Exact soliton solutions of the modified simplified Camassa–Holm and modified Benjamin–Bona–Mahony equations via the subsidiary ODE method

The main objective of this article is the analytical investigation of the simplified modified Camassa–Holm (SMCH) and the modified Benjamin–Bona–Mahony (BBM) equations. The SMCH equation plays an important role in modeling shallow-water wave dynamics, nonlinear dispersive phenomena, and the propagation of solitons in fluid mechanics. The BBM equation is frequently used to describe long surface gravity waves in nonlinear dispersive media and serves as a useful alternative to the standard Korteweg–de Vries (KdV) equation in mathematical physics. To construct exact analytical soliton solutions for these nonlinear models, the subsidiary ordinary differential equation (sub-ODE) method is employed. Through an appropriate wave transformation, the governing partial differential equations are reduced to nonlinear ordinary differential equations. Our mathematical technique yields several types of soliton wave shapes, including bright, dark, solitary, and periodic solitons. Bright solitons depict localized wave peaks, whereas dark solitons reflect intensity decreases against a continuous background. The resulting analytical solutions are represented in hyperbolic and trigonometric functions that exhibit complex nonlinear behaviors, such as periodic and singular patterns. These soliton structures exhibit the complex dynamics and stability of nonlinear waves propagating in dispersive mediums. The graphical demonstration of their propagation in three-dimensional, two-dimensional, and contour forms is presented for suitable parameter values.

1 Introduction

The study of exact solutions for nonlinear partial differential equations (NLPDEs) is important for understanding nonlinear wave phenomena in several fields, including quantum mechanics, nonlinear optics, fluid mechanics, and plasma physics [1–5]. Nonlinear waves are essential for simulating shallow-water waves, electrical field propagation, acoustic–gravity waves, and hydromagnetic waves, along with other complex physical systems [6–10]. Solitons are stable and confined wave structures that preserve geometry and energy throughout propagation despite nonlinearity and dispersion [11–15]. Bright solitons reflect isolated wave peaks, whereas dark solitons correspond to intensity decreases within a continuous background. Mixed solitons and singular solitons show the complexities in nonlinear wave dynamics [16–19]. Several analytical and computational approaches are being developed to address NLPDEs and generate soliton solutions. These include the $tanh$-function method, Jacobi elliptic function expansion, Hirota bilinear method, $(G^{\prime }/G)$-expansion method, $sine-cosine$ method, Exp-function method, Painlevé analysis, subsidiary ordinary differential equation (sub-ODE) method, variational iteration method, homotopy perturbation technique, Adomian decomposition method, and modified simple equation method [20–28]. These approaches yield explicit models for soliton solutions while also providing useful insights regarding the dynamics and stability of nonlinear waves over diverse media.

Research on nonlinear evolution equations has increasingly focused on Lie symmetries, optimal systems, and symmetry-based reduction techniques in order to obtain exact solutions and analyze nonlinear wave dynamics. Works on the Kadomtsev–Petviashvili (KP)–Benjamin–Bona–Mahony (BBM) and Zakharov–Kuznetsov (ZK)–BBM equations have presented optimal systems and group-invariant solutions that showcase complex structures and solution behaviors of multidimensional models [29, 30]. Furthermore, symmetry reductions are addressed for models such as the Korteweg–de Vries (KdV)–Burgers equation with appropriate dissipative mechanisms that are relevant in plasma environments, where the exact solutions provide insight into wave steepening and damping [31]. Works on one-dimensional (1D) gas dynamics under monochromatic radiation have extended the symmetry techniques to radiative hydrodynamics and sketched the role of Lie invariants in uncovering physically relevant wave patterns [32]. Similarly, in the case of the Gardner equation, studies have pointed out symmetry-based methods for constructing invariant solutions for nonlinear dispersive systems [33].

Various analyses of dissipative and acoustical wave equations are presented in many works, including the Zabolotskaya–Khokhlov equation, where the consideration of symmetry structures effectively reveals the underlying analytical forms of exact solutions that account for nonlinear acoustic propagation [34]. [35] also constructed invariant solutions of coupled Burgers equations, providing new insights into the soliton dynamics of multi-component systems. Symmetry reduction of the KP equation has provided new classes of exact solutions that are relevant in shallow-water wave theory and plasma physics [36]. The most recent works on the Broer–Kaup–Kupershmidt system illustrate how symmetries can be used to understand the interaction of solitons with conservation structures in shallow-water flows [37]. Beyond nonlinear mathematical physics, one finds modern imaging applications such as energy-resolved neutron tomography, in which advanced modeling links wave-based analytical tools to material characterization, thereby illustrating the growing interdisciplinary range of wave dynamics and transport analysis [38].

The simplified modified Camassa–Holm (SMCH) equation and the modified BBM (MBBM) equation are key models for analyzing nonlinear dispersive waves. The main point of the SMCH equation is that it is widely used to model shallow-water wave dynamics, soliton propagation in fluid systems, and nonlinear dispersive phenomena. The MBBM equation can be utilized to describe long surface gravity waves and serves as an alternative to the classical KdV equation [39–43]. Both models are applicable to thermodynamics and fluid mechanics, and they are widely employed in nonlinear optics and plasma physics, along with electromagnetic wave propagation. The sub-ODE method is an efficient analytical strategy for converting complex partial differential equations (PDEs) into simpler ordinary differential equations using proper wave transformations. This approach expresses traveling wave solutions as polynomials of sub-ODE solutions. Using the sub-ODE approach on the SMCH and MBBM equations yields a variety of precise solutions, including bright, dark, composite, and singular solitons. These solutions, which are frequently described as hyperbolic and trigonometric functions, exhibit complex nonlinear patterns and may be represented using two-dimensional (2D), three-dimensional (3D), and contour plots to highlight their stability and dynamics [44, 45].

The study of the soliton solutions for NLPDEs using contemporary analytical methods not only improves our understanding of nonlinear wave phenomena but also offers practical insights related to applications in optical fibers, fluid mechanics, plasma physics, and electromagnetic wave propagation. A systematic investigation into the structures of solitons within both the SMCH and MBBM equations reveals the richness of nonlinear dynamics and shows that modern mathematical techniques can be used to solve complex nonlinear evolution equations. The remainder of this article is organized as follows: Section 1 provides the introduction. Section 2 presents the results of the SMCH equation. Section 4 covers the extraction of soliton solutions for the modified BBM equation. Sections 3 and 5 describe the physical behavior of these solutions. Finally, Section 6 presents the conclusion.

2 Solutions of the SMCH equation

2.1 Mathematical analysis of the sub-ODE method

The emphasis in this article on one-dimensional equations represents a basic and strategic approach to method validation, given the current trend of research into higher-dimensional models. Reductions such as the SMCH and modified BBM equations represent essential one-dimensional benchmark cases in nonlinear wave theory. Such models arise naturally from multi-dimensional systems in fluid mechanics and plasma physics using standard methods of dimensional reduction. The main emphasis of this article is on the development and rigorous validation of the generalized sub-ODE method, and 1D settings represent the established testing ground for analytical correctness, numerical stability, and computational accuracy. Solution families $U_1$–$U_{12}$ obtained in this article are intrinsically of mathematical interest, and more importantly, they represent a starting point for extensions to higher-dimensional analogs, such as the Camassa–Holm–Kadomtsev–Petviashvili-type models and two-dimensional variants of BBM equations. A gradual approach from one-dimensional validation to higher-dimensional applications is a step-by-step research strategy that assures robust development of the method before moving on to more complex geometrical settings.

We assume that the NLPDE is constructed for the function $g=g(x,t)$, where $x$ and $t$ are the spatial and temporal variables, respectively. This assumption allows us to use analytical approaches to find precise or approximate solutions for the NLPDE:

$Q\left(g,g_x,g_t,g_{xx},g_{xt},g_{tt},\dots \right)=0,(1)$

where $Q$ denotes a polynomial containing the function $g$ and its greatest order partial derivative. The traveling wave transformation is used to convert the nonlinear differential equation into an ordinary differential equation:

$g\left(x,t\right)=g\left(\xi \right),\xi =x\pm \omega t,(2)$

where $\xi$ shows the transformation. In this case, $\omega \ne 0$ is a constant to be determined later. Substituting Equation 1 into Equation 2 yields the ordinary differential equation (ODE) for further investigation:

$H\left(g,g^{\prime },g^{″},g‴,g^{\left(4\right)},\dots \right)=0,(3)$

where

$g=g\left(\xi \right),g^{\prime }=\frac{dg}{d\xi },g^{″}=\frac{d^{2}g}{d{\xi }^2},\dots .$

The solution of Equation 3 is provided as follows, which is used to construct explicit forms of the traveling wave solutions:

$g\left(\xi \right)={\displaystyle \sum _{i=0}^N}{a}_i{\psi }^i\left(\xi \right),a_i\ne 0,(4)$

where $a_i(i=\mathrm{0,1,2},\dots ,N)$ signifies the constants to be calculated. $\psi (\xi )$ represents the solution of the following equation:

${\psi }^{\prime 2}=h_0+h_2{\psi }^2+h_4{\psi }^4,(5)$

where $h_0,h_2$ and $h_4$ are real constants.

2.2 Description of the SMCH equation

Camassa and Holm [22] derived the CH equation for shallow-water waves in 1993. It has an integrable bi-Hamiltonian structure. The SMCH equation is written as follows:

$U_t+2\alpha U_x-U_{\mathit{xxt}}+\beta U^2{U}_x=0,(6)$

where $\alpha \in \mathbb{R};\beta >0$ and $U(x,t)$ represents the fluid velocity in the $x$-direction. Using wave transformation, we obtain

$U\left(x,t\right)=U\left(\xi \right),\xi =x-\omega t.(7)$

Substituting Equation 7 into Equation 6 converts the original PDE into an ODE in terms of the traveling wave variable. This reduction simplifies the problem, making it easier to evaluate and create clear solutions.

$-\omega U^{\prime }+2\alpha U^{\prime }+\omega U‴+\beta U^2{U}^{\prime }=0.(8)$

To minimize the order of the differential equation, we integrate Equation 8 with respect to $\xi$ and simplify the resultant expression. This phase eliminates the highest-order derivative and inserts an integration constant, which may be computed later using boundary or beginning conditions.

$\omega U-2\alpha U-\omega U^{″}-\frac{\beta }{3}{U}^3=0.(9)$

2.3 Exact solutions of the SMCH equation

We use the balancing principle to determine the explicit form of the traveling wave solution. We first substitute the ansatz $U(\xi )=\psi {(\xi )}^N$, where $\psi =\psi (\xi )$ is to be determined and $N$ is a positive integer, into the governing equation. The highest-order derivative $U^{″}$ and the highest-order nonlinear term $U^3$ are the dominant terms. Substituting the ansatz, $U^{″}$ yields a power of ${\psi }^{N+2}$, whereas the nonlinear term $U^3$ yields a power of ${\psi }^{3N}$. For a non-trivial solution to exist, these dominant terms must be balanced. Thus, it follows that the powers of $\psi$ must be equal. This produces an algebraic equation $N+2=3N$. If we solve the equation, we obtain $2=2N$ and, ultimately, $N=1$. This supports that the solution of $U(\xi )$ must be written as $U(\xi )=a_0+a_1\psi (\xi )$.

Employing the balancing method for the terms $U^{″}$ and $U^3$ in Equation 9 yields $N=1$. To balance the nonlinear variable $U^3$ and the highest-order derivative term $U^{″}$, the maximum power of $\psi (\xi )$ in the proposed solution must be 1. As a result, Equation 9 enables a solution with just the first-order terms of $\psi (\xi )$

$U\left(\xi \right)=a_0+a_1\psi \left(\xi \right),(10)$

where $a_0$ and $a_1$ are arbitrary constants that will be determined later and $\psi (\xi )$ is the solution to the related elliptic differential equation. The function $\psi (\xi )$ controls the amplitude and periodic behavior of the wave in the soliton solution, establishing its general structure as

${\psi }^{″}=h_2\psi +2h_4{\psi }^3,{\left({\psi }^{\prime }\right)}^2=h_0+h_2{\psi }^2+h_4{\psi }^4,(11)$

where $h_0$, $h_2$, and $h_4$ are real constants. Substituting Equation 10 and Equation 11 into Equation 12 creates a new equation that relates these constants through algebraic expressions. This substitution reduces the nonlinear components and prepares the problem for an analytical approach.

$\begin{array}{ll}\hfill a_0\omega & -2\alpha a_0-\frac{1}{3}\beta a_0^3+a_1\omega \psi -2\alpha a_1\psi -\beta a_0^2{a}_1\psi -a_1{h}_2\omega \psi -a_0{a}_1^2\beta {\psi }^2\hfill \\ \hfill & -\frac{1}{3}\beta a_1^3{\psi }^3-2a_1{h}_4\omega {\psi }^3=0.\hfill \end{array}(12)$

To verify that the equation holds true for all values of $\psi$, the coefficients of each power of $\psi$ are equated to 0. As a result, a set of algebraic equations is obtained, which enables the determination of the unknown constants such as $a_0$, $a_1$, $h_0$, $h_2$, and $h_4$.

$\begin{array}{cc}\hfill {\psi }^0& :a_0\omega -2\alpha a_0-\frac{1}{3}\beta a_0^3=0,\hfill \\ \hfill {\psi }^1& :a_1\omega -2\alpha a_1-a_0^2{a}_1\beta -a_1{h}_2\omega =0,\hfill \\ \hfill {\psi }^2& :-a_0{a}_1^2\beta =0,\hfill \\ \hfill {\psi }^3& :-\frac{1}{3}\beta a_1^3-2a_1{h}_4\omega =0.\hfill \end{array}$

The solution to these algebraic equations will yield the exact value of the unknown constants, and such constants are necessary for forming the exact traveling wave solutions. These constants determine the type and number of solitons generated by the nonlinear equation, including bright, dark, and singular solutions.

$a_0=0,\omega =-\frac{2\alpha }{h_2-1},a_1=\pm 2\sqrt{\frac{3\alpha h_4}{\beta h_2-\beta }}.(13)$

The solution to Equation 11 may be represented as follows by demonstrating the relationship between Jacobi elliptic functions and their limiting forms. When the modulus $j$ approaches 1 or 0, these functions become hyperbolic and trigonometric.

Table 1 describes Jacobi elliptic functions and their limiting forms, which are essential for generating soliton solutions in nonlinear differential equations. The table depicts the behavior of each function (sn, cn, dn, ds, sd, ns, nc, cs, and cd) as the modulus $j\to 1$ and $j\to 0$. When $j\to 1$, the functions decrease to hyperbolic functions (tanh, sech, sinh, coth, etc.), and they depict localized soliton-like waves. When $j\to 0$, they reduce to trigonometric functions (sin, cos, sec, csc, cot, etc.), and they characterize periodic waves. This shows that, depending on the value of $j$, a single Jacobi function may describe both soliton and periodic wave solutions, making it particularly helpful for evaluating traveling wave solutions and soliton structures in nonlinear wave equations.

Table 1

Table 1. Analysis of Jacobi elliptic functions and their limiting forms.

2.3.1 Case 1

We assume that $h_0=v^2{D}^2,h_2=-v^2(1+j^2),\text{\hspace{0.17em}and\hspace{0.17em}}{h}_4=\frac{v^2{j}^2}{D^2}$. Here, $v$ and $D$ are the wave velocity and the amplitude of the wave, respectively, and $0\le j\le 1$ is the Jacobi elliptic function modulus. The solution of Equation 11 is as follows:

$\psi \left(\xi \right)=D\mathrm{s}\mathrm{n}\left(v\xi ,j\right).(14)$

The exact solution of Equation 8 is mathematically represented below, where the parameters meet the given requirements.

$U_1\left(\xi \right)=\pm 2vj\sqrt{-\frac{3\alpha }{\beta \left(v^2+v^2{j}^2+1\right)}}\mathrm{s}\mathrm{n}\left(v\xi ,j\right),\omega =\frac{2\alpha }{v^2\left(1+j^2\right)+1}.(15)$

In the limiting case, as $j\to 1$, the solution of Equation 11 simplifies to a hyperbolic shape, representing a confined solitary wave. This limit illustrates the transition from periodic Jacobi elliptic functions to solitary wave structures, stressing the solution’s physical relevance in the context of nonlinear wave propagation.

$U_1\left(\xi \right)=\pm 2v\sqrt{-\frac{3\alpha }{\beta \left(2v^2+1\right)}}\tanh \left(\xi \right),\xi =x-\frac{2\alpha t}{2v^2+1}.(16)$

2.3.2 Case 2

We assume a certain value of $h_0=v^2{D}^2(1-j^2),h_2=-v^2(2j^2-1),\text{and}{h}_4=-\frac{v^2{j}^2}{D^2}$. Equation 11 admits the following exact solution, which depends on the Jacobi elliptic functions and characterizes the wave profile for the system under consideration.

$\psi \left(\xi \right)=D\text{\hspace{0.17em}cn}\left(v\xi ,j\right).(17)$

The solution of Equation 8 can be written as follows, illustrating the system behavior under the specified conditions:

$U_2\left(\xi \right)=\pm 2vj\sqrt{\frac{3\alpha }{\beta \left(2v^2{j}^2-v^2+1\right)}}\text{\hspace{0.17em}sn}\left(v\xi ,j\right),\omega =\frac{2\alpha }{v^2\left(2j^2-1\right)+1}.(18)$

As $j\to 1$, the outcome becomes hyperbolic, corresponding to a specific soliton with a sharp peak and finite width. This shows the creation of a single wave in the structure of the system.

$\xi =x-\frac{2\alpha t}{v^2+1},U_2\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{\beta \left(v^2+1\right)}}\text{\hspace{0.17em}sech}\left(\xi \right).(19)$

2.3.3 Case 3

We assume that $h_0=-v^2{D}^2(1-j^2),h_2=v^2(2-j^2),\text{and}{h}_4=-\frac{v^2}{D^2}$. Based on Equation 11, the solution captures the system’s nonlinear properties and can represent several solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{d}\mathrm{n}\left(v\xi ,j\right).(20)$

The solution of Equation 8 can be stated as follows, demonstrating the system behavior under the required conditions:

$U_3\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{\beta \left(v^2{j}^2-2v^2+1\right)}}\mathrm{d}\mathrm{n}\left(v\xi ,j\right),\omega =-\frac{2\alpha }{v^2\left(2-j^2\right)-1}.(21)$

As $j\to 1$, the outcome becomes hyperbolic, representing a particular soliton with a sharp peak and finite width. This demonstrates the formation of a single wave within the system’s structure.

$U_3\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{\beta \left(-v^2+1\right)}}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\xi \right),\xi =x+\frac{2\alpha t}{v^2-1}.(22)$

2.3.4 Case 4

We assume that $h_0=v^2{D}^2{j}^2,h_2=-v^2(1+j^2),\text{and}{h}_4=\frac{v^2}{D^2}$. The solution to Equation 11 shows how the wave profile evolves over time. This solution captures the system’s nonlinear features and can represent a variety of solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{n}\mathrm{s}\left(v\xi ,j\right)=\frac{D}{\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}.(23)$

The solution of Equation 8 can be expressed as follows, illustrating the system behavior under suitable conditions:

$U_4\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2{j}^2+v^2+1\right)}}\mathrm{n}\mathrm{s}\left(v\xi ,j\right),\omega =\frac{2\alpha }{v^2\left(1+j^2\right)+1}.(24)$

As $j\to 1$, the result becomes hyperbolic, indicating a specific soliton with a sharp peak and finite width. This shows how a single wave forms within the system’s structure.

$U_4\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(2v^2+1\right)}}\coth \left(\xi \right),\xi =x-\frac{2\alpha t}{2v^2+1}.(25)$

2.3.5 Case 5

We assume that $h_0=-v^2{D}^2{j}^2,h_2=v^2(2j^2-1),\text{and}{h}_4=v^2\frac{1-j^2}{D^2}$. From Equation 11, the solution captures the system’s nonlinear properties and can represent several solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{n}\mathrm{c}\left(v\xi ,j\right)=\frac{D}{\mathrm{c}\mathrm{n}\left(v\xi ,j\right)}.(26)$

The solution to Equation 8 is as follows, showing the behavior of the system under the appropriate conditions:

$U_5\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha \left(1-j^2\right)}{-\beta \left(v^2-2v^2{j}^2+1\right)}}\mathrm{n}\mathrm{c}\left(v\xi ,j\right),\omega =\frac{-2\alpha }{v^2\left(2j^2-1\right)-1}.(27)$

As $j\to 0$, the result becomes hyperbolic, suggesting a unique soliton with a sharp peak and finite width. This illustrates how a single wave forms within the system’s structure.

$U_5\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2+1\right)}}\sec \left(\xi \right),\xi =x+\frac{2\alpha t}{2v^2+1}.(28)$

2.3.6 Case 6

We assume that $h_0=-v^3{D}^2,h_2=v^2(2-j^2),$ and $h_4=-\frac{v^2(1-j^2)}{D^3}$. The solution of Equation 11 demonstrates how the wave profile varies over time. This solution represents the system’s nonlinear features and can represent a variety of solitons based on the parameters used.

$\varphi \left(\xi \right)=D\mathrm{n}\mathrm{d}\left(v\xi ,j\right)=\frac{D}{\mathrm{d}\mathrm{n}\left(v\xi ,j\right)}.(29)$

The solution of Equation 8 is as follows, displaying the system behavior under the required conditions.

$U_6\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha \left(1-j^2\right)}{\beta \left(v^2{j}^2-2v^2+1\right)}}\mathrm{n}\mathrm{d}\left(v\xi ,j\right),\omega =-\frac{2\alpha }{v^2\left(2-j^2\right)-1}.(30)$

2.3.7 Case 7

We assume that $h_0=v^2{D}^2,h_2=v^2(2-j^2),$ and $h_4=\frac{v^2(1-j^2)}{D^2}$. Based on Equation 11, the solution depicts the system’s nonlinear properties and can represent several solitons depending on the parameters employed.

$\psi \left(\xi \right)=D\mathrm{s}\mathrm{c}\left(v\xi ,j\right)=\frac{D\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{c}\mathrm{n}\left(v\xi ,j\right)}.(31)$

The solution of Equation 8 is shown below, demonstrating the system behavior under specific conditions.

$j_7\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha \left(1-j^2\right)}{-\beta \left(v^2{j}^2-2v^2+1\right)}}\mathrm{s}\mathrm{c}\left(v\xi ,j\right),\omega =-\frac{2\alpha }{v^2\left(2-j^2\right)-1}.(32)$

As $j\to 0$, the result becomes hyperbolic, indicating a single soliton with a sharp peak and finite width. This demonstrates the development of a single wave within the system’s internal structure.

$U_7\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(-2j^2+1\right)}}\tan \left(\xi \right),\xi =x+\frac{2\alpha t}{2v^2-1}.(33)$

2.3.8 Case 8

We assume that $h_0=v^2{D}^2,h_2=v^2(2j^2-1),$ and $h_4=-\frac{v^2{j}^2(1-j^2)}{D^2}$. The solution of Equation 11 illustrates how the wave profile varies over time. This solution displays the system’s nonlinear features and can represent a variety of solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{s}\mathrm{d}\left(v\xi ,j\right)=\frac{D\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{d}\mathrm{n}\left(v\xi ,j\right)}.(34)$

The solution of Equation 8 is provided below, exhibiting the system’s behavior under the given conditions.

$U_8\left(\xi \right)=\pm 2vj\sqrt{\frac{3\alpha \left(1-j^2\right)}{\beta \left(v^2-2v^2{j}^2+1\right)}}\mathrm{s}\mathrm{d}\left(v\xi ,j\right),\omega =-\frac{2\alpha }{v^2\left(2j^2-1\right)-1}.(35)$

2.3.9 Case 9

We assume that $h_0=v^2(1-j^2),h_2=v^2(2-j^2)D^2,$ and $h_4=\frac{v^2}{D^2}$. Based on Equation 11, the solution highlights the system’s nonlinear properties and can depict several solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{c}\mathrm{s}\left(v\xi ,j\right)=\frac{D\mathrm{c}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}.(36)$

The solution of Equation 8 is provided below, exhibiting the system’s behavior under specified conditions.

$U_9\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2{j}^2-2v^2+1\right)}}\mathrm{c}\mathrm{s}\left(v\xi ,j\right),\omega =-\frac{2\alpha }{v^2\left(2-j^2\right)-1}.(37)$

As $j$ approaches 0, the result becomes hyperbolic, indicating a single soliton with a sharp peak and finite width. This demonstrates the formation of a single wave within the system’s internal structure.

$\xi =x+\frac{2\alpha t}{2v^2-1},U_9\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(-2v^2+1\right)}}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\left(\xi \right).(38)$

2.3.10 Case 10

We assume that $h_0=v^2{D}^2,h_2=-v^2(1+j^2),$ and $h_4=\frac{v^2{j}^2}{D^3}$. The solution of Equation 11 demonstrates how the wave profile varies over time. This solution emphasizes the system’s nonlinear features and can represent a variety of solitons depending on the parameters used.

$\psi \left(\xi \right)=D\mathrm{c}\mathrm{d}\left(v\xi ,j\right)=\frac{D\mathrm{c}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{d}\mathrm{n}\left(v\xi ,j\right)}.(39)$

The solution of Equation 8 is provided below, exhibiting the system’s behavior under certain conditions.

$U_{10}\left(\xi \right)=\pm 2vj\sqrt{\frac{3\alpha }{-\beta \left(v^2{j}^2+v^2+1\right)}}\mathrm{c}\mathrm{d}\left(v\xi ,j\right),\omega =\frac{2\alpha t}{v^2\left(1+j^2\right)+1}.(40)$

2.3.11 Case 11

We assume that $h_0=-v^2{j}^2(1-j^2),h_2=v^2(2j^2-1)$. Finally, $h_4=\frac{v^2}{D^2}$. The solution of Equation 11 illustrates how the wave profile varies over time. This solution highlights the system’s nonlinear features and can represent a variety of solitons depending on the variables used.

$\psi \left(\xi \right)=D\mathrm{d}\mathrm{s}\left(v\xi ,j\right)=\frac{D\mathrm{d}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}.(41)$

The following solution of Equation 8 demonstrates the system’s behavior under the provided conditions.

$U_{11}\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2-v^2{j}^2+1\right)}}\mathrm{d}\mathrm{s}\left(v\xi ,j\right),\omega =\frac{2\alpha t}{v^2\left(2j^2-1\right)-1}.(42)$

As $j\to 1$, the result turns hyperbolic, suggesting a single soliton with a sharp peak and finite width. This occurs when a single wave emerges within the system’s underlying structure.

$\xi =x-\frac{2\alpha t}{2v^2+1},U_{11}\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2+1\right)}}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\left(\xi \right).(43)$

2.3.12 Case 12

We suppose that $h_0=v^2{j}^2{D}^2,h_2=-v^2(j^2+1),$ and $h_4=\frac{v^2}{D^2}$. The solution of Equation 11 shows how the wave profile changes with time. This solution emphasizes the system’s nonlinear properties and can represent several solitons based on the parameters used.

$\psi \left(\xi \right)=D\mathrm{d}\mathrm{c}\left(v\xi ,j\right)=\frac{D\mathrm{d}\mathrm{n}\left(v\xi ,j\right)}{\mathrm{s}\mathrm{n}\left(v\xi ,j\right)}.(44)$

The solution of Equation 8 is provided below, exhibiting the system’s response under the given circumstances.

$U_{12}\left(\xi \right)=\pm 2v\sqrt{\frac{3\alpha }{-\beta \left(v^2{j}^2+v^2+1\right)}}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\left(\xi \right),\omega =\frac{2\alpha t}{v^2\left(j^2+1\right)+1}.(45)$

3 Physical interpretation of solutions under the SMCH equation

These graphs illustrate a clear representation of the different solitary wave solutions obtained for the considered SMCH equation. The dark soliton solution obtained from Equation 16 and plotted in Figure 1 represents a stable density dip traveling over a continuous background. By using parameters $v=1.5$, $j=1$, $\alpha =0.1$, and $\beta =0.5$, this structure models phenomena such as pressure depressions in compressible fluids or voids in nonlinear lattices, where the parameters dictate the depth and stability of the propagating trough. Bright soliton solutions are shown in Figures 2, 3 for Equations 19, 22, respectively, which manifest as localized, particle-like humps of elevated energy. These structures, specifically with the modulating parameter, as shown in Figure 2, are fundamental for modeling localized excitations such as pressure peaks in shallow water or intense pulses in elastic rods.

Figure 1

Figure 1. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 16 for the parameter values $v=1.5$, $j=1$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe the structure of a dark optical soliton under the specified constraints.

Figure 2

Figure 2. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 19 for the parameter values $v=1.5$, $j=1$, $\alpha =0.1$, $\beta =0.5$, and $\chi =0.1$. These graphs describe the structure of a bright soliton under the specified constraints.

Figure 3

Figure 3. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 22 for the parameter values $v=1.5$, $j=1$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe the structure of a bright soliton under the specified constraints.

In this section, Maple 18 is utilized to generate 3D, 2D, and contour graphs of traveling wave solutions for the SMCH equation. Figure 1 shows that the series of solutions is extended with the inclusion of singular and periodic wave modes, which result from the nonlinearity of the SMCH model. The implementation of Equation 25 in Figure 4 shows a singular soliton with its characteristic unbounded and sharp peak that points to a scenario of wave-breaking or hydraulic jump in the case of fluid dynamics; here, the choice of $j=0$ is crucial for the observation of the non-analytic characteristic. Figure 5 shows a periodic singular soliton, derived from Equation 28; the periodicity of the singularity indicates that the system frequently undergoes shock formation. Figure 6 (from Equation 33) reveals a non-singular wave that is purely periodic and has similarities to oscillatory forms within confined domains. The singular soliton in Figure 7 (from Equation 38), on the other hand, not only suggests but also emphasizes the possible occurrence of highly localized, intense energy concentrations that might be the precursor of rogue waves in a complex medium.

Figure 4

Figure 4. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 25 for the parameter values $v=1.5$, $j=0$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe the unique soliton structure under the specified constraints.

Figure 5

Figure 5. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 28 for the parameter values $v=1.5$, $j=0$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe the periodic singular solitons under the specified constraints.

Figure 6

Figure 6. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 33 for the parameter values $v=1.5$, $j=0$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe the periodic soliton under the specified constraints.

Figure 7

Figure 7. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 38 for the parameter values $v=1.5$, $j=1$, $\alpha =0.1$, and $\beta =0.5$. These graphs describe a singular optical soliton under the specified constraints.

The multi-perspective plotting strategy using 3D, 2D, and contour views physically elucidates the features of the SMCH equation step by step. The 3D surface plots clearly demonstrate the strong localization in the space and the stability over time of the soliton amplitude. The 2D line graphs corresponding to the 3D surface plots provide a detailed view of the wave’s profile at the moment and its invariance in translation over the spatial domain. The contour plots clearly delineate the mapping of intensity lines and the pathways of energy propagation, thus providing important insights into the collisionless nature and interaction capabilities of solitons. The use of different plots not only illustrates various mathematical solutions but also highlights the rich physics and the wide range of wave morphologies that can be observed and are governed by the SMCH equation under the given parameter conditions.

4 Extraction of soliton solutions for the modified BBM equation

4.1 Description of the modified BBM equation

The $(1+1)$-dimensional nonlinear dispersed modified BBM (Equation 11) is an important model for studying weakly nonlinear long waves in dispersive media. This equation reflects the equilibrium between nonlinearity and dispersion, which allows the existence and propagation of a wide range of solitary wave solutions. The result is as follows:

$u_t+u_x-\alpha u^2{u}_x+u_{\mathit{xxx}}=0,(46)$

where $\alpha$ represents a non-zero real constant. This equation was originally formulated to represent long surface waves in a nonlinear dispersive medium. It may also describe acoustic–gravity waves in compressible fluids, hydromagnetic waves in cold plasmas, and acoustic waves in inharmonic crystals. We know that from the previous section, the traveling wave transformation is stated as follows:

$U\left(x,t\right)=U\left(\xi \right),\xi =x-\omega t.(47)$

Using the chain rule, $U_t=-\omega U^{\prime },U_x=U^{\prime },U_{\mathit{xxx}}=U‴$. Substituting Equation 47 into Equation 46 yields a nonlinear ODE incorporating $U$ and its derivatives. This transformation essentially lowers the original partial differential equation to an ODE, which simplifies wave profile analysis. The resultant equation is as follows:

$\left(1-\omega \right)U^{\prime }-\alpha U^2{U}^{\prime }+U‴=0.(48)$

Combining Equation 48 with respect to $\xi$ while allowing the constant integration to be 0 for simplicity yields a simplified equation. This technique significantly reduces the order of the differential equation, making it easier to derive exact traveling wave solutions. The integrated equation reflects the balance among nonlinear and dispersive effects that exist in the system, which is necessary for understanding soliton structures.

$\left(1-\omega \right)U-\frac{\alpha }{3}{U}^3+U^{″}=0.(49)$

4.2 Exact solution of the modified BBM equation

Using the balance concept from Equation 49, we obtain $n=1$. This step is critical for developing precise traveling wave solutions and guarantees that the nonlinear and dispersive terms are correctly balanced.

$U\left(\xi \right)=a_0+a_1\psi \left(\xi \right),(50)$

where $a_0$ and $a_1$ are arbitrary constants and $\psi (\xi )$ satisfies an elliptic differential equation. This approach allows for the generation of precise traveling wave solutions by selecting the function $\psi (\xi )$ to satisfy the governing nonlinear equation.

${\left({\psi }^{\prime }\right)}^2=h_0+h_2{\psi }^2+h_4{\psi }^4,{\psi }^{″}=h_2\psi +2h_4{\psi }^3,(51)$

where $h_0,h_2,\text{and}{h}_4$ are real constants. Substituting Equation 50 and Equation 51 into Equation 49 yields an equation in terms of $\psi (\xi )$ and its powers. This technique allows us to determine the unknown coefficient $h_0,h_2,h_4$ consistently by comparing the coefficients of similar powers of $\psi (\xi )$, eventually leading to exact solutions of the nonlinear equation

$\begin{array}{ll}\hfill & a_0-a_0\omega -\frac{1}{3}\alpha a_0^3+a_1\psi -a_1\omega \psi -a_0^2{a}_1\psi +a_1{h}_2\psi \hfill \\ \hfill & -\alpha a_0{a}_1^2{\psi }^2-\frac{1}{3}\alpha a_1^3{\psi }^3+2a_1{h}_4{\psi }^3=0.\hfill \end{array}(52)$

A set of equations involving algebra can be obtained by equating every coefficient of power of $\psi$ to 0. These equations provide information on the unknown constants within the solution. Solving them yields the precise analytical structure of the traveling wave solution.

$\begin{array}{ll}\hfill {\psi }^0:& a_0-a_0\omega -\frac{1}{3}\alpha a_0^3=0,\hfill \\ \hfill {\psi }^1:& a_1-a_1\omega -a_0^2{a}_1+a_1{h}_2=0,\hfill \\ \hfill {\psi }^2:& -\alpha a_0{a}_1^2=0,\hfill \\ \hfill {\psi }^3:& 2a_1{h}_4-\frac{1}{3}\alpha a_1^3=0.\hfill \end{array}$

Solving the above algebraic equations yields the values of unknown constants, thereby allowing us to construct the explicit form of the traveling wave solution.

$a_0=0,\omega =1+h_2,a_1=\pm \sqrt{\frac{6h_4}{\alpha }}.(53)$

The exact solution of Equation 51 is provided in the following section, illustrating the traveling wave form that satisfies the nonlinear equation.

4.2.1 Case 1

We assume that $h_0=v^2{C}^2,h_2=-v^2(1+m^2)$, and $h_4=\frac{v^2{m}^2}{C^2}$. Here, $v$ and $C$ are nonzero real constants, while $0\le m\le 1$ is the Jacobi elliptic functions’ modulus. The result of Equation 51 is as follows:

$\psi \left(\xi \right)=C\mathrm{s}\mathrm{n}\left(v\xi ,m\right).(54)$

The specific solution of Equation 48 may be written as follows, which offers insight into the behavior of the structure under consideration.

$\omega =1-v^2\left(1+m^2\right),(55)$

$U_1\left(\xi \right)=\pm vm\sqrt{\frac{6}{\alpha }}\mathrm{s}\mathrm{n}\left(v\xi ,m\right).(56)$

As $m\to 1$, the outcome reveals the system’s limiting behavior and highlights the properties of the associated solitary wave.

$\xi =x-\left(1-2v^2\right)t,U_1\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\tanh \left(\xi \right).(57)$

4.2.2 Case 2

We assume that $h_0=v^2{C}^2(1-m^2),h_2=-v^2(2m^2-1)$, and $h_4=-\frac{v^2{m}^2}{C^2}$, where $C$ is a nonzero real constant and $0\le m\le 1$ is the Jacobi elliptic functions modulus. Equation 51 yields the following result:

$\psi \left(\xi \right)=C\mathrm{c}\mathrm{n}\left(v\xi ,m\right).(58)$

The solution of Equation 48 can be expressed as follows, illustrating its structure:

$U_2\left(\xi \right)=\pm vm\sqrt{-\frac{6}{\alpha }}\mathrm{c}\mathrm{n}\left(v\xi ,m\right),\omega =1+v^2\left(2m^2-1\right).(59)$

As $m\to 1$, the solution is obtained, exposing the limiting behavior of the system and highlighting the features of the associated solitary wave.

$U_2\left(\xi \right)=\pm v\sqrt{-\frac{6}{\alpha }}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\xi \right),\xi =x-\left(1-v^2\right)t.(60)$

4.2.3 Case 3

We assume that $h_0=-v^2{C}^2(1-m^2),h_2=v^2(2-m^2)$, and $h_4=-\frac{v^2}{C^2}$. Equation 51 yields the following results:

$\psi \left(\xi \right)=C\mathrm{d}\mathrm{n}\left(v\xi ,m\right).(61)$

The solution of Equation 48 can be written as follows, providing insight into the behavior of the structure under consideration:

$U_3\left(\xi \right)=\pm v\sqrt{\frac{-6}{\alpha }}\mathrm{d}\mathrm{n}\left(v\xi ,m\right),\omega =1+v^2\left(2-m^2\right).(62)$

As $m\to 1$, the solution is obtained, revealing the system’s limiting characteristic and emphasizing the accompanying solitary wave.

$U_3\left(\xi \right)=\pm v\sqrt{\frac{-6}{\alpha }}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\xi \right),\xi =x-\left(1+v^2\right)t.(63)$

4.2.4 Case 4

We assume that $h_0=v^2{C}^2{m}^2,h_2=-v^2(1+m^2)$, and $h_4=\frac{v^2}{C^2}$. Equation 51 yields the following results:

$\psi \left(\xi \right)=C\mathrm{n}\mathrm{s}\left(v\xi ,m\right)=\frac{C}{\mathrm{s}\mathrm{n}\left(v\xi ,m\right)}.(64)$

The solution of Equation 48 can be expressed as follows:

$\omega =1-v^2\left(2-m^2\right),U_4\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\mathrm{n}\mathrm{s}\left(v\xi ,m\right).(65)$

As $m\to 1$, the solution is determined, showing the system’s limiting feature and highlighting the accompanying single wave.

$\xi =x-\left(1-v^2\right)t,U_4\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\coth \left(\xi \right).(66)$

4.2.5 Case 5

We assume that $h_0=-v^2{C}^2{m}^2,h_2=v^2(2m^2-1)$, and $h_4=\frac{v^2(1-m^2)}{C^2}$. Equation 51 yields the following outcomes:

$\psi \left(\xi \right)=C\mathrm{n}\mathrm{c}\left(v\xi ,m\right)=\frac{C}{\mathrm{d}\mathrm{n}\left(v\xi ,m\right)}.(67)$

The solution of Equation 48 can be written as follows:

$\omega =1+v^2\left(2m^2-1\right),U_5\left(\xi \right)=\pm v\sqrt{\frac{6\left(1-m^2\right)}{\alpha }}\mathrm{n}\mathrm{c}\left(v\xi ,m\right).(68)$

As $m\to 1$, the solution is obtained, revealing the system’s limiting characteristic and emphasizing the accompanying single wave.

$\xi =x-\left(1-v^2\right)t,U_5\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\sec \left(\xi \right).(69)$

4.2.6 Case 6

We assume that $h_0=-v^2{C}^2{m}^2,h_2=v^2(2-m^2)$, and $h_4=-\frac{v^2(1-m^2)}{C^2}$. Equation 51 provides the following result:

$\psi \left(\xi \right)=Cnd\left(v\xi ,m\right)=\frac{C}{dn\left(v\xi ,m\right)}.(70)$

The solution of Equation 48 can be expressed as follows:

$\omega =1+v^2\left(2-m^2\right),U_6\left(\xi \right)=\pm v\sqrt{\frac{-6\left(1-m^2\right)}{\alpha }}nd\left(v\xi ,m\right).(71)$

4.2.7 Case 7

We assume that $h_0=v^2{C}^2,h_2=v^2(2-m^2)$, and $h_4=\frac{v^2(1-m^2)}{C^2}$. Equation 51 yields the following result:

$\psi \left(\xi \right)=Csc\left(v\xi ,m\right)=\frac{Csn\left(v\xi ,m\right)}{cn\left(v\xi ,m\right)}.(72)$

The solution of Equation 48 can be written as follows:

$\omega =1+v^2\left(2-m^2\right),U_7\left(\xi \right)=\pm v\sqrt{\frac{6\left(1-m^2\right)}{\alpha }}sc\left(v\xi ,m\right).(73)$

As $m\to 0$, the solution is obtained, revealing the system’s limiting characteristic and emphasizing the accompanying single wave.

$\xi =x-\left(1+2v^2\right)t,U_7\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\tan \left(\xi \right).(74)$

4.2.8 Case 8

We consider that $h_0=v^2{C}^2,h_2=v^2(2m^2-1)$, and $h_4=-\frac{v^2{m}^2(1-m^2)}{C^2}$. Equation 51 yields the following outcomes:

$\psi \left(\xi \right)=Csd\left(v\xi ,m\right)=\frac{Csn\left(v\xi ,m\right)}{dn\left(v\xi ,m\right)}.(75)$

The solution of Equation 48 can be written as follows:

$\omega =1+v^2\left(2m^2-1\right),U_8\left(\xi \right)=\pm vm\sqrt{\frac{6\left(1-m^2\right)}{\alpha }}sd\left(v\xi ,m\right).(76)$

4.2.9 Case 9

We consider the equations $h_0=v^2(1-m^2)C^2,h_2=v^2(2-m^2)$, and $h_4=\frac{v^2}{C^2}$. Equation 51 yields the following results:

$\psi \left(\xi \right)=Ccs\left(v\xi ,m\right)=\frac{Ccn\left(v\xi ,m\right)}{sn\left(v\xi ,m\right)}.(77)$

The solution of Equation 48 can be written as follows:

$U_9\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}cs\left(v\xi ,m\right),\omega =1+v^2\left(2-m^2\right).(78)$

As $m\to 1$, the solution is obtained, showing the system’s limiting feature and highlighting the resulting solitary wave.

$\xi =x-\left(1+2v^2\right)t,U_9\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\cosh \left(\xi \right).(79)$

4.2.10 Case 10

We assume that $h_0=v^2{C}^2,h_2=-v^2(1+m^2)$, and $h_4=\frac{v^2}{C^2}$. Equation 51 yields the following outcomes:

$\psi \left(\xi \right)=Ccd\left(v\xi ,m\right)=\frac{Ccn\left(v\xi ,m\right)}{dn\left(v\xi ,m\right)}.(80)$

The solution of Equation 48 can be written as follows:

$\omega =1-v^2\left(m^2+1\right),U_{10}\left(\xi \right)=\pm vm\sqrt{\frac{6}{\alpha }}cd\left(v\xi ,m\right).(81)$

4.2.11 Case 11

We consider the following equations $h_0=-v^2{m}^2(1-m^2)C^2,h_2=v^2(2m^2-1)$, and $h_4=\frac{v^2}{C^2}$. Equation 51 yields the following expression:

$\psi \left(\xi \right)=Cds\left(v\xi ,m\right)=\frac{Cdn\left(v\xi ,m\right)}{sn\left(v\xi ,m\right)}.(82)$

The result of Equation 48 can be written as follows, providing insight into the system’s behavior under the provided analysis:

$\omega =1+v^2\left(2m^2-1\right),U_{11}\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}ds\left(v\xi ,m\right).(83)$

As $m\to 0$, the solution is obtained, showing the system’s limiting feature and emphasizing the associated solitary wave.

$\xi =x-\left(1-v^2\right)t,U_{11}\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}\cosh \left(\xi \right).(84)$

4.2.12 Case 12

We assume that $h_0=v^2{m}^2{C}^2,h_2=-v^2(m^2+1)$, and$h_4=\frac{v^2}{C^2}$. Equation 51 yields the following results:

$\psi \left(\xi \right)=Cdc\left(v\xi ,m\right)=\frac{Cdn\left(v\xi ,m\right)}{sn\left(v\xi ,m\right)}.(85)$

The solution of Equation 48 can be written as follows:

$U_{12}\left(\xi \right)=\pm v\sqrt{\frac{6}{\alpha }}dc\left(v\xi ,m\right),\omega =1-v^2\left(m^2+1\right).(86)$

Table 2 offers a comparative analysis between classical analytical techniques and the more contemporary SMCH and BBM solution families. This comparison highlights the relative advantages and methodological developments within the field.

Table 2

Table 2. Comparison of classical methods with SMCH and BBM solution families.

5 Physical interpretation of solutions under the BBM equation

In this section, the sub-ODE method is used to derive an array of analytical solutions with varying degrees of accuracy for the BBM equation, which is a fundamental model for long-wave propagation in nonlinear dispersive media. The solutions thus obtained are interpreted through the creation of 3D, 2D, and contour plots in Maple 18, which serve to display their dynamic features. A dark soliton solution obtained from Equation 57 is depicted in Figure 8, where the values of the parameters $v=1.3$ and $\alpha =0.2$ are provided. This wave can be observed as a stable reduction in intensity over the continuous background, which physically corresponds to the context of either a density void or a wave of depression that can be represented in terms of shallow-water waves or plasma physics. The velocity parameter $v$ and the nonlinear coefficient $\alpha$ determine the soliton’s speed and the depth of the trough in the intensity profile, respectively. On the other hand, Figures 9, 10 show the bright soliton solutions from Equations 60, 63, respectively. The solutions are represented as particles or oscillating humps of energy, with $\alpha =0.8$ and $v=0.02$, and $\alpha =0.8$ and $v=0.2$, respectively. The difference in the velocity parameter between the figures is $v$, and this provides an opportunity for comparing how the speed of propagation affects the amplitude and width of the bright, stable pulses. These pulses are ubiquitous in optical fiber communications and hydrodynamics.

Figure 8

Figure 8. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 57 for the parameter values $v=1.3,\alpha =0.2$. These graphs describe the dark optical soliton under the specified constraints.

Figure 9

Figure 9. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 60 for the parameter values $\alpha =0.8;v=0.02$. These graphs describe the bright optical soliton under the specified constraints.

Figure 10

Figure 10. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 63 for the parameter values $\alpha =0.8;v=0.02$. These graphs describe the bright optical soliton under the specified constraints.

Furthermore, the investigation into the BBM equation reveals that solutions with richer topological features may also be supported. Figure 11 illustrates a singular soliton solution from Equation 66, with parameters $\alpha =0.01$ and $v=0.5$. This solution comprises a sharp, unbounded peak that is indicative of a wave-breaking scenario or, in other words, the formation of a shock-like structure within a dissipativeless medium. The strong nonlinearity of the solution due to the much smaller value of $\alpha$ compared with the dispersion expresses itself as this steep, singular profile. Complementary to the above, Figure 12 illustrates a periodic singular soliton solution from Equation 79 for $\alpha =0.001$ and $v=0.3$. The intriguing wave structure involves a periodic recurrence of singularities, suggesting a regime of coherent, repeating blow-up events, which could model phenomena in driven nonlinear lattices or certain unstable wave regimes. Regarding the development of each solution, the parameters were carefully chosen in order to ensure numerical stability and definitely isolate each type of soliton.

Figure 11

Figure 11. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 66 for the parameter values $\alpha =0.01;v=0.5$. These graphs describe the singular optical soliton under the specified constraints.

Figure 12

Figure 12. Three different graphical representations, namely, 3D surface (left panel), contour plot (middle panel), and 2D profile (right panel), of the solution of Equation 79 for the parameter values $\alpha =0.001;v=0.3$. These graphs describe the periodic singular soliton under the specified constraints.

The comprehensive graphical representation, systematically presenting the 3D, 2D, and contour plots for each solution, offers a multifaceted analysis of the wave dynamics inherent to the BBM equation. Each 3D surface plot allows for a vivid depiction of the temporal evolution and robust spatial localization of each soliton, thereby demonstrating their stability during propagation. Corresponding 2D line graphs provide an exact cross-section view of the instantaneous amplitude profile of the wave, allowing one to compare waveforms such as dark depression and bright peak clearly. Finally, the contour plots map the propagation pathways and regions of energy concentration, providing insight into the wave’s interaction potential and dispersive properties. The BBM equation’s ability to model complex wave phenomena in dispersive media is substantiated by the diverse range of wave morphologies, which are not merely dark and bright but also singular and periodic, which the equation supports. The importance of the BBM equation is thus reaffirmed through the use of these visualizations, which not only illustrate mathematical functions but also confirm the existence of wave morphologies that are very varied in nature.

This work systematically classifies its contributions to clearly delineate their novelty against established literature. Our results include solutions matching classical solitons, such as the standard bright ($sech$-type) and dark ($tanh$-type) solitary waves, which serve to validate our methods through the recovery of known results. Importantly, we generalized several known families by deriving solutions with extended parametric pre-factors, such as a velocity-dependent scaling in the periodic $tan$-type solution, allowing enhanced control over the soliton dynamics. The core novelty, however, is in the new solutions reported herein, which are inclusive of the $sec$-type singular solution $U_5$ and the suite of previously unreported Jacobi elliptic solutions $U_6,U_8,U_{10},U_{11},U_{12}$ with fully specified dispersion relations. Critical demonstration of their validity and generality is also derived from their correct reduction to known classical limits; in other words, as the modulus $m\to 1$, the elliptic solutions here correctly reduce to hyperbolic $sech/tanh$ solitons, and as $m\to 0$, they simplify into trigonometric periodic waves. Hence, this work extends the known landscape of analytical solutions, offering both a broader unifying framework and specific, novel waveforms for future application. Table 3 summarizes the obtained soliton solutions and provides a dedicated assessment of their stability. So, this table present a consolidated view of both the derived solution sand their dynamical robustness.

Table 3

Table 3. Summary of soliton solutions and stability analysis.

6 Conclusion

In this study, the sub-ODE method is effectively used to derive and analyze several forms of soliton solutions for the nonlinear SMCH and BBM equations. New traveling wave solutions involving hyperbolic, exponential, and trigonometric functions have been obtained for these nonlinear models. This approach is well-structured and effective for producing analytical solutions to nonlinear partial differential equations. Specific 3D, 2D, and contour graphs are used to illustrate the physical behaviors of the SMCH and BBM equations using Maple 18. The exact solutions obtained include dark, bright, single, and periodic solitons. Both the SMCH and BBM equations are important in the study of nonlinear wave propagation as they provide insight into the behavior of solitons in a variety of physical systems. By studying these equations and their solutions, researchers can gain a deeper understanding of fundamental nonlinear phenomena and develop innovative technologies that use soliton properties. These solutions are extremely useful, with extensive applications in engineering, optical fibers, applied mathematics, and nuclear physics.

The present work has successfully advanced the field of nonlinear wave dynamics by systematically deducing a wide spectrum of new analytical soliton solutions for two key model equations. The key novelty of the results presented lies not only in the application of the sub-ODE method to derive such solutions but also in the comprehensive characterization of their stability—a crucial step that is often overlooked in similar analytical studies. We have moved beyond simple solution generation and provided a comprehensive physical classification, confirming the existence of stable bright and dark solitons, which are essential in optical communication systems, while also identifying and simultaneously for determining unstable and singular structures that define the operation limits of such systems. The clear link drawn between specific ranges of the parameters and soliton stability is a significant contribution, providing a practical roadmap through which experimentalists can achieve these waveforms in laboratory conditions. The discovery of unique, conditionally stable soliton structures will further expand the known catalog of waveforms and suggest new directions for theoretical investigation. This work incorporates detailed visual analytics with rigorous stability criteria, hence bridging an important gap between abstract mathematical solutions and tangible physical applicability, strongly positioning our findings as a meaningful and predictive contribution to the ongoing research within the context of integrable systems and applied mathematical physics.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

TZ: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Acknowledgements

The author sincerely appreciates the editor and the reviewers for their time, insightful comments, and constructive feedback, which have significantly enhanced the quality of this manuscript.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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