t ) by taking a = 0.993 , b = 0.992 , c = 3.99 , r 1 = 1.998 , r 2 = 2.991 , r 3 = 1.98 , r 4 = 0.234 .

Class 3.

r 4 = r 1 r 2 r 3 .

Proceeding as before, we obtain

(33) S 2 , 3 ( η ) = a 2 ( r 1 2 + r 3 2 ) ( η r 1 + r 2 ) 2 + η r 3 + r 1 r 2 r 3 2 + r 5 − 4 ( η r 1 2 + η r 3 2 + 2 r 2 r 1 ) 2 ( η r 1 + r 2 ) 2 + η r 3 + r 1 r 2 r 3 2 + r 5 2 .

It follows that the solution in this case is given by

(34) R 2 , 3 ( x , t ) = N 2 , 3 D 2 , 3 ,

where

(35) N 2 , 3 = a ( 2 ( r 1 2 + r 3 2 ) ( ( r 2 − r 1 ( b + c t − x ) ) 2 + r 1 r 2 r 3 − r 3 ( b + c t − x ) 2 + r 5 − 4 ( r 1 2 ( b + c t − x ) + r 3 2 ( b + c t − x ) − 2 r 2 r 1 ) 2 )

(36) D 2 , 3 = ( r 2 − r 1 ( b + c t − x ) ) 2 + r 1 r 2 r 3 − r 3 ( b + c t − x ) 2 + r 5 2 .

Figure 8(a) shows the surface plot of the solution function R 2 , 3 ( x , t ) for some values of the model parameters. Meanwhile, Figure 8(b) provides a contour plot of the solution.

Figure 8

Graphs of the three-dimensional surface and two-dimensional projection of the solution R 2 , 3 ( x , t ) by taking a = 0.093 , b = 0.002 , c = 0.0009 , r 1 = 0.9945 , r 2 = 0.000094 , r 3 = 0.0098 , and r 5 = 0.91 .

Before we close this subsection, observe that the solutions derived in this stage correspond to two peaks propagating in the same direction. Physically, these solutions may represent the propagation of solitary rogue waves on the surface of the ocean. It is well known that these waves may occur in the open sea under suitable circumstances. The fact that they are twin peaks in all of the cases makes these quite interesting in view that peaks of water are usually found alone. This fact leads to the interesting question of whether such solutions can be found in the laboratory or in the oceanic environment.