Integrability, Hirota D-operator expression, multi solitons, breather wave, and complexiton of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation

1.   Introduction

As known, nonlinear partial differential (NLPD) equations have found wide applications in modeling nonlinear phenomena in vast areas of scientific disciplines, such as optics, plasma physics, and fluid mechanics. Owing to such capabilities, researchers have devoted an incredible amount of their lives to creating and exploring such models. Nowadays, scholars deal with a wide range of specific nonlinear waves for NLPD equations such as multi-soliton waves, breather waves, complexiton waves, etc. Wazwaz ^[1], by applying the Hirota method, extracted multi-soliton waves of KP equations. The author in ^[2] found breather waves of Kairat-Ⅱ and Kairat-X equations using an ansatz composed of trigonometric and hyperbolic functions. Hosseini et al. ^[3] employed a systematic method to construct complexiton waves of a generalized KdV equation. Papers ^[4,5,6,7] include more details on NLPD equations and their nonlinear waves.

One of the key characteristics of NLPD equations, which is usually explored in Mathematical Physics, is integrability. Although the literature does not provide a unified definition for integrability, an integrable NLPD equation includes multi-soliton waves, bilinear Bäcklund transformation (BBT), etc. An effective method for analyzing the integrability of NLPD equations is the Painlevè method ^[8]. Ma et al. ^[9] assessed the integrability of the Sakovich equation using the Painlevè method. Chu et al. ^[10], in a comprehensive study, applied the Painlevè method to check the integrability of a 2D KdV equation with variable coefficient. Zhang et al. ^[11] explored the integrability of a variable coefficient Boussinesq equation via the Painlevè method.

According to the classical Bell polynomials theory ^[12] proposed in 1934, Lambert et al. ^[13] introduced a generalized Bell's polynomials to establish a systematic procedure for discovering the bilinear form, bilinear Bäcklund transformation, and Lax pairs for NLPD equations. The Bell polynomial approach ^{[14,15,16,17]} has extensively been used in recent decades to deal with NLPD equations. Some authors have tried to apply such an effective method to deal with a series of well-known NLPD equations. For example, Hosseini et al. ^[18] utilized the BPA to acquire the bilinear representation of a generalized KdV equation. Umar et al. ^[19] found the BBT of a 2D generalized KP equation using the BPA. Asadi et al. ^[20] employed the BPA to construct Lax pairs and conservation laws of a 3D extended BLMP equation.

Wazwaz in ^[21] proposed the Korteweg-de Vries–Caudrey Dodd Gibbon equation, i.e.,

which has been composed of KdV and CDG equations, and constructed its multi solitons using the Hirota method. Later, some researchers conducted a complete study on the KdV–CDG equation and its different wave structures. For instance, Biswas et al. ^[22] applied the $F$-expansion method to acquire solitons of the KdV–CDG equation. Ma et al. ^[23] constructed hybrid solutions of the KdV–CDG equation using some particular operations. Hosseini et al. ^[24] employed systematic methods to extract solitons and complexiton of the KdV–CDG equation. Almusawa and Jhangeer ^[25] obtained invariant solutions of the KdV–CDG equation using the Lie method.

In this paper, we aim to conduct a detailed exploration of the following generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation

with some applications in modeling water waves in the ocean. More precisely, we:

● Examine the integrability of the gKdV–CDG equation using the Painlevè method;

● Establish the Hirota $D$-operator expression of the gKdV–CDG equation by employing the Bell polynomial approach;

● Obtain multi solitons of the gKdV–CDG equation through exerting the Hirota method;

● Construct breather and complexiton waves of the gKdV–CDG equation using distinct methods.

The paper's structure is as follows: In Section 2, an integrability test of the governing equation is carried out based on the Painlevè method. In Section 3, a short review of Bell's polynomials is presented, and the Hirota $D$-operator expression for the gKdV–CDG equation is constructed. In Section 4, multi solitons along with breather and complexiton waves to the gKdV–CDG equation are derived by serving distinct ansatzes. Additionally, in Section 4, we provide several figures positioned two- and three-dimensionally to illustrate the dynamic features of nonlinear waves. The results are summarized at the end of the paper.

2.   Integrability test of the governing equation

Owing to the efforts of Weis et al. ^[8], the Painlevé property of the governing equation can be formally investigated. The key idea of their method is to seek the solution of Eq (1.1) as follows

where $\mathrm{\Phi }\left(x, t\right)$ is a singular manifold, ${u}_{j}\left(x, t\right)$ are expansion coefficients, and $\alpha$ is a pole order of the solution $u(x, t)$.

The Painlevé test, however, can rarely be performed directly using the above method because of numerous complications. To overcome this shortcoming, we use Kruskal's simplification ^[26,27], which employs a specific form for the singular manifold function as

The Painlevé's test consists of three steps as follows:

First step: It involves determining the leading-order terms in Eq (1.1). To achieve this, substituting

into Eq (1.1) results in the values of $\alpha$ and ${u}_{0}\left(t\right)$ as

● 1st branch: $\alpha = 2, {u}_{0} = -60{\psi }_{x}^{2}$;

● 2nd branch: $\alpha = 2,$ ${u}_{0} = -30{\psi }_{x}^{2}$.

Second step: For the above-specified values of $\alpha$ and ${u}_{0}$, the non-negative integer values of $j$ referred to as resonances are computed. To this end, setting

in Eq (1.1) gives

● resonances of 1st branch: $j = -2, -1, 5, 6, 12$;

● resonances of 2nd branch:$j = -1, 2, 3, 6, 10$.

Third step: The expansion

is substituted into Eq (1.1). For the first branch, it is found that

whereas for the second branch

Since ${u}_{5}$, ${u}_{6}$ and ${u}_{12}$ in the first branch and ${u}_{2}{, u}_{3},$ ${u}_{6}$, and ${u}_{10}$ in the second branch are arbitrary functions of $t$, the necessary condition for the integrability of Eq (1.1) is satisfied. These show that Eq (1.1) satisfies the Painlevé test for the integrability.

3.   Bell polynomials and the Hirota D-operator expression

As early as the 20th century, Bell ^[12] established his polynomial in the form

The above definition leads to the following results

The classical Bell polynomial, which was generalized by Lambert et al. ^[13], for $f({x}_{1}, \dots, {x}_{l})$ is

where the outcomes of such a generalization when $f = f\left(x, t\right)$ are

Following Bell's definition, the binary Bell polynomial is

where ${r}_{k} = \mathrm{0, 1}, \dots, {n}_{k}, k = \mathrm{0, 1}, \dots, l$. It is easy to check that

Theorem 1 (See ^[13]). For ${\mathcal{Y}}_{{n}_{1}{x}_{1}, \dots, {n}_{l}{x}_{l}}\left(v, w\right)$ and ${D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}F.G$, we have

where ${n}_{1}+{n}_{2}+\dots +{n}_{l}\ge 1$, and

Equation (3.1) for $F = G$ is

where

Theorem 2. If $u = 30{\left(\mathrm{ln}\left(G\right)\right)}_{xx},$ then the Hirota $D$-operator expression of Eq (1.1) is

Proof. To prove such an assertion, we introduce

with $c\left(t\right)$ is a function to connect Eq (1.1) with P-polynomials. Inserting Eq (3.2) into Eq (1.1) gives

Integrating with respect to $x$ yields

A direct result of comparing $\frac{1}{15}c{\left(t\right)}^{2}{q}_{2x}^{3}+c\left(t\right){q}_{2x}{q}_{4x}+{q}_{6x}$ with ${P}_{6x}\left(q\right)$ is $c\left(t\right) = 15$. Now, the above equation can be expressed as

Through the transformation

and Eq (3.3), then the Hirota $D$-operator expression of Eq (1.1) is given by

with the following bilinear representation

4.   The gKdV-CGD equation and its nonlinear waves

In this section, we focus on the governing model in order to establish its multi solitons, breather wave, and complexiton. The first kind of such results is supported by analyzing the three-soliton condition as a criterion for the existence of a triple-soliton wave.

4.1.   Multi solitons of the governing equation

To arrive at the single-soliton wave, we assume

and insert it into

After simplifying, we have

A direct result of the above expression gives the dispersion relation as

and so, the phase variable ${\theta }_{i}$ can be constructed as

Now, the single-soliton wave of Eq (1.1) is

where

The double-soliton wave of the governing equation can be determined by inserting

where

into Eq (1.1) and discovering the phase shift ${a}_{12}$ through some systematic computations as

Hence, the double-soliton wave of the governing equation is

where

In order to have the triple-soliton wave, the three-soliton condition ^[28,29,30]

in which

must be zero. The results show that

where

is zero. For Eq (1.1), the triple-soliton wave exists as a result of the above condition.

Now, the triple-soliton wave of Eq (1.1) is

where

The single, double, and triple solitons for (a) $\left\{{c}_{1} = 1, {c}_{2} = 1, {k}_{1} = 0.7\right\}$, (b) $\left\{{c}_{1} = 1, {c}_{2} = 1, {k}_{1} = 0.7, {k}_{2} = 1\right\}$, and (c) $\left\{{c}_{1} = 1, {c}_{2} = 1, {k}_{1} = \frac{4}{5}, {k}_{2} = 1, {k}_{3} = \frac{9}{7}\right\}$ have been portrayed in Figure 1. Using the above parameter regimes, the height and width of such waves can be assessed.

4.2.   Breather and complexiton waves of the governing model

To obtain the breather wave of the governing equation, we apply an ansatz as

where

and $k$, ${b}_{0}$, $h$, ${b}_{1}$, and ${a}_{i}$, $i = \mathrm{1, 2}, \dots, 6$ are unknowns. Inserting Eq (4.1) into Eq (3.4) results in

whose solution is

As a consequence, the breather wave of the governing equation is

where

Figure 2 represents the dynamical characteristics of the breather wave for

in three- and two-dimensional postures. This figure demonstrates the height and width of such a wave for the above parameter regime.

To derive the complexiton wave, the following assumptions are taken:

Due to $P\left(\mu, \nu \right) = 0$, we obtain a nonlinear system of algebraic equations as

The solution set for the above system is

Additionally, the phase shift can be found as

Now, the complexiton wave of the governing equation is

where

The dynamical feature of the complexiton wave positioned three- and two-dimensionally for

has been given in Figure 3. Through the above parameter regime, the height and width of such a wave can be analyzed. More details about the origin of the complexiton wave can be found in ^[31].

4.3.   Other nonlinear waves of the governing equation

Other nonlinear waves of the governing equation can be derived by serving the following ansatzes ^[32]:

1) $u = A{\mathrm{S}\mathrm{N}}^{2}\left(x-wt, k\right),$

2) $u = A{\mathrm{C}\mathrm{N}}^{2}\left(x-wt, k\right),$

3) $u = A{\mathrm{D}\mathrm{N}}^{2}\left(x-wt, k\right),$

where $A$ and $w$ are unknowns.

By setting the first ansatz in Eq (1.1), we have a nonlinear system as

for which the solution set is

As a result, the Jacobi elliptic solution of the governing equation is

Considering $k = 1$, it gives the following bright soliton

Additionally, by substituting the second ansatz in Eq (1.1), we arrive at the following nonlinear system

whose solution is

Accordingly, the Jacobi elliptic solution of the governing equation is

Assuming $k = 1$, we yield the following soliton wave

Now, setting the third ansatz in Eq (1.1) leads to a nonlinear system as

with the following solution set

As a consequence, the Jacobi elliptic solution of the governing equation is

Letting $k = 1$, it can obtain the following soliton wave

The first continuous wave and its corresponding bright wave have been depicted in three- and two-dimensional postures in Figure 4 when (a) $\left\{{c}_{1} = 1, {c}_{2} = 1, k = 0.1\right\}$ and (b) $\left\{{c}_{1} = 1, {c}_{2} = 1\right\}.$ This figure illustrates the height and width of such waves for the above parameter regimes.

5.   Conclusions

In this paper, we presented an in-depth study of specific oceanic waves based on a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation. As a starting point, we constructed the Hirota $D$-operator expression of the gKdV–CDG equation by using the Bell polynomial approach. Based on the Painlevè analysis, the governing model was tested for integrability, and multi solitons were formally retrieved. As a result of symbolic computations, breather and complexiton waves were derived from the gKdV–CDG equation based on distinct ansatzes. For the dynamical features of nonlinear waves, a few representations positioned two- and three-dimensionally have been provided. Our findings suggest useful ways for assessing the height and width of nonlinear waves in the ocean. In light of the fact that some other waves are missing in this paper, future work can be devoted to finding these waves.

Author contributions

Kamyar Hosseini: Supervision, Conceptualization, Writing–original draft; Farzaneh Alizadeh: Writing–original draft, Methodology, Investigation; Sekson Sirisubtawee: Supervision, Methodology, Investigation, Writing–original draft, Funding acquisition; Chaiyod Kamthorncharoen: Methodology, Investigation, Writing–review & editing; Samad Kheybari: Writing – review & editing; Kaushik Dehingia: Writing–review & editing.

All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have used Artificial Intelligence (AI) tools (Grammarly and Wordtune) to improve their writing quality.

Acknowledgments

This research budget was allocated by National Science, Research and Innovation Fund (NSRF), and King Mongkut's University of Technology North Bangkok (Project no. KMUTNB-FF-68-B-18).

Conflict of interest

The authors declare that they have no conflicts of interest.