Analyzing Sensitivity and Solitonic Behavior Using the Dullin-Gottwald-Holm Model in Shallow Water Waves

Scientists and engineers are becoming ever more intrigued by the research of nonlinear wave phenomena in a modern global era of science and technology. In many scientific disciplines, nonlinear partial differential equations (NLPDEs) are essential for comprehending nonlinear wave phenomena that constitute our day-to-day challenges. Due to its various uses and applications throughout the past few decades, NLPDEs have therefore attracted a lot of attention in the field of nonlinear sciences. The NLPDEs are crucial in describing the dynamical, physical, and physical processes in many different scientific domains, including ocean engineering, optical fibers, fluid mechanics, plasma physics, quantum physics, geology, biology, and many othersr [1], [2], [3], [4], [5], [6]. The behavior of waves in several domains is explained by NLPDEs. In many different branches of science, particularly the study of complex nonlinear wave phenomena, the precise solution of NPDEs is crucial to understanding many physical processes. Since these effective computing packages make it easier to perform complex and laborious algebraic computations, finding the exact solutions has recently grown in importance among scholars [7], [8], [9], [10], [11]. The exact solutions of NLPDEs, especially the solitary wave solutions, have a special significance in the soliton theory of mathematical physics. These NLPDE solutions provide more convincing evidence of its physical structures [12], [13], [14]. As a result, numerous strong and effective algorithms were developed to derive the exact results for highly nonlinear models. The new auxiliary scheme, out of all of these techniques, is the most effective and reliable for obtaining the precise solution to NLPDEs [15], [16], [17].

Analytical exploration and discussion of the $DGH$ system [18] are presented in this research. It has been addressed from a variety of angles, despite not having been analytically addressed yet. The single and combined forms of wave solutions are retrieved under specific parameter conditions, making it pertinent to investigate them analytically. Additionally, the sensitivity analysis is addressed to evaluate how a physical system behaves under stable beginning conditions. The following is how the (1+1)-dimensional $DGH$-model for shallow water waves is expressed:

where the velocity of the fluid of the aforementioned system is denoted by $\mathcal{Q}(x,t)$ in the spatial direction $x$. The coefficients $\varpi^{2}(\varpi>0)$ and $\theta/\Lambda$ mention squares of length scales, while $\theta=\sqrt{gh}$ (where $\theta=2\omega$) denotes the linear wave speed for undisturbed water at rest at spatial infinity. The remarkable repercussions of this $DGH$ system is similar to those of other unidirectional shallow water wave equations.

The $DGH$ equation is a well-known nonlinear evolution equation that is the subject of this article. Many scientists have carefully studied the equation under consideration in order to explain its many aspects. This model has so far been the subject of work. Mustafa [19] has identified the existence and uniqueness of low regularity patterns. Zhou et al. [20] investigated peakon-anti peakon interaction with direct computation. Xiao et al. [21] discussed exact results by employing the traveling-wave transformation and the exp-function technique. The use of qualitative planar systems to describe limited exact traveling wave solutions was examined [22]. Periodic wave solutions were discovered by Meng et al. [23] using integral bifurcation and semi-inverse techniques. To assure a single solution, the ansatz method [24] was also applied. Yu [25] provided examples of the dynamical behavior of traveling wave solutions and related bifurcations in a variety of parameter ranges. da Silva [26] proposed the categorization of bounded traveling wave solutions.

Although this equation has been solved using a variety of analytical and numerical techniques, the new auxiliary equation method (NAEM) has not been performed to examine the governing model before. This approach has also been used to explore different models in multiple studies [27]. On the other hand, this approach makes it much simpler to solve the $DGH$ problem to find novel solutions. Since its beginnings, this approach has gained support from the assessment profession for its straightforward estimation procedure. Future nonlinear scientific discussions will benefit from the findings of this study by having accessibility to excellent research. Future study on this topic may make use of the $DGH$ equation in a fractional sense by applying fractional derivatives because many nowadays researches are focused on solving NLPDEs formulated in the sense of fractional calculus [28], [29], [30].

The layout of this article is as follows: Section (2) provides the fundamental idea of the technique. Soliton's solutions have been taken out for the equation in Section (3). Section (4) presents the findings and a discussion of the graphical depiction of the solutions. The sensitive performance of the system is represented in Section (5). Section (6) concludes with final comments.

The solitary wave solutions to the equation under explored are determined by using an analytical scheme that is outlined in this section. A general example of how NLPDE is constructed is the statement that follows:

Regarding a specific variable, $P$ is a polynomial function. To transform Eq. (2) into an essential sense of NLODE, take propagational transformation as $\mathcal{Q}(x,t)=U(\zeta)$, where, $\zeta=x-\nu~t$, then

The $U$ superscripts signify $U$'s derivative with respect to $\zeta$, thus a polynomial with both linear and nonlinear terms makes up the function $R$. Using the idea of NAEM, it is now possible to assume the initial solution of Eq. (2) as:

which satisfy the auxiliary equation

where, $\mathfrak{f}_{0}$, $\mathfrak{f}_{1}$, $\mathfrak{f}_{2}$, ..., $\mathfrak{f}_M$ are unknown coefficients such that $\mathfrak{f}\neq0$. We can estimate the value of $M$ in accordance with balancing principle by equating the highest nonlinear element in Eq. (3) with the higher level derivative. Here, the various scenarios for possible solutions to Eq. (5) are described.

Case 1: For $\Omega\neq0$ and $\Phi^{2}-4\Theta\Omega<0$,

or

Case 2: When $\Phi^{2}-4\Theta\Omega>0$ and $\Omega\neq0$,

or

Case 3: When $\Phi^{2}+4\Theta^{2}<0$, $\Omega\neq0$ and $\Omega=-\Theta$,

or

Case 4: When $\Phi^{2}+4\Theta^{2}>0$, $\Omega\neq0$ and $\Omega=-\Theta$,

or

Case 5: When $\Phi^{2}-4\Theta^{2}<0$ and $\Omega=\Theta$,

or

Case 6: When $\Phi^{2}-4\Theta^{2}>0$ and $\Omega=\Theta$,

or

Case 7: When $\Phi^{2}=4\Theta\Omega$,

Case 8: For $\Theta\Omega<0$, $\Phi=0$ and $\Omega\neq0$,

or

Case 9: When $\Theta=-\Omega$ with $\Phi=0$,

Case 10: For $\Theta=\Omega=0$,

Case 11: For $\Theta=\Phi=K$, $\Omega=0$,

Case 12: When $\Omega=\Phi=K$ and $\Theta=0$,

Case 13: When $\Phi=\Theta+\Omega$,

Case 14: When $\Phi=-\Theta-\Omega$,

Case 15: When $\Theta=0$,

Case 16: When $\Theta=\Phi=\Omega\neq0$,

Case 17: When $\Phi=\Omega=0$,

Case 18: When $\Phi=\Theta=0$,

Case 19: For $\Theta=\Omega$ and $\Phi=0$,

Case 20: For $\Omega=0$,

The objective of this section is to explore into the problem's current traveling wave solutions. We begin with wave transformation to resolve the aforementioned system as $\mathcal{Q}(x,t)=U(\zeta)$ where $\zeta=x-\nu~t$ such that $\nu\neq 0$. The transformation is substituted into Eq. (1) to produce the following form:

Eq. (33) is once integrated with respect to $\zeta$ with an integration constant of zero and after some simplification, the above equation becomes

We just apply the homogeneous balance principle to Eq. (34) in order to determine $M$, resulting in $M=2$. Now, Eq. (4) has the following shape:

By putting Eq. (35) with Eq. (5) into Eq. (34), a system of equations is established by putting all coefficients of different powers of} $\lambda^{\psi(\zeta)}$ to zero.

$\begin{aligned} & \left(\lambda^{\psi(\zeta)}\right)^0:-2 \varpi^2 \Theta^2 \nu \mathfrak{f}_2+2 \varpi^2 \Theta^2 \mathfrak{f}_0 \mathfrak{f}_2-\Theta \Phi \Lambda \mathfrak{f}_1-3 / 2 \mathfrak{f}_0^2+\nu \mathfrak{f}_0-\mathfrak{f}_0 a_0-2 \Theta^2 \Lambda \mathfrak{f}_2+ \\ & 1 / 2 \varpi^2 \Theta \nu^2 \mathfrak{f}_1{ }^2-\varpi^2 \Theta \Phi \mathfrak{f}_1+\varpi^2 \Theta \Phi \mathfrak{f}_0 \mathfrak{f}_1, \\ & \left(\lambda^{\psi(\zeta)}\right)^1: 4 \varpi^2 \Theta^2 \mathfrak{f}_1 \mathfrak{f}_2-6 \varpi^2 \Theta \nu \Phi \mathfrak{f}_2-2 \varpi^2 \Theta \nu \Omega \mathfrak{f}_1+6 \varpi^2 \Theta \Phi \mathfrak{f}_0 \mathfrak{f}_2+2 \varpi^2 \Theta \Phi \mathfrak{f}_1{ }^2+ \\ & 2 \varpi^2 \Theta \Omega \mathfrak{f}_0 \mathfrak{f}_1-\varpi^2 \nu \Phi^2 \mathfrak{f}_1+\varpi^2 \Phi^2 \mathfrak{f}_0 \mathfrak{f}_1-6 \Theta \Phi \Lambda \mathfrak{f}_2-2 \Theta \Omega \Lambda \mathfrak{f}_1-\Phi^2 \Lambda \mathfrak{f}_1+ \\ & \nu \mathfrak{f}_1-3 \mathfrak{f}_0 \mathfrak{f}_1-\mathfrak{f}_1 a_0, \\ & \left(\lambda^{\psi(\zeta)}\right)^2: 4 \varpi^2 \Theta^2 \mathfrak{f}_2{ }^2+3 / 2 \varpi^2 \Phi^2{\mathfrak{f}_1}^2-4 \Phi^2 \Lambda \mathfrak{f}_2-8 \varpi^2 \Theta \nu \Omega \mathfrak{f}_2+11 \varpi^2 \Theta \Phi \mathfrak{f}_1 \mathfrak{f}_2+ \\ & 8 \varpi^2 \Theta \Omega \mathfrak{f}_0 \mathfrak{f}_2-3 \varpi^2 \nu \Phi \Omega \mathfrak{f}_1+3 \varpi^2 \Phi \Omega \mathfrak{f}_0 \mathfrak{f}_1-3 / 2 \mathfrak{f}_1^2+\nu \mathfrak{f}_2-\mathfrak{f}_2 a_0- \\ & 3 \mathfrak{f}_0 \mathfrak{f}_2+3 \varpi^2 \Theta \Omega \mathfrak{f}_1{ }^2-4 \varpi^2 \nu \Phi^2 \mathfrak{f}_2+4 \varpi^2 \Phi^2 \mathfrak{f}_0 \mathfrak{f}_2-8 \Theta \Omega \Lambda \mathfrak{f}_2-3 \Phi \Omega \Lambda \mathfrak{f}_1, \\ & \left(\lambda^{\psi(\zeta)}\right)^3: 10 \varpi^2 \Theta \Phi \mathfrak{f}_2{ }^2+14 \varpi^2 \Theta \Omega \mathfrak{f}_1 \mathfrak{f}_2-10 \varpi^2 \nu \Phi \Omega \mathfrak{f}_2-2 \varpi^2 \nu \Omega^2 \mathfrak{f}_1+7 \varpi^2 \Phi^2 \mathfrak{f}_1 \mathfrak{f}_2+ \\ & 10 \varpi^2 \Phi \Omega \mathfrak{f}_0 \mathfrak{f}_2+4 \varpi^2 \Phi \Omega \mathfrak{f}_1^2+2 \varpi^2 \Omega^2 \mathfrak{f}_0 \mathfrak{f}_1- \\ & 10 \Phi \Omega \Lambda \mathfrak{f}_2-2 \Omega^2 \Lambda \mathfrak{f}_1-3 \mathfrak{f}_1 \mathfrak{f}_2, \\ & \left(\lambda^{\psi(\zeta)}\right)^4: 6 \varpi^2 \Phi^2{\mathfrak{f}_2}^2+5 / 2 \varpi^2 \Omega^2 \mathfrak{f}_1{ }^2-6 \Omega^2 \Lambda \mathfrak{f}_2+17 \varpi^2 \Phi \Omega \mathfrak{f}_1 \mathfrak{f}_2-3 / 2 \mathfrak{f}_2{ }^2+ \\ & 12 \varpi^2 \Theta \Omega \mathfrak{f}_2^2-6 \varpi^2 \nu \Omega^2 \mathfrak{f}_2+6 \varpi^2 \Omega^2 \mathfrak{f}_0 \mathfrak{f}_2, \\ & \left(\lambda^{\psi(\zeta)}\right)^5: 14 \varpi^2 \Phi \Omega \mathfrak{f}_2{ }^2+10 \varpi^2 \Omega^2 \mathfrak{f}_1 \mathfrak{f}_2, \\ & \left(\lambda^{\psi(\zeta)}\right)^6: 8 \varpi^2 \Omega^2{\mathfrak{f}_2}^2 . \\ & \end{aligned}$

By using the Maple software to solve the provided system, the following acceptable solution is produced:

The following is the result of inserting Eq. (36) into Eq. (35):

Eq. (37) yields a variety of surface wave solutions when the solutions identified by Eq. (5) are substituted:

Case 1: When $\Phi^{2}-4\Theta\Omega<0$ and $\Omega\neq0$,

or

Case 2: When $\Phi^{2}-4\Theta\Omega>0$ and $\Omega\neq0$,

or

Case 3: When $\Phi^{2}+4\Theta\Omega<0$, $\Omega\neq0$ and $\Omega=-\Theta$,

or

Case 4: When $\Phi^{2}+4\Theta\Omega>0$, $\Omega\neq0$ and $\Omega=-\Theta$,

or

Case 5: When $\Phi^{2}-4\Theta^{2}<0$ and $\Omega=\Theta$,

or

Case 6: When $\Phi^{2}-4\Theta^{2}>0$ and $\Omega=\Theta$,

or

Case 7: When $\Phi^{2}=4\Theta\Omega$,

Case 8: $\Theta \Omega<0$, $\Phi=0$ and $\Omega\neq0$,

or

Case 9: When $\Phi=0$ and $\Theta=-\Omega$,

Case 12: When $\Omega=\Phi=K$ and $\Theta=0$,

Case 13: When $\Theta+\Omega=\Phi$,

Case 14: When $-(\Theta+\Omega)=\Phi$,

Case 15: When $\Theta=0$,

Case 16: When $\Phi=\Theta=\Omega\neq0$,

Case 19: When $\Theta=\Omega$ and $\Phi=0$,

We show a visual representation of these few wave structures that are discovered in the system under study in this section. We employ NAEM to acquire twenty-two precise solutions, and when we compare these solutions, it is clear that the suggested method provides more solutions than the one currently used in the literature for the solution of the $DGH$ equation [19], [26]. By using the proposed technique, wave structures such as kink, rational, hyperbolic, and singular-type wave results are obtained and visually shown via $2D$, $3D$, and their contours plot. These waveforms have various physical significance, as the diagrams demonstrate. For instance, hyperbolic functions like the hyperbolic tangent are used in the computation and speed of special relativity, while the Langevin function for magnetic polarization involves the hyperbolic cotangent [31]. This relationship to the governing model can therefore be explained using the results presented in this work. The findings of this investigation will contribute as motivation and inspiration for the next nonlinear scientific discussions. Our approach is simpler to calculate than previous approaches and is more straightforward, succinct, and effective, and in comparison, to the standard direct methods, it offers more accurate solutions. Figure 1 depicts bright-singular solitons, Figure 2 and Figure 3 showcase bright solitons, Figure 4 illustrates kink solitons, Figure 5 represents singular solitons, Figure 6 and Figure 7 display dark-singular solitons, and Figure 8 and Figure 9 also illustrate kink solitons.

The sensitive behavior of the suggested model after being transformed into a system is covered in this section. This analysis determines how sensitive our system is to the specified initial and parametric conditions. The system is said to be insensitive if a little change in the beginning conditions only results in a tiny change in the system. Since the system will alter tremendously if the initial conditions vary even slightly, it is highly sensitive. The major objective of this inquiry is to accurately quantify the output disruption caused by changes in input. Here, we'll use the idea of sensitivity analysis to examine Eq. (1). We convert Eq. (31) into the planer dynamical system as a result of studying Galilean transformation:

where, $A=\big(\varpi^{2}(U-c)-\Lambda\big)$, $B=\big(\theta-\nu\big)$, $C=\frac{3}{2}$ and $D=\frac{1}{2}\varpi^{2}$. The above system of equations contains initial conditions namely as: $(0.3, 0.4)$, $(0.3, 0.45)$, $(0.25, 0.45)$, and $(0.3, 0.4)$. The analysis results may be exhibited to show how little changes in the input can cause large variations in the outcome. The results are displayed using a variety of parametric values. Figure 10 reflects the sensitivity of the system taking first initial conditions and Figure 11 shows the behavior of the system when we take a little change in initial conditions and some parameters.

The dynamics of novel solitons to the $DGH$ equation has been discussed in this article. In addition to several unique solutions, such as the well-known traveling wave solutions, the achieved solutions include hyperbolic, kink, brilliant, singular, and rational function solutions. NAEM, which is more efficient and has families of solutions, is the technique used to find the solutions. By choosing the appropriate values for the parameter, we also drew some plots to represent the physical behavior of the solutions that were presented. Physically, it can be seen that wave behavior has reported their estimated wave distributions and propagation in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, and Figure 9. The calculations also show how useful this method is for locating the exact results from a wider perspective. By selecting various initial conditions and parametric values, the sensitivity analysis to the considered system is also thoroughly examined in Figure 10 and Figure 11. The findings of this study add to the analysis of the $DGH$ equation. The findings of this study will provide future nonlinear scientific conversations with high-quality literature. One may use $DGH$ equation in a fractional sense applying new fractional derivatives to extend this work in the future [32], [33], [34], [35].