The oblique soliton waves along the interface between layers of different densities in stratified fluids

1 Introduction

The nonlinear partial differential equations are fundamental mathematical tools used to describe and analyze systems where quantities depend on multiple independent variables, typically space and time. They appear in science, engineering, economics, and more. Non-linear factors appear in many processes, and the nature of dependencies between variables is more intricate. In the last few decades, the wave phenomenon involving the surface of water has been the subject of considerable interest among scholars from diverse disciplines. Abnormal waves do not only contribute to the understanding of ocean and coastal engineering. Still, they are also significant in all these fields of science and engineering: communications, plasma, tsunami waves, fluids, and many more [1], [2], [3], [4], [5]. Thus, the behavior of such complex phenomena is described with great detail and clarity by soliton solutions that help to explain nonlinear wave actions. Scientists and analysts at the crossroads of mathematics and physics have extensively applied several analytical and computational tools that can be used to confront NLEEs, using solutions that have pushed forward the limits of their practical applicability to estimate and manage these problematic and widespread phenomena. Soliton, lumps, stability analysis, and modulation instability for extended (2 + 1)-dimensional Boussinesq modeling shallow water was proved by Fazal et al. [6]. Lump, periodic, traveling, semi-analytical solutions, and stability analysis for the Ito integro-differential equation arising in shallow water waves were established by Bakhash et al. [7]. The Gardner-Kadomtsev-Petviashvili (Gardner-KP) equation is a nonlinear differential equation, which itself is a uni-tensor extension of the Korteweg-de Vries (KdV) equation and a bi-tensor generalization of the Kadomtsev-Petviashvili (KP) equation. This equation describes the wave character in cases where nonlinear dispersion plays a significant role and where the spatial relations of the wave field are accounted for, at the very least, in two coordinates. The Gardner-KP equation is an improved variant of the KdV equation that adds higher-order nonlinearity found in the KP equation, together with the multi-dimensional aspect. The general form of the Gardner-KP equation is given [8]:

The Gardner-KP equation helps to analyze waves in physical situations involving fluids, plasmas, and optics. The nonlinear arms represent the main nonlinear actions in the wave field, while the dispersive term takes care of the dispersion of the waves. The multi-dimensional approach allows one to study the behavior of waves in more realistic situations where the wave field depends on the coordinates in many directions.

Solutions of linear PDEs can be combined to produce other solutions, but this property does not apply to nonlinear equations, which makes them extremely challenging to solve and analyze. Some of the frequently used analytical technique are the inverse scattering transform method, tanh-expansion method, new auxiliary equation method, Hirota’s direct method, extended direct algebraic technique, and (G/G′)-expansion approach, which provide analytical exact solutions that give deep insight into the underlying dynamics. Almatrafi et al. [9], [10], [11], [12] have studied many physical models and derived classical and fractional soliton solutions via diverse analytical methods. Nonlinear equations are rather significant in diversified fields of science and engineering. Some among them are optical fibers, thermodynamics and heat transfer, elasticity and plasticity, momentum and energy, measuring instruments, fluids and solids, sound and light, space, time, and matter [13], [14], [15], [16], [17].

In mathematics, solitons are explicit solutions of the modeling of integrable systems [18], [19], [20], [21] usually characterized by permanent, intense waveforms that carry an unchanging profile and velocity. The remarkable phenomenon known as solitary waves may be caused to collide, allowing the solitons to pierce one another and continue on their journey. Sometimes the pair just uses a change in phase to signal the meeting. Since the groundbreaking work of Zabusky and Kruskal [22] that first demonstrated the soliton’s unique properties within the framework of the KdV equation, their collisional behaviors have been the focus of investigations and have remained of interest. Thus, Zabusky and Gardner’s work provided a solid foundation for contemporary studies on solitons and contributed to the discovery of mathematical structures, including the inverse scattering transform and further advancement in various fields, from fluid dynamics to optical fiber communications. Many other fields of modern sciences can also emerge in this filed which are more significant [23], [24], [25], [26] A soliton consists of a single ridge and trough, maintaining its size and speed indefinitely because of a balance between distortion and dispersion. This equilibrium allows solitons to traverse vast distances without shape degradation. Recent methodological advances have expanded solution techniques for fractional-order systems, enhancing our understanding of nonlinear wave dynamics across disciplines [27].

This ongoing research discloses the intimate relationship between solitons and integrable systems and enhances our understanding of nonlinear dynamics and mathematical physics. Soliton is a term used to describe a certain sort of solution to a specific nonlinear. It consists of one ridge and trough and is non-breaking or shallow. It remains the same size and moves at the same rate for an extended time, even when encountering other waves. This is because nonlinearity, which tends to distort the soliton, tends to be countered by dispersive effects, which tend to spread the soliton out. This balance enables solitons to cover large distances and, at the same time, preserve their shape; for this reason, they differ from other types of waves. Please include information on the research on solitons and their properties, which are very useful for different scientific topics. The autonomous multiple wave solutions and hybrid phenomena to a (3 + 1) − dimensional Boussinesq-type equation in fluid media were proved by Hajar et al. [28]. Construction of degenerate lump solutions for the (2 + 1)-dimensional Yu-Toda-Sasa-Fukuyama equation was established by Li and Li in [29]. Using an extended version of physics-informed neural networks (PINNs) with interface zones, conducted studies on data-driven localized wave generation and parameter detection in the massive significant models reported [30], [31], [32], [33], [34].

Some useful techniques used to find soliton solutions are; the F-expansion method [35], the Hirota bilinear approach [36], 37], Kudryashov technique [38], translation method [39], G/G′-expansion method [40], tanh-function technique [41], fractional dual-functional approach [42], exponential-function approach [43], sine-cosine method [44], inverse scattering transform method [45], homogeneous balance method [46], ϕ ⁶-model expansion technique [47], Sardar sub-equation improved scheme [48], generalized exponential rational function methods [49] and direct algebraic methods [50]. These are just a few examples of techniques for solving nonlinear differential equations. Each method provides different benefits for simplifying complex equations to obtain exact solutions. As a result, they have been extensively employed in mathematical and physical analysis to find solutions for issues like nonlinear dynamics, wave propagation, etc. Thus, algorithms based on them can be quite handy within the range of methodologies for solving wide classes of nonlinear.

This work is motivated by the literature discussed above and aims to transform the Gardner-KP partial differential equation into an ODE using an appropriate wave transformation. This approach enhances our understanding of the complex PDE, facilitating its study and solution. Many dynamical aspects of the considered model were mystery and not been discussed. Thus, in order to demonstrate these insights, the Hamiltonian function generated to develop the planer dynamical system that help to visualized the chaotic, periodic and quasi-periodic and also studied the sensitivity of the system for testing different factors. We then employ the newly developed direct extended algebraic equation method to obtain exact soliton solutions for the resulting ODE. Soliton solutions are highly valuable due to their non-dissipative nature and ability to maintain their shape and velocity upon interaction with other solitons or waves. These solutions are particularly useful for understanding the nonlinear dispersive characteristics of waves in multi-dimensional media.

The rest of the paper is organized as follows: In Section 2, the new direct algebraic method is introduced and described in detail, followed by a rigorous explanation of its formulation and implementation. Section 3 reduces the PDE to an ODE through a suitable transformation and then analyzes it. In Section 4, the soliton solution of the ODE is derived and then accompanied by a graphical representation describing the shape and properties of the soliton. The paper is concluded in Section 5 with a discussion of findings and perspectives for future research.

2 Description of new extended direct algebraic technique

The new extended direct algebraic technique [51] is a powerful and systematic method of finding exact solutions for nonlinear differential equations (PDEs). By incorporating extra algebraic structures and transformations, the technique goes beyond classical algebraic methods and becomes more effective in solving complicated nonlinear equations. The following describes this method in great detail: Assume a general partial differential equation,

using the transformation,

Inserting Eq. (3) into Eq. (2) yields,

Assume that the solution of a general ODE (4) is,

where,

where, γ, α and β are real constants.

The general solutions with respect to parameters α, β and γ are:

(Case 1): When β ² − 4αγ < 0 and γ ≠ 0. Then, the solutions of Eq. (6) are given by

(Case 2): When β ² − 4αγ > 0 and γ ≠ 0. Then, the solutions of Eq. (6) are given by

(Case 3): When αγ > 0 and β = 0. Then, the solutions of Eq. (6) are given by

(Case 4): When αγ < 0 and β = 0. Then, the solutions of Eq. (6) are given by

(Case 5): When β = 0 and α = γ. Then, the solutions of Eq. (6) are given by

(Case 6): When β = 0 and γ = −α. Then, the solutions of Eq. (6) are given by

(Case 7): When β ² = 4αγ. Then, the solutions of Eq. (6) are given by

(Case 8): When β = p, α = pq, (q ≠ 0) and γ = 0. Then, the solutions of Eq. (6) are given by

(Case 9): When β = γ = 0. Then, the solutions of Eq. (6) are given by

(Case 10): When β = α = 0. Then, the solutions of Eq. (6) are given by

(Case 11): When α = 0 and β ≠ 0. Then, the solutions of Eq. (6) are given by

(Case 12): When β = p, γ = pq, (q ≠ 0 and α = 0). Then, the solutions of Eq. (6) are given by

The deformation parameters m, n > 0.

3 The formation of ordinary differential equation

The general procedure of transforming a PDE into an ODE usually entails reducing the number of independent variables used in the PDE through a specific transformation. The Gardner-KP is a nonlinear differential equation of the Korteweg-de Vries (KdV) type and the Kadomtsev-Petviashvili (KP) equations. It simulates wave behavior in scenarios when effects such as no nonlinear dispersion is essential and when spatial dependencies of the wave field are considered at least in two coordinates.

Using the transformation,

Eq. (1) is reduced into ODE as,

Integrating Eq. (45) and setting the constant of integration one time equal to zero yield,

where ϵ is the constant of integration.

Eq. (46) is often used in cases involving fluid dynamics, nonlinearity, and other systems where wave interactions are essential. Thus, the present approach describes how a function U is modified by experiencing nonlinear dispersion over a domain and force applications.

It may depict the evolution of an optical field U under the impact of both linear and nonlinear processes in the setting of nonlinear light propagation. The dispersive term represents the group velocity dispersion, whereas the quadratic and cubic factors may represent self-phase modulation and other nonlinear processes. The dispersion term indicates the dispersion of the wave crest, therefore the equation may represent the motion of waves in a fluid. In plasma dynamics, this equation seemed to explain the wave packets in a plasma medium with prominent nonlinear reactions and dispersion effects. It can describe the phenomenon of solitary waves or solitons, which are stable wave packets in which the shape of the individual waves does not change during the wave’s progression. One can obtain stable soliton solutions depending on the relative contribution of the nonlinear terms, which tend to steepen the wave, and dispersive terms, which tend to spread the wave out.

3.1 The formulation of Hamiltonian function for Eq. (46)

The framework offered by the Hamiltonian function is profound and adaptable for comprehending and resolving differential systems. It is an essential instrument in contemporary research because of its uses in physics, stability analysis, numerical simulations, and even quantum mechanics [52]. Thus, the implementation of Galilean transformation yields system from Eq. (46), let us take U ′ = S ,

(47) d U d ϖ = S , d S d ϖ = − 1 κ 4 ( κ ω + λ 2 ) U + 3 κ 2 U 2 + 2 κ 2 U 3 + ϵ .

Interestingly, the first order system (47) is a planer Hamiltonian system. The dynamical system (47)’s Hamiltonian function on the integration of system (47) may be confirmed.

(48) H ( U , S ) = S 2 2 + 1 κ 4 ( κ ω + λ 2 ) 2 U 2 + κ 2 U 3   + κ 2 2 U 4 + ϵ U = h .

From Eq. (48), one can confirm that

(49) d U d ϖ = ∂ H ∂ S   and   d S d ϖ = − ∂ H ∂ U .

The planner Hamiltonian nature of system (47) is evident from Eq. (49). The phase trajectories generated by the ecosystems will reflect all of the soliton solutions of Eq. (46) since system (47) is conservative. Based on the energy level h of Eq. (48), the level curves H h ( U , S ) may be defined as follows.

H h = { ( U , S ) ∈ R × R : H ( U , S ) = h } .

Every energy level h in the phase portrayal has an orbit. Firstly, establish a connection to examine the correlation between the system’s closed orbits and energy level. For instance,

(50) Ψ h ( U ) = h − 1 κ 4 ( κ ω + λ 2 ) 2 U 2 + κ 2 U 3 + κ 2 2 U 4 + ϵ U .

(51) S = 2 h − 2 κ 4 ( κ ω + λ 2 ) 2 U 2 + κ 2 U 3 + κ 2 2 U 4 + ϵ U .

It is evident that Ψ h ( V ) = S 2 2 (Figure 1).

Figure 1:

Different energy levels at various scales.

3.2 Chaos analysis

The article analyzes three types of dynamical system patterns through this segment. Gradual system visualization becomes possible by utilizing the perturbation factor g ₀ cos(d, t) within system (47), which represents a Hamiltonian planner dynamical system. When taken together, system described in system (47) finds expression through the following equation:

(52) d U d ϖ = S , d S d ϖ = − 1 κ 4 ( κ ω + λ 2 ) U + 3 κ 2 U 2 + 2 κ 2 U 3 + ϵ + g 0 ⁡ cos ( d , t ) .

The perturbation frequency stands as g ₀ while having its strength denoted by d. The perturbed system (52) has superficial power added into its structure, whereas system (47) does not contain this component.

The research investigates frequency effects on the governing model through static physical parameter conditions. The analysis reveals how these two perturbation factors affect the system’s behavior. The three-dimensional representation together with the two-dimensional view and time-series section of the perturbed dynamical planner system (52) can be observed in Figure 2, when d = 2.1π, g ₀ = 0.1, and the initial condition (0.05, 0.001) is applied.

Figure 2:

The periodic behavior with the perturbation factor of a dynamical planner Hamiltonian system. (a) Phase orbit with perturbation factor. (b) Phase-space diagram. (c) Alteration between S versus U. (d) Time series section.

This system depicts a quasi-periodic behavior when d = 0.01π and g ₀ = 10.1 using the same initial condition (0.05, 0.001) as shown in Figure 3.

Figure 3:

The quasi-periodic behavior with the perturbation factor of a dynamical planner Hamiltonian system. (a) Phase orbit with perturbation factor. (b) Phase-space diagram. (c) Alteration between S versus U. (d) Time series section.

The time-series section and two-dimensional graphs together with three-dimensional graphs for d = 0.1π, g ₀ = 10.1 with the same initial condition (0.05, 0.001) display chaotic behavior in the perturbed dynamical planner shown in Figure 4.

Figure 4:

The chaotic behavior with the perturbation factor of the dynamical planner Hamiltonian system. (a) Phase orbit with perturbation factor. (b) Phase-space diagram. (c) Alteration between S versus U. (d) Time series section.

3.3 Sensitivity analysis

In this part, the planner dynamical system’s sensitivity of system (47) will be shown. The case studies demonstrate how even small adjustments to the initial values may cause propagating waves to fluctuate dramatically. It illustrates the system’s sensitivity to its initial value (Figures 5–14).

Figure 5:

Sensitivity shown visually.

Figure 6:

Sensitivity shown visually.

Figure 7:

Sensitivity shown visually.

Figure 8:

Sensitivity shown visually.

Figure 9:

Sensitivity shown visually.

Figure 10:

Sensitivity shown visually at κ = 10.

Figure 11:

Sensitivity shown visually at κ = 1.111.

Figure 12:

Sensitivity shown visually at κ = 0.667.

Figure 13:

Sensitivity shown visually at κ = 0.344.

Figure 14:

Sensitivity shown visually at κ = 0.111.

4 Construction of analytical exact soliton solution

The considered partial differential equation (PDE) has been transformed into an ordinary differential equation (ODE) by applying the traveling wave transformation. By implementing this transformation, the complexity of the original PDE is reduced, allowing it to be analyzed as an ODE. Subsequently, a new approach to auxiliary equation will be employed to derive exact analytical solutions. This method systematically tackles the transformed equation, leading to precise solutions that provide deeper insights into the system’s behavior. The effectiveness of this approach highlights its potential for solving a wide range of non-linear systems.