Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed

Raj Kumar; Ravi Shankar Verma

doi:10.1016/j.joes.2022.04.015

Journal of Ocean Engineering and Science >

2024 , Vol. 9 >Issue 2: 154 - 163

DOI: https://doi.org/10.1016/j.joes.2022.04.015

Research article

Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed

Raj Kumar ^,^* ,
Ravi Shankar Verma

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Department of Mathematics, Faculty of Engineering & Technology, Veer Bahadur Singh Purvanchal University, Jaunpur 222003, India

^*E-mail addresses: rsoniraj2@gmail.com (R. Kumar)

rsverma747@gmail.com (R.S.Verma).

Received date: 2022-02-12

Revised date: 2022-04-15

Accepted date: 2022-04-18

Online published: 2022-04-22

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Abstract

The system of (1+1 )-coupled Drinfel'd-Sokolov-Wilson equations describes the surface gravity waves travelling horizontally on the seabed. The objective of the present research is to construct a new variety of analytical solutions for the system. The invariants are derived with the aid of Killing form by using the optimal algebra classification via Lie symmetry approach. The invariant solutions involve time, space variables, and arbitrary constants. Imposing adequate constraints on arbitrary constants, solutions are represented graphically to make them more applicable in designing sea models. The behavior of solutions shows asymptotic, bell-shaped, bright and dark soliton, bright soliton, parabolic, bright and kink, kink, and periodic nature. The constructed results are novel as the reported results [26,28,29,30,33,38,42,49] can be deduced from the results derived in this study. The remaining solutions derived in this study, are absolutely different from the earlier findings. In this study, the physical character of analytical solutions of the system could aid coastal engineers in creating models of beaches and ports.

Highlights

● System of coupled Drinfel'd-Sokolov-Wilson equations describes waves travelling horizontally on the seabed.

● The killing form is used to get the invariants.

● One dimensional optimal sub algebras are classified and analytical solutions are attained via one parameter Lie symmetry analysis.

● The solutions are asymptotic, bell-shaped, bright and dark soliton, bright soliton, parabolic, bright and kink, kink, and periodic in nature.

Key words： Coupled Drinfel'd-Sokolov-Wilson model; Invariants; Killing form; Optimal sub algebra; Lie symmetry reductions

Cite this article

Raj Kumar , Ravi Shankar Verma . Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed[J]. Journal of Ocean Engineering and Science, 2024 , 9(2) : 154 -163 . DOI: 10.1016/j.joes.2022.04.015

1. Introduction

1.1. Scope

In the physical sciences, nonlinear partial differential equations (NPDEs) are frequently employed to describe complicated processes. Thus, the problems/models arising in ocean science [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], oceanography [24], mathematical physics [25], complex fluid flows [26], [27], plasma physics [28], electromagnetic theory, fluid dynamics [29], [30], nuclear physics, chemical physics, to name a few [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69] are not easily solvable and finding their analytical solutions is critical. Reviewing the existing literature [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], the authors are motivated to solve the following (

1 + 1

)-coupled Drinfel'd-Sokolov-Wilson equations (CDSWEs) system analytically. The system is governed by

(1)

\begin{matrix} u_{t} + a v v_{x} = 0, \\ v_{t} + b v_{x x x} + c u v_{x} + d v u_{x} = 0, \end{matrix}

where

a

b

c

and

d

are non-zero parameters,

x

t

are space and time variables, respectively, while

u

and

v

depict the components of nonlinear surface gravity waves travelling horizontally on the seabed.

Gravity waves are formed in a fluid medium or at the interface of two media when gravity or buoyancy trying to restore equilibrium. The contact between the atmosphere and the ocean, which causes wind waves, is an example of such an interaction. In deeper water, a long water wave moves faster. In deeper water, gravity waves have a faster phase speed than in shallow water. Shallow water waves (SWW) are helpful to classify the marine environment, investigate ocean dynamics, and model equatorial tsunami waves. During their propagation, SWW are influenced by the ocean floor, causing the orbital motion of water to be disrupted. As a result, they may cause underwater earthquakes, and unimaginable damage to the coastal ecology. On the other hands, the heights, wave lengths, and time durations of gravity waves frequently vary as they advance in different directions [1]. Gravity waves in shallow water are dispersion (water waves) and non-dispersive as the depth is substantially smaller than the wavelength. Nonlinear interactions between triads of wave components with frequencies

(ω)

and vector wave numbers

(k)

satisfying following conditions which affect ocean surface gravity waves [2].

(2)

\begin{matrix} | ω_{1} \pm ω_{2} \pm ω_{3} | & = δ ω, \\ | k_{1} \pm k_{2} \pm k_{3} | & = δ k . \end{matrix}

1.2. Origin of the problem

It is essential to consider the following form Hirota et al. [3] of coupled Korteweg-de Vries (KdV):

(3)

\begin{matrix} 2 u_{t} - (u_{x x x} + 6 u u_{x}) = 4 b v v_{x}, \\ v_{t} + v_{x x x} + 3 u v_{x} = 0, \end{matrix}

which represents the interaction of two long waves with distinct dispersion relations, in which

b

is a non-zero parameter and

u

and

v

are similar as described in CDSWEs (1).

The system (3) is integrable and can be reformulated as six-reductions of the Kadomtsev-Petviashvili hierarchy. The general construction of the KdV system (3) can be deduced using affine Lie algebras [3], while Drinfel'd and Sokolov [4] established that the general construction of the KdV system (3) can be derived using affine Lie algebras (ALA). Wilson established a link between the Kac-Moody Lie algebra [5] and KdV (3). Wilson [6] used ALA to obtain a particular variant of Eq. (1) with

a = - 3

b = - 2

c = - 2

, and

d = - 1

as inputs. One of the universal models is the CDSWEs (1), which have an infinite number of conservation laws and an integrable system [3].

1.3. Literature review

The authors are inspired by the characteristics of CDSWEs (1). The research community [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50] solved the system (1) using different approaches and addressing different concerns. Singh et al. [25] employed a fractional time derivative and a homotopy perturbation technique to get a solitary wave solution in the Caputo sense. The travelling wave solutions were obtained by Khan et al. [26] using a modified simple equation approach. Lu et al. [27] have employed first integral technique and obtained exponential and rational functions types solutions. Homotopy analysis has been used by Arora and Kumar [28] to establish a convergent series with two initial conditions. To get a solitary wave solution, Arnous et al. [29] applied two approaches to obtain a solitary wave solution which were the Bäcklund transformation of the Riccati equation and the trial function approach. Using the F-expansion method, Akbar and Ali [30] derived hyperbolic, rational, single soliton, and periodic solutions.

For solving time-fractional DSW systems and obtaining a solution in the form of a truncated series, Chen et al. [31] compared the coupled fractional reduced differential transform method with the residual power series method. Tariq and Seadawy [32] tackled the equation using an auxiliary equation and derived traveling and solitons solutions. The Jacobi elliptic function method was employed by Yao [33] and attained kink, bell-shaped, singular, and periodic solutions. Lu et al. [34] exploited the extended auxiliary and equation mapping methods to obtain a solution in the form of a moving and solitary wave for the same system of equations with odd and even order partial derivatives. For the CDSWEs (1), Wen et al. [35] applied bifurcation and qualitative analysis of dynamical systems to obtain solitary wave, periodic, kink shaped solutions.

Javeed et al. [36] used the first integral approach to construct a nontrivial solution to a coupled space-time fractional DSW system with Riemann-Liouville and conformable types of derivatives. For the space-time fractional derivative of the classical DSW system, Shehata et al. [37] employed the Kudryashov method to obtain a travelling wave solution. In order to determine unique periodic and solitary wave solutions for CDSWEs (1), Zhang [38] employed a semi-inverse variational approach, whereas Bibi et al. [39] used the Tanh and expanded Tanh methods and generated kink and soliton solutions. Bhatter et al. [40] exploited fractional homotopy analysis and obtained bell-shaped solutions to the nonlinear CDSWEs using a fractional operator. The

\frac{G^{'}}{G}

-expansion method was used by Matjila et al. [41], Shi et al. [42], and Khan et al. [43] who obtained different forms of the solutions like travelling waves, hyperbolic, and trigonometric, rational, bell-shaped soliton, kink, and solitary waves.

Hirota et al. [3] established a special type of solitons for CDSWEqs. (1) i.e. static solitons, if

a = - 3

b = c = - 2

, and

d = 1

. They [3] claimed that when using Painlevé to deform CDSWEs into Bilinear form, such solitons can not be deformed when interacting with other moving solitons, whereas Ali Akbar et al. [44] used the modified alternative

\frac{G^{'}}{G}

-expansion method to extract travelling wave solutions for the same form of equations.

Using

a = 3

b = c = 2

, and

d = 1

in system (1), Zhao and Zhi [45] utilised the F-expansion approach to get Jacobi elliptic, rational, solitary, and periodic solutions; and Ullah et al. [46] used the optimal homotopy method to provide doubly periodic wave solutions. The Adomian decomposition method was used by Inc [47] to determine the system's Jacobi elliptic solution

A different form of CDSWEs is examined for specific values of

a = 2

b = - a

c = 3 k

, and

d = 3 b

in Eq. (1), Morris and Kara [48] established a travelling wave solution using Lie symmetry and conservation laws. Using Ibragimov's technique, Zhang and Zhao [49] created a one-dimensional optimal system. They [49] solved the additional reductions using the simple equation approach and then extracted the polynomial solutions, whereas Zhao et al. [50] applied Lie symmetry analysis and conservation laws to the identical form of DSWEs but did not solve the reductions further. This paved the way for more solutions of DSWEs.

1.4. Motivation

Making use of the similarity transformations method (STM) via one-parameter Lie symmetry analysis, the authors derived several types of novel analytical solutions. Sophus Lie created the Lie symmetry analysis as an ad-hock integration technique for the system of PDEs. The system of differential equations is still invariant after each similarity reduction. In a system of PDEs, the STM decreases the number of independent variables by one. Using the STM repeatedly can obviously reduce the system to an equivalent system of ordinary differential equations (ODEs). Due to the invariance of STM, a system of PDEs might result in a linear system of new over-determining system of PDEs that enables one to find infinitesimal generators that are functions of the independent and dependent variables. Such adjustments, which employ Lagrange's equation as a tool, result in similarity variables and similarity functions. Substituting the value of the dependent variable in terms of a similarity function can reduce the given system of PDEs to a new system. This process is repeated until one gets a system of ODEs. On solving it, it provides solutions to the system. For a description of the STM and its uses, one might look through the extensive literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69] and the references therein.

1.5. Outline

This article is structured as follows: In the next section, invariants are derived by killing form and then one-dimensional optimal sub algebras are generated. In Section 3, a new class of analytical solutions is constructed by using Lie symmetry analysis. Section 4 depicts the comparison with the reported results till now. Section 5 comprises a detailed discussion about the nature of solutions and their physical analysis. Finally, a summary of the whole article is given in the conclusion.

2. Invariants by using Killing form

The essential terminology for the methodology is presented in the following. The one-parameter

(ϵ)

Lie symmetry transformations can be considered as:

(4)

\begin{matrix} x^{*} & = x + ϵ ξ (χ) + O (ϵ^{2}), t^{*} = t + ϵ τ (χ) + O (ϵ^{2}), \\ u^{*} & = u + ϵ η^{(1)} (χ) + O (ϵ^{2}), and \\ v^{*} & = v + ϵ η^{(2)} (χ) + O (ϵ^{2}) \end{matrix}

where

ξ

τ

η^{(1)}

and

η^{(2)}

are the infinitesimals of the variables

x

t

u

, and

v

respectively. The notation

x^{*}

means the value of

x

in transformed space. The notation

(χ)

is equivalent to (

x

t

u

v

). The vector field V generated by the infinitesimal transformations

(5)

\begin{matrix} V = ξ \frac{\partial}{\partial x} + τ \frac{\partial}{\partial t} + η^{(1)} \frac{\partial}{\partial u} + η^{(2)} \frac{\partial}{\partial v} . \end{matrix}

The invariance conditions (termed in Bluman and Cole [68], Olver [69]) for the CDSWEs (1) can be written as:

(6)

\begin{matrix} {P r}^{(1)} V [u_{t} + a v v_{x}] = 0, and \\ {P r}^{(3)} V [v_{t} + b v_{x x x} + c u v_{x} + d v u_{x}] = 0, \end{matrix}

where the first prolongation

{P r}^{(1)}

and the third prolongation

{P r}^{(3)}

are

(7)

\begin{matrix} {P r}^{(1)} V & = V + [η_{t}^{(1)}] \frac{\partial}{\partial (u_{t})} + [η_{x}^{(2)}] \frac{\partial}{\partial (v_{x})}, and \\ {P r}^{(3)} V & = V + [η_{t}^{(2)}] \frac{\partial}{\partial (v_{t})} + [η_{x}^{(1)}] \frac{\partial}{\partial (u_{x})} + [η_{x}^{(2)}] \frac{\partial}{\partial (v_{x})} \\ + [η_{x x x}^{(2)}] \frac{\partial}{\partial (v_{x x x})}, \end{matrix}

where

\begin{matrix} [η_{t}^{(1)}] & = & \frac{\partial η^{(1)}}{\partial t} + \frac{\partial η^{(1)}}{\partial u} θ_{t} + [\frac{\partial η^{(1)}}{\partial v} - \frac{\partial τ}{\partial t}] ϕ_{t} - \frac{\partial ξ}{\partial t} ϕ_{x} \\ - \frac{\partial ξ}{\partial u} θ_{t} ϕ_{x} - \frac{\partial τ}{\partial u} θ_{t} ϕ_{t} - \frac{\partial ξ}{\partial v} ϕ_{x} ϕ_{t} - \frac{\partial τ}{\partial v} ϕ_{t}^{2} and \\ [η_{x}^{(2)}] & = & \frac{\partial η^{(2)}}{\partial x} + \frac{\partial η^{(2)}}{\partial u} θ_{x} + [\frac{\partial η^{(2)}}{\partial v} - \frac{\partial ξ}{\partial x}] ϕ_{x} - \frac{\partial τ}{\partial x} ϕ_{t} \\ - \frac{\partial ξ}{\partial u} θ_{x} ϕ_{x} - \frac{\partial τ}{\partial u} θ_{x} ϕ_{t} - \frac{\partial ξ}{\partial v} ϕ_{x}^{2} - \frac{\partial τ}{\partial v} ϕ_{x} ϕ_{t}, etc. \end{matrix}

are extensions.

Then over-determining equations are

(8)

\begin{matrix} ξ_{t} & = ξ_{u} = ξ_{v} = τ_{x} = τ_{t t} = τ_{u} = τ_{v} = 0, \\ ξ_{x} & = \frac{1}{3} τ_{t}, η^{(1)} = - \frac{2}{3} u τ_{t}, and η^{(2)} = - \frac{2}{3} v τ_{t} . \end{matrix}

By solving them, one can get

(9)

\begin{matrix} ξ = a_{1} + a_{2} x, τ = a_{3} + 3 a_{2} t, \\ η^{(1)} = - 2 a_{2} u, and η^{(2)} = - 2 a_{2} v, \end{matrix}

where

a_{1}

a_{2}

, and

a_{3}

are arbitrary constants.

The Lie symmetry algebra for CDSWEs (1) can be generated by

\begin{matrix} X_{1} & = \frac{\partial}{\partial x}, X_{2} = x \frac{\partial}{\partial x} + 3 t \frac{\partial}{\partial t} - 2 u \frac{\partial}{\partial u} - 2 v \frac{\partial}{\partial v}, and X_{3} = \frac{\partial}{\partial t} . \end{matrix}

Lemma 2.1. Let

X = \sum_{i = 1}^{3} a_{i} X_{i}

be an arbitrary element of

L^{3}

, where

a_{1}

a_{2}

a_{3}

\in

R

. Then Symmetry algebra admits an arbitrary invariant function of the form

F (a_{2})

, where

F

is an arbitrary function.

Proof. Let

ψ

be an invariant function in Lie algebra

L^{3}

then,

ψ [A d (g) X] = ψ (X)

\forall

X \in L^{3}

and

g \in G

, where

G

stands for the symmetry group. Now

X = \sum_{i = 1}^{3} a_{i} X_{i}

and

W = \sum_{i = 1}^{3} b_{i} X_{i}

, then

\begin{matrix} A d (e x p (ϵ W)) X & = e^{- ϵ W} X e^{ϵ W} \\ = X - ϵ [W, X] + \frac{ϵ^{2}}{2} [W, [W, X]] - \dots \\ = (a_{1} X_{1} + a_{2} X_{2} + a_{3} X_{3}) - (Θ_{1} X_{1} \\ + Θ_{2} X_{2} + Θ_{3} X_{3}) + 0 (ϵ^{2}) . \end{matrix}

Now, with the assistance of a commutator Table 1, one can get

(10)

\begin{matrix} Θ_{1} = b_{1} a_{2} - b_{2} a_{1}, Θ_{2} = 0, and Θ_{3} = - 3 b_{2} a_{3} + 3 b_{3} a_{2} . \end{matrix}

Then the corresponding system of linear DEs recasts as:

(11)

\begin{matrix} a_{2} \frac{\partial ψ}{\partial a_{1}} = 0, a_{1} \frac{\partial ψ}{\partial a_{1}} + 3 a_{3} \frac{\partial ψ}{\partial a_{3}} = 0, and 3 a_{2} \frac{\partial ψ}{\partial a_{3}} = 0 . \end{matrix}

□

Table 1 Commutator table for (1).

*	$X_{1}$	$X_{2}$	$X_{3}$
$X_{1}$	0	$X_{1}$	0
$X_{2}$	$- X_{1}$	0	$- 3 X_{3}$
$X_{3}$	0	$3 X_{3}$	0

Lemma 2.2. The killing form is

K (X, X) = 10 a_{2}^{2}

, where

X = \sum_{i = 1}^{3} a_{i} X_{i}

for Lie algebra

L^{3}

Proof. The Lie algebra's killing form is provided by

K (X, X) = T r a c e [a d X a d X]

, where

\begin{matrix} A d X = (\begin{matrix} a_{2} & - a_{1} & 0 \\ 0 & 0 & 0 \\ 0 & - 3 a_{3} & 3 a_{2} \end{matrix}) \end{matrix}

So, the killing form is

K (X, X) = 10 a_{2}^{2}

. □

The adjoint relation is calculated using a Lie series that follows

A d (e x p (ϵ W)) X = X - ϵ [W, X] + \frac{1}{2} ϵ^{2} [W, [W, X]] - \dots

Lemma 2.3.

B = a_{2}

is an invariant.

Proof. Table 3 is a direct consequence of the proof. □

Table 2 Adjoint table for (1).

$A d$	$X_{1}$	$X_{2}$	$X_{3}$
$X_{1}$	$X_{1}$	$X_{2}$ - $ϵ X_{1}$	$X_{3}$
$X_{2}$	$e^{ϵ} X_{1}$	$X_{2}$	$e^{3 ϵ} X_{3}$
$X_{3}$	$X_{2}$	$X_{2} - 3 ϵ X_{3}$	$X_{3}$

Table 3 Construction of invariant function for (1).

Coeff of $\to$	$X_{1}$	$X_{2}$	$X_{3}$
$A d_{(e x p ϵ X_{1})} X$	$a_{1} - a_{2} ϵ$	$a_{2}$	$a_{3}$
$A d_{(e x p ϵ X_{2})} X$	$e^{ϵ} a_{1}$	$a_{2}$	$e^{3 ϵ} a_{3}$
$A d_{(e x p ϵ X_{3})} X$	$a_{1}$	$a_{2}$	$a_{3} - 3 ϵ a_{2}$

Lemma 2.4. I is an invariant, where

I = {\begin{matrix} 1, & if a_{2}^{2} + a_{3}^{2} \neq 0 \\ 0 & otherwise. \end{matrix}

Proof. Table 3 shows that the adjoint action

A d (e x p (ϵ X_{1}))

keeps invariant to the coefficients of the

X_{2}

and

X_{3}

. Hence I is an invariant.

(12)

\begin{matrix} {\bar{a}}_{2}^{2} + {\bar{a}}_{3}^{2} = a_{2}^{2} + {(a_{3} e^{3 ϵ})}^{2} \end{matrix}

(13)

\begin{matrix} {\bar{a}}_{2}^{2} + {\bar{a}}_{3}^{2} = a_{2}^{2} + {(a_{3} - 3 ϵ a_{2})}^{2} \end{matrix}

From Eqs. (12), (13), it is clear that

{\bar{a}}_{2} = {\bar{a}}_{3} = 0

iff

a_{2} = a_{3} = 0

, where

{\bar{a}}_{i}

i = 2

, 3 is an adjoint action. □

Lemma 2.5. Let

S = {\begin{matrix} s g n (a_{3}), & a_{3} \neq 0, a_{2} = 0 \\ 0 & otherwise \end{matrix}

where

S

is a sgn function.

Then

S (X) = S (A d (exp (ϵ X))),

for any vector

X = \sum_{i = 1}^{3} a_{i} X_{i}

with

a_{2} = 0

and

A d (exp (ϵ X_{i})) = \sum_{i = 1}^{3} a_{i} X_{i}

with

{\bar{a}}_{2} = 0

for

i = 1, 2, 3

Proof. The adjoint action

A d (exp (ϵ X_{1}))

do not the change sgn of

a_{3}

i . e .

the coefficient of

X_{3}

. □

Lemma 2.6. Let

K = {\begin{matrix} s g n (a_{1}), & a_{1} \neq 0, a_{2} = 0 \\ 0 & otherwise. \end{matrix}

where

K

is a sgn function.

Then

K (X) = K (A d (e x p (ϵ X))),

for

X

with

a_{2} = 0

and

A d (e x p (ϵ X_{i})) = \sum_{i = 1}^{3} a_{i} X_{i}

with

{\bar{a}}_{2} = 0

and

i = 1, 2, 3

Proof. By Lemma 2.5, that can be proved. □

Construction of adjoint matrix:

Apply the adjoint action of

X_{1}

to X,

\begin{matrix} A d (exp (ϵ X_{1})) X & = a_{1} X_{1} + a_{2} X_{2} + a_{3} X_{3} \\ = (a_{1} + a_{2} + a_{3}) A_{1} {(X_{1} + X_{2} + X_{3})}^{T}, \end{matrix}

where

\begin{matrix} A_{1} = (\begin{matrix} 1 & 0 & 0 \\ - ϵ_{1} & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) . \end{matrix}

The matrices

A_{2}

and

A_{3}

can be calculated as:

\begin{matrix} A_{2} = (\begin{matrix} e^{ϵ_{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & e^{3 ϵ_{2}} \end{matrix}), A_{3} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - 3 ϵ_{3} \\ 0 & 0 & 1 \end{matrix}) \end{matrix}

and

(14)

\begin{matrix} A = A_{1} A_{2} A_{3} = (\begin{matrix} e^{ϵ_{2}} & 0 & 0 \\ - ϵ_{1} e^{ϵ_{2}} & 1 & - 3 ϵ_{3} \\ 0 & 0 & e^{3 ϵ_{2}} \end{matrix}) \end{matrix}

A is termed as global adjoint matrix.

Lemma 2.7. The one-dimensional optimal system of Lie algebra

L^{3}

for CDSWEs (1) is

P_{1} = X_{2}

, and

P_{2} = X_{1} + a_{3} X_{3}

where,

a_{3} \in {1, - 1, 0}

Proof. The adjoint transformation can be calculated as

(15)

\begin{matrix} ({\bar{a}}_{1}, {\bar{a}}_{2}, {\bar{a}}_{3}) = (a_{1}, a_{2}, a_{3}) A \end{matrix}

It recasts as

(16)

\begin{matrix} {\bar{a}}_{1} = a_{1} e^{ϵ_{2}} - a_{2} ϵ_{1} e^{ϵ_{2}}, {\bar{a}}_{2} = a_{2}, and {\bar{a}}_{3} = - 3 ϵ_{3} a_{2} + a_{3} e^{3 ϵ_{2}} . \end{matrix}

Further, solutions to (16) for

ϵ_{1}

ϵ_{2}

, and

ϵ_{3}

can be obtained as:

Step (I): If

a_{2} = 1

, taking

ϵ_{2} = 0

, for

{\bar{a}}_{1} = 0

{\bar{a}}_{2} = 1

, and

{\bar{a}}_{3} = 0

, then the system has a solution

ϵ_{1} = a_{1}

, and

ϵ_{3} = \frac{a_{3}}{3}

, thus the representative element is

P_{1} = X_{2}

Step (II): If

a_{2} = 0

, and

a_{1} \neq 0

, then the authors took

a_{1} = 1

, and

ϵ_{2} = 0

, then system (16) follows three representative as

X_{1} + a_{3} X_{3}

, where

a_{3} \in {1, - 1, 0}

. i.e.

P_{2} = X_{1} + X_{3}

P_{3} = X_{1} - X_{3}

, and

P_{4} = X_{1}

□

3. Similarity solutions

This section shows how to derive invariant solutions and similarity reductions for different classes of optimum sub algebra. All of the cases can be categorized as:
Case (1) For

P_{1} = X_{2} = x \frac{\partial}{\partial x} + 3 t \frac{\partial}{\partial t} - 2 u \frac{\partial}{\partial u} - 2 v \frac{\partial}{\partial v}

, the characteristics equations can be written as

(17)

\begin{matrix} \frac{d x}{x} = \frac{d t}{3 t} = - \frac{d u}{2 u} = - \frac{d v}{2 v}, \end{matrix}

which yields the similarity variable

X = x t^{- \frac{1}{3}}

and similarity functions as

u = F (X) t^{- \frac{2}{3}}

and

v = G (X) t^{- \frac{2}{3}}

. Thus CDSWEs (1) reduces into an equivalent form

(18)

\begin{matrix} X F^{'} - 3 a G G^{'} + 2 F = 0, and \end{matrix}

(19)

\begin{matrix} 3 b G^{″'} + 3 c F G^{'} + 3 d F^{'} G - X G^{'} - 2 G = 0 . \end{matrix}

Prime shows the derivative of a function with respect to the indicated variable.
Multiply Eq. (18) by

G (X)

and by

F (X)

in Eq. (19), then adding them we have an ODE. In which taking

F = G

, and after twice integration, one can get

(20)

\begin{matrix} \frac{b}{2} {(F^{'})}^{2} + \frac{d + c - a}{6} F^{3} = C_{1} F + C_{2}, \end{matrix}

where

C_{1}

and

C_{2}

are constants of integration.
If

C_{2} = 0

, then solution for Eq. (1) can be explored as

(21)

\begin{matrix} u_{1} (x, t) = v_{1} (x, t) = F t^{- \frac{2}{3}}, \end{matrix}

where F can be expressed as:

\begin{matrix} E l l i p t i c F (\sqrt{1 + \frac{F}{α}}, \frac{1}{\sqrt{2}}) = (\mp x t^{- \frac{1}{3}} + α + C_{3}) \sqrt{C_{1} / b}, \end{matrix}

where

α = \sqrt{\frac{6 C_{1}}{d + c - a}}

and

C_{3}

is a constant of integration.
Another particular solution of (18) and (19) can be obtained treating

F

and

G

F = \frac{X}{c + d}

and

G = \pm \frac{1}{\sqrt{a (c + d)}} X

,

then solution of the system (1) is

(22)

\begin{matrix} u_{2} (x, t) = \frac{x}{(c + d) t}, and v_{2} (x, t) = \frac{1}{\sqrt{a (c + d)}} \frac{x}{t} . \end{matrix}

Case (2) For

P_{2} = X_{1} \pm X_{3} = \frac{\partial}{\partial x} \pm \frac{\partial}{\partial t}

, the corresponding Lagrange's characteristic equation is

(23)

\begin{matrix} d x = \pm d t = \frac{d u}{0} = \frac{d v}{0} . \end{matrix}

It yields

X_{1} = x \mp t

and

u = F (X_{1})

v = G (X_{1}) .

Then, similarity reduction of the CDSWEs (1) is

(24)

\begin{matrix} a G G^{'} \mp F^{'} = 0, \end{matrix}

(25)

\begin{matrix} b G^{″'} + c F G^{'} + d F^{'} G \mp G^{'} = 0 . \end{matrix}

The authors have used double signs (

\pm

\mp

) wherever in the whole manuscript, which are considered upper sign with upper and lower sign with lower.
Taking Eq. (24) and integrating it yields

(26)

\begin{matrix} F = \pm \frac{a}{2} G^{2} + C_{4}, \end{matrix}

where

C_{4}

is a constant of integration.
From Eqs. (25), (26), getting

(27)

\begin{matrix} b G^{″'} \pm \frac{α_{1}}{2} G^{2} G^{'} + α_{2} G^{'} = 0, \end{matrix}

where

α_{1} = a c + 2 a d

and

α_{2} = c C_{4} - 1

are constants and twice integrating yields

(28)

\begin{matrix} {(G^{'})}^{2} \pm \frac{α_{1}}{12 b} G^{4} + \frac{α_{2}}{b} G^{2} - C_{5} G = C_{6}, \end{matrix}

where

C_{5}

and

C_{6}

are constants of integration.

Case (2a) For

α_{1} = \pm 3 k^{3}

b = k

α_{2} = - \frac{1}{4} k^{3}

, and

C_{5} = C_{6} = 0

, the solution for the CDSWEs (1) is

(29)

\begin{matrix} u_{3} (x, t) & = \pm \frac{a}{2} s e c h^{2} [\frac{k (x \mp t)}{2} + C_{7}] \\ + C_{4}, and \\ v_{3} (x, t) & = s e c h [\frac{k (x \mp t)}{2} + C_{7}] . \end{matrix}

Case (2b) Assuming

α_{1} = \mp 3 k^{5}

b = k

α_{2} = - \frac{1}{2} k^{3}

C_{5} = 0

, and

C_{6} = \frac{1}{4}

, the solution for the system (1) can be obtained as

(30)

\begin{matrix} u_{4} (x, t) & = \pm \frac{a}{2 k^{2}} \tan^{2} [\frac{k (x \mp t)}{2} + C_{8} k] \\ + C_{4}, and \\ v_{4} (x, t) & = \frac{1}{k} \tan [\frac{k (x \mp t)}{2} + C_{8} k] . \end{matrix}

Case (2c) For

α_{1} = \mp 3 k^{5}

b = k

α_{2} = \frac{1}{4} k^{5}

, and

C_{5} = C_{6} = 0

, the solution for the system (1) are

(31)

\begin{matrix} u_{5} (x, t) & = \pm \frac{a}{2} \sec^{2} [\frac{k^{2} (x \mp t)}{2} + C_{9}] \\ + C_{4}, and \\ v_{5} (x, t) & = \sec [\frac{k^{2} (x \mp t)}{2} + C_{9}]; \end{matrix}

(32)

\begin{matrix} u_{6} (x, t) & = \pm \frac{a}{2} c s c^{2} [\frac{k^{2} (x \mp t)}{2} + C_{10}] \\ + C_{4}, and \\ v_{6} (x, t) & = \csc [\frac{k^{2} (x \mp t)}{2} + C_{10}] . \end{matrix}

Case (2d) For

α_{1} = 3 k^{3}

b = - k

, and

α_{2} = C_{5} = C_{6} = 0

, the solution is

(33)

\begin{matrix} u_{7} (x, t) & = \pm \frac{a}{2} [\frac{2}{\pm k (x \mp t) + C_{11}}]^{2} + C_{4}, and \\ v_{7} (x, t) & = \frac{- 2}{\pm k (x \mp t) + C_{11}} . \end{matrix}

Case (2e) Putting

α_{1} = \mp 3 k^{5}

b = k

α_{2} = \frac{1}{2} k^{3}, C_{5} = 0

, and

C_{6} = \frac{1}{4}

, the solutions can be expressed as

(34)

\begin{matrix} u_{8} (x, t) & = \pm \frac{a}{2 k^{2}} \tanh^{2} [\pm \frac{k (x \mp t)}{2} + C_{12}] \\ + C_{4}, and \\ v_{8} (x, t) & = \pm \frac{1}{k} \tanh [\pm \frac{k (x \mp t)}{2} + C_{12}]; \end{matrix}

(35)

\begin{matrix} u_{9} (x, t) & = \pm \frac{a}{2} [\frac{2}{1 - \exp (k (x \mp t) + C_{13})} - 1]^{2} \\ + C_{4}, and \\ v_{9} (x, t) & = \frac{1}{k} [\frac{2}{1 - \exp (k (x \mp t) + C_{13})} - 1] . \end{matrix}

Case (2f) Assuming

α_{1} = 3 k^{3}

b = \mp k

α_{2} = \pm \frac{1}{4} k^{3}

, and

C_{5} = C_{6} = 0

, the solution of CDSWEs is

(36)

\begin{matrix} u_{10} (x, t) & = \pm \frac{a}{2} c s c h^{2} [\frac{k (x \mp t)}{2} + C_{14}] \\ + C_{4}, and \\ v_{10} (x, t) & = c s c h [\frac{k (x \mp t)}{2} + C_{14}] . \end{matrix}

Case (2g) For

c = 2 k_{1}

d = - k_{1}

α_{2} = k^{2}

C_{5} = 0

C_{6} = \frac{k^{2}}{b}

, and

b \neq 0

, the solution of system (1) is

(37)

\begin{matrix} u_{11} (x, t) & = \pm \frac{a}{2} \sin^{2} [\frac{k (x \mp t)}{\sqrt{b}} + C_{15}] \\ + C_{4}, and \\ v_{11} (x, t) & = \sin [\frac{k (x \mp t)}{\sqrt{b}} + C_{15}]; \end{matrix}

(38)

\begin{matrix} u_{12} (x, t) & = \pm \frac{a}{2} \cos^{2} [\frac{k (x \mp t)}{\sqrt{b}} + C_{16}] \\ + C_{4}, and \\ v_{12} (x, t) & = \cos [\frac{k (x \mp t)}{\sqrt{b}} + C_{16}] . \end{matrix}

Case (2h) Assuming

c = 2 k_{1}

d = - k_{1}

α_{2} = - k^{2}

C_{5} = 0

C_{6} = \frac{k^{2}}{b}

, and

b \neq 0

, the solution of Eq. (1) is

(39)

\begin{matrix} u_{13} (x, t) & = \pm \frac{a}{2} \sinh^{2} [\frac{k (x \mp t)}{\sqrt{b}} + C_{17}] \\ + C_{4}, and \\ v_{13} (x, t) & = \sinh [\frac{k (x \mp t)}{\sqrt{b}} + C_{17}] . \end{matrix}

Case (2i) Taking

c = 2 k_{1}

d = - k_{1}

α_{2} = - k^{2}

C_{5} = 0

C_{6} = - \frac{k^{2}}{b}

, and

b \neq 0

, the solution of CDSWEs is

(40)

\begin{matrix} u_{14} (x, t) & = \pm \frac{a}{2} \cosh^{2} [\frac{k (x \mp t)}{\sqrt{b}} + C_{18}] \\ + C_{4}, and \\ v_{14} (x, t) & = \cosh [\frac{k (x \mp t)}{\sqrt{b}} + C_{18}] . \end{matrix}

Case (3) : For

P_{4} = X_{1} = \frac{\partial}{\partial x}

, the characteristic equation is

(41)

\begin{matrix} d x = \frac{d t}{0} = \frac{d u}{0} = \frac{d v}{0} . \end{matrix}

The trivial solution for system (1) is

(42)

\begin{matrix} u_{15} (x, t) = v_{15} (x, t) = C_{19} . \end{matrix}

4. Comparison with results reported earlier

The novelty of the results can be confirmed as some of the reported results [26], [28], [29], [30], [33], [38], [42], [49] are derived from the solutions attained in this article. Other solutions are absolutely different from earlier findings as far as authors are aware.

1. If

C_{4} = C_{7} = 0

is substituted in the expressions of

u_{3}

, and

v_{3}

, then Eqs. (12) and (13) of Arora and Kumar [28], Eq. (52) of Yao [33], and Eqs. (20) and (21) of Zhang [38] can be retrieved.

2. Taking

C_{4} = 0

u_{3}, v_{3}

u_{10}, v_{10}

u_{5}, v_{5}

u_{6}, v_{6}

u_{8}, v_{8}

u_{4}, v_{4}

, so the results represented by Eqs. (35)-(48) of Arnous et al. [29] can be obtained respectively.

3. Eqs. (3.21) and (3.23) of Khan et al. [26], Eq. (52) of Yao [33], and Eq. (37) of Shi et al. [42] can be obtained by replacing

C_{4} = C_{12} = 0

u_{8}

, and

v_{8}

of this study.

4. Putting

C_{4} = C_{12} = 0

u_{8}

and

v_{8}

of this study, one can get Eqs. (3.21) and (3.23) of Khan et al. [26], Eq. (52) of Yao [33], and Eq. (37) of Shi et al. [42].

5. For

C_{4} = C_{11} = 0

u_{7}

and

v_{7}

, then

u_{2} (ξ)

and

v_{2} (ξ)

in Eq. (36) of Zhang and Zhao [49] can be obtained.

6. Taking

C_{4} = C_{8} = 0

u_{4}

, and

v_{4}

, then Eq. (36)

u_{3} (ξ)

v_{3} (ξ)

of Zhang and Zhao [49], and

u_{12}

v_{23, 24}

of Akbar and Ali [30] can be deduced.

7. Eqs. (28) and (29) of Zhang [38] can be proved replacing

C_{4} = C_{9} = 0

u_{5}

and

v_{5}

respectively.

5. Physical analysis of solutions and discussions

The graphical illustration of the mathematical expressions (21), (22), (29)-(40) and (42) of invariant solutions might make them more significant. These solutions are different from those available in the existing literature [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50]. Using the MATLAB code in the space range

- 20 \leq x \leq 20

, and selecting appropriate values of arbitrary variables and parameters such as

a = 0.9134

k = 0.1419

, and

C_{1} = 0.4218

. Variations in

u

, and

v

can be observed via dominant behaviour frames of animations created with respect to changes in space and time.

6. Conclusions

Lie symmetry analysis is exploited successfully to produce one-dimensional optimum sub algebras of CDSWEs (1) in this research. The killing form has been used to derive invariants. The analytical solutions of CDSWEs (1) are represented by Eqs. (21), (22), (29)-(40) and (42). Some of the existing results [26], [28], [29], [30], [33], [38], [42], [49] can be derived from our solutions, demonstrating the originality of the results. The remaining solutions in this study are completely different from those established in the published works in Singh et al. [25], Khan et al. [26], Lu et al. [27], Arora and Kumar [28], Arnous et al. [29], Akbar and Ali [30], Chen et al. [31], Tariq and Seadawy [32], Yao [33], Lu et al. [34], Wen et al. [35], Javeed et al. [36], Shehata et al. [37], Zhang [38], Bibi and Tauseef [39], Bhatter et al. [40], Matjila et al. [41], Shi et al. [42], Khan et al. [43], Akbar et al. [44], Zhao and Zhi [45], Ullah et al. [46], Inc [47], Morris and Kara [48], Zhang and Zhao [49], Zhao et al. [50]. Asymptotic, bell-shaped, bright and dark soliton, bright soliton, parabolic, bright and kink, kink, and periodic nature of the solutions are shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13. Over a horizontal seafloor, CDSWEs (1) depict the dispersion of nonlinear surface gravity waves whose amplitude varies with time and space. Coastal engineers may get benefit from the analytical solutions of CDSWEs (1) with physical nature found in this study.

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Fig.1 Asymptotic nature of $u_{2}$ and $v_{2}$ with $c = 0.6324$ and $d = 0.0975$ .

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Fig.2 Bell-shaped profiles of the expressions $u_{3}$ and $v_{3}$ with constant $C_{7} = 0.9157$ .

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Fig.3 Bright and dark solitons for $u_{4}$ and $v_{4}$ at arbitrary constant $C_{8} = 0.9595$ .

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Fig.4 Parabolic profiles of the $u_{5}$ and $v_{5}$ taking $C_{9} = 0.1712$ .

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Fig.5 Bright and kink profiles of the solutions represented as $u_{6}$ and $v_{6}$ with choice of $C_{10} = 0.0971$ .

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Fig.9 Bright and kink soliton profile for $u_{10}$ and $v_{10}$ at $C_{14} = 0.2785$ .

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Fig.10 Periodic nature of $u_{11}$ and $v_{11}$ choosing $C_{15} = 0.9134$ .

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Fig.11 Periodic of $u_{12}$ and $v_{12}$ with $C_{16} = 0.7431$ .

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Fig.12 Asymptotic nature profiles of $u_{13}$ and $v_{13}$ taking $C_{17} = 0.7577$ .

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Fig.13 Asymptotic of $u_{14}$ and $v_{14}$ treating $C_{18} = 0.6787$ .

Availability of data and materials

Data sharing does not apply to this article.

Funding

Not applicable.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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模态框（Modal）标题

Abstract

Cite this article

1. Introduction

1.1. Scope

1.2. Origin of the problem

1.3. Literature review

1.4. Motivation

1.5. Outline

2. Invariants by using Killing form

Table 1 Commutator table for (1).

Table 2 Adjoint table for (1).

Table 3 Construction of invariant function for (1).

3. Similarity solutions

4. Comparison with results reported earlier

5. Physical analysis of solutions and discussions

6. Conclusions

Fig.1 Asymptotic nature of u2 and v2 with c=0.6324 and d=0.0975 .

Fig.2 Bell-shaped profiles of the expressions u3 and v3 with constant C7=0.9157 .

Fig.3 Bright and dark solitons for u4 and v4 at arbitrary constant C8=0.9595 .

Fig.4 Parabolic profiles of the u5 and v5 taking C9=0.1712 .

Fig.5 Bright and kink profiles of the solutions represented as u6 and v6 with choice of C10=0.0971 .

Fig.6 Bright soliton for u7 and v7 with C11=0.8235 .

Fig.7 Kink profile of the u8 and v8 with C12=0.8147 .

Fig.8 Bright and kink profiles for u9 and v9 with C13=0.1270 .

Fig.9 Bright and kink soliton profile for u10 and v10 at C14=0.2785 .

Fig.10 Periodic nature of u11 and v11 choosing C15=0.9134 .

Fig.11 Periodic of u12 and v12 with C16=0.7431 .

Fig.12 Asymptotic nature profiles of u13 and v13 taking C17=0.7577 .

Fig.13 Asymptotic of u14 and v14 treating C18=0.6787 .

Availability of data and materials

Funding

Declaration of Competing Interest

References

Links

Fig.1 Asymptotic nature of $u_{2}$ and $v_{2}$ with $c = 0.6324$ and $d = 0.0975$ .

Fig.2 Bell-shaped profiles of the expressions $u_{3}$ and $v_{3}$ with constant $C_{7} = 0.9157$ .

Fig.3 Bright and dark solitons for $u_{4}$ and $v_{4}$ at arbitrary constant $C_{8} = 0.9595$ .

Fig.4 Parabolic profiles of the $u_{5}$ and $v_{5}$ taking $C_{9} = 0.1712$ .

Fig.5 Bright and kink profiles of the solutions represented as $u_{6}$ and $v_{6}$ with choice of $C_{10} = 0.0971$ .

Fig.6 Bright soliton for $u_{7}$ and $v_{7}$ with $C_{11} = 0.8235$ .

Fig.7 Kink profile of the $u_{8}$ and $v_{8}$ with $C_{12} = 0.8147$ .

Fig.8 Bright and kink profiles for $u_{9}$ and $v_{9}$ with $C_{13} = 0.1270$ .

Fig.9 Bright and kink soliton profile for $u_{10}$ and $v_{10}$ at $C_{14} = 0.2785$ .

Fig.10 Periodic nature of $u_{11}$ and $v_{11}$ choosing $C_{15} = 0.9134$ .

Fig.11 Periodic of $u_{12}$ and $v_{12}$ with $C_{16} = 0.7431$ .

Fig.12 Asymptotic nature profiles of $u_{13}$ and $v_{13}$ taking $C_{17} = 0.7577$ .

Fig.13 Asymptotic of $u_{14}$ and $v_{14}$ treating $C_{18} = 0.6787$ .