Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Chaudry Masood Khalique

doi:10.1515/phys-2021-0002

Open Access Published by De Gruyter Open Access February 24, 2021

Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Chaudry Masood Khalique

From the journal Open Physics

https://doi.org/10.1515/phys-2021-0002

Abstract

In this article, a generalized Hirota–Satsuma coupled Korteweg–de Vries (KdV) system is investigated from the group standpoint. This system represents an interplay of long waves with distinct dispersion correlations. Using Lie’s theory several symmetry reductions are performed and the system is reduced to systems of non-linear ordinary differential equations (NLODEs). Subsequently, the simplest equation method is invoked to find exact solutions of the NLODE systems, which then provides the solitary wave solutions for the system under discussion. Finally, we construct conservation laws of generalized Hirota–Satsuma coupled KdV system with the aid of general multiplier approach.

Keywords: generalized Hirota–Satsuma coupled KdV system; Lie’s theory; simplest equation method; conservation laws; multiplier method

1 Introduction

Many physical phenomena in numerous disciplines, for instance, plasma physics, engineering, relativity, fluid and classical mechanics, control theory and geochemistry, are modeled by nonlinear evolution equations (NEEs). For some of the recent work published in the literature, see for example refs [1–11], in which conservation laws, multi-soliton, bright, dark and Gaussons optical solutions are presented. These equations are studied by various scientists and engineers from different aspects. One of the important aspects is the integrability of such NEEs. Because of the importance of NEEs, in the last several decades, a number of techniques were developed by researchers to establish exact closed-form solutions of such NEEs. Some of the predominant techniques that exist in the literature include the sine-Gordon expansion method [12], Bäcklund transformations [13], tanh–coth technique [14], inverse scattering transform technique [15], Hirota’s bilinear technique [16], the homogeneous balance of undetermined coefficient technique [17], Darboux transformation technique [18], simplest equation technique [19,20], extended simplest equation technique [21], Kudryashov’s method [22], bifurcation technique [23], the first integral method [24], Lie’s theory [25,26,27, 28,29] and F-expansion method [30].

It is common knowledge that the renowned Korteweg–de Vries (KdV) equation [31]

u t + 6 u u x + u x x x = 0

models the attributes of solitary waves. It was first formulated as an equation that governed the shallow water waves in channels. However, subsequently it was discovered that it models an extensive variety of natural phenomena, especially those demonstrating solitons and travelling waves.

A coupled KdV system [32]

(1.1a) u t − 1 2 u x x x + 3 u u x − 6 v v x = 0 ,

(1.1b) v t + v x x x − 3 u v x = 0 ,

was introduced by Hirota and Satsuma in 1981. In the literature, this system is called the Hirota–Satsuma coupled KdV system and portrays an interplay of two long waves that has distinct dispersion relations. Subsequently, the authors of ref. [33] introduced a generalized version of (1.1), called the generalized Hirota–Satsuma coupled system of KdV equations (gHS-KdVes) that are given by

(1.2a) Q 1 ≡ u t − 3 ( v w ) x − 1 2 ( u x x x − 6 u u x ) = 0 ,

(1.2b) Q 2 ≡ v t − 3 u v x + v x x x = 0 ,

(1.2c) Q 3 ≡ w t − 3 u w x + w x x x = 0 .

This new system received a great deal of attention from the researchers and has been extensively studied. For instance, the decomposition technique [34], the modified extended tanh function technique [35], the homotopy analysis method [36], the differential transform method [37] and the ( G ′ / G ) -expansion method [38] were utilized to study the gHS-KdVes (1.2).

In this study, an entirely distinct approach is used to investigate the gHS-KdVes (1.2). Lie’s theory combined with the simplest equation technique is employed to find explicit closed-form solutions of (1.2), Subsequently, conserved quantities, using the general multiplier approach, will be computed.

During the eighteenth and nineteenth century, one of the fundamental problems of differential equations (DEs) was to find their closed-form explicit solutions. Perhaps the first explicit solution was the travelling wave solution of the second-order linear wave equation given by d’Alembert in 1747. Fourier developed the separation of variable method during his work on heat conduction problems. Likewise, several other eminent mathematicians contributed to finding exact solutions to linear and nonlinear DEs of physics. S. Lie (1842–1899) and F. Klein (1849–1925) worked on DEs from the point of view of transformation groups that left the DEs unchanged. However, Lie further went onto developing the theory of continuous transformation groups and its applications to DEs [25,26,27, 28,29]. Today, Lie’s theory is widely used by scientists to find exact closed-form solutions of DEs that arise in countless fields of research. See for example refs [5,6, 7,8,9, 10,11].

Conservation laws are pivotal in the investigation of solutions to DEs. These are laws of nature that are expressed as mathematical expressions. For instance, we have conservation of momentum, energy, angular momentum, charge, just to mention a few. Conservation laws have been utilized to determine the existence, uniqueness and moreover, stability of DEs. Conservation laws are exploited in the investigation of numerical techniques and can also be used in the reduction of order and solution process of DEs [39,40,41, 42,43,44].

This article is organized as follows. In Section 2, we calculate symmetries and perform symmetry reductions of gHS-KdVes (1.2) using Lie’s theory and according to the optimal systems of one-dimensional Lie subalgebras of (1.2). We then invoke the simplest equation technique to obtain exact explicit solutions of (1.2). Thereafter, with the aid of general multiplier method, conservation laws are obtained in Section 3. Finally, in Section 4 we put forward concluding remarks.

2 Symmetry reductions and exact solutions

We first compute symmetry group of gHS-KdVes (1.2). This is the classical group of point transformations of the dependent and independent variables that map solutions of (1.2) into new solutions of (1.2). Assume that the symmetry group is generated by the vector field

(2.3) S = ξ 1 ∂ ∂ t + ξ 2 ∂ ∂ x + η 1 ∂ ∂ u + η 2 ∂ ∂ v + η 3 ∂ ∂ w ,

where the infinitesimals ξ ¹, ξ ², η ¹, η ², η ³ depend on t, x, u, v, w. The generator (2.3) is a point symmetry of (1.2) if Lie’s invariance condition

S [ 3 ] Q 1 Q 1 = Q 2 = Q 3 = 0 = 0 , S [ 3 ] Q 2 Q 1 = Q 2 = Q 3 = 0 = 0 , S [ 3 ] Q 3 Q 1 = Q 2 = Q 3 = 0 = 0

holds. Here S [ 3 ] denotes third prolongation of S [28]. The above three equations, on expanding, give a system of linear PDEs, whose solution yields the following four symmetries of (1.2):

S 1 = ∂ ∂ x , S 2 = ∂ ∂ t , S 3 = − v ∂ ∂ v + w ∂ ∂ w , S 4 = 3 t ∂ ∂ t + x ∂ ∂ x − 2 u ∂ ∂ u − 4 v ∂ ∂ v .

2.1 Optimal system of Lie subalgebras

We first determine optimal system of one-dimensional Lie subalgebras for (1.2). The adjoint representations are given by ref. [27]

Ad ( exp ( ϵ S i ) ) S j = S j − ϵ [ S i , S j ] + 1 2 ϵ 2 [ S i , [ S i , S j ] ] − ⋯ ,

where [ S i , S j ] denotes the commutator of S i and S j and is defined as

[ S i , S j ] = S i S j − S j S i .

After computing commutators of all symmetries of (1.2), we display the results in Table 1. We then calculate adjoint representations and display them in Table 2.

Table 1

Commutators of Lie algebra of (1.2)

	S 1	S 2	S 4
S 1	0	0	S 1
S 2	0	0	3 S 2
S 3	0	0	0
S 4	− S 1	− 3 S 2	0

Table 2

Adjoint commutators of Lie algebra of (1.2)

	S 1	S 2	S 3	S 4
S 1	S 1	S 2	S 3	− ϵ S 1 + S 4
S 2	S 1	S 2	S 3	− 3 ϵ S 2 + S 4
S 3	S 1	S 2	S 3	S 4
S 4	e ϵ S 1	e 3 ϵ S 2	S 3	S 4

Using Tables 1 and 2, and following the method described in ref. [27], we obtain optimal system of Lie subalgebras as

(2.4) { S 3 + a 4 S 4 , a 1 S 1 + a 2 S 2 + S 3 , S 4 , S 2 , ν S 1 + S 2 } ,

where ν = ±1, a ₁, a ₂ and a ₄ are constants with a ₄ ≠ 0.

2.2 Symmetry reductions of (1.2)

We now use each element of the set (2.4) and reduce gHS-KdVes (1.2) to ordinary differential equation (ODE) systems.

Case 1

S 3 + a 4 S 4

The symmetry S 3 + a 4 S 4 provides the similarity transformation

(2.5) u = t − 2 / 3 E ( ρ ) , v = t − ( 1 + 4 a 4 ) / 3 a 4 F ( ρ ) , w = t 1 / 3 a 4 G ( ρ ) ,

with ρ = t ^−1/3 x as an invariant of S 3 + a 4 S 4 . Substituting these values of u, v, w from (2.5) into (1.2), we obtain the ODEs

9 a 4 E ( ρ ) G ′ ( ρ ) − 3 a 4 G ‴ ( ρ ) + a 4 ρ G ′ ( ρ ) − G ( ρ ) = 0 , 9 a 4 E ( ρ ) F ′ ( ρ ) − 3 a 4 F ‴ ( ρ ) + a 4 ρ F ′ ( ρ ) + F ( ρ ) + 4 a 4 F ( ρ ) = 0 , 3 E ‴ ( ρ ) − 18 E ( ρ ) E ′ ( ρ ) + 2 ρ E ′ ( ρ ) + 4 E ( ρ ) + 18 G ( ρ ) F ′ ( ρ ) + 18 F ( ρ ) G ′ ( ρ ) = 0 .

Case 2

a 1 S 1 + a 2 S 2 + S 3

The symmetry generator a 1 S 1 + a 2 S 2 + S 3 provides the similarity transformation

(2.6) u = E ( ρ ) , v = exp ( − t / a 2 ) F ( ρ ) , w = exp ( t / a 2 ) G ( ρ ) ,

where ρ = (a ₂ x − a ₁ t)/a ₂ is an invariant of a 1 S 1 + a 2 S 2 + S 3 and E, F and G solves

3 a 2 E ( ρ ) F ′ ( ρ ) − a 2 F ‴ ( ρ ) + a 1 F ′ ( ρ ) + F ( ρ ) = 0 , 3 a 2 E ( ρ ) G ′ ( ρ ) − a 2 G ‴ ( ρ ) + a 1 G ′ ( ρ ) − G ( ρ ) = 0 , a 2 E ‴ ( ρ ) + 2 a 1 E ′ ( ρ ) − 6 a 2 E ( ρ ) E ′ ( ρ ) + 6 a 2 G ( ρ ) F ′ ( ρ ) + 6 a 2 F ( ρ ) G ′ ( ρ ) = 0 .

Case 3

S 4

The symmetry generator S 4 furnishes us with similarity transformation

(2.7) u = t − 2 / 3 E ( ρ ) , v = t − 4 / 3 F ( ρ ) , w = G ( ρ ) ,

where invariant of S 4 is ρ = t ^−1/3 x and E, F, G solve

9 E ( ρ ) G ′ ( ρ ) − 3 G ‴ ( ρ ) + ρ G ′ ( ρ ) = 0 , 9 E ( ρ ) F ′ ( ρ ) − 3 F ‴ ( ρ ) + ρ F ′ ( ρ ) + 4 F ( ρ ) = 0 , 3 E ‴ ( ρ ) − 18 E ( ρ ) E ′ ( ρ ) + 2 ρ E ′ ( ρ ) + 4 E ( ρ ) + 18 G ( ρ ) F ′ ( ρ ) + 18 F ( ρ ) G ′ ( ρ ) = 0 .

Case 4

S 2

The symmetry generator S 2 provides ρ = x as an invariant, and consequently, similarity transformation is

(2.8) u = E ( ρ ) , v = F ( ρ ) , w = G ( ρ ) ,

with E, F, G satisfying

E ‴ ( ρ ) − 6 E ( ρ ) E ′ ( ρ ) + 6 G ( ρ ) F ′ ( ρ ) + 6 F ( ρ ) G ′ ( ρ ) = 0 , F ‴ ( ρ ) − 3 E ( ρ ) F ′ ( ρ ) = 0 , G ‴ ( ρ ) − 3 E ( ρ ) G ′ ( ρ ) = 0 .

Case 5

ν S 1 + S 2

Finally, the symmetry ν S 1 + S 2 gives ρ = x − νt as an invariant and hence the invariant solution is

(2.9) u = E ( ρ ) , v = F ( ρ ) , w = G ( ρ ) ,

where E, F, G solve

(2.10a) E ‴ ( ρ ) + 2 ν E ′ ( ρ ) − 6 E ( ρ ) E ′ ( ρ ) + 6 G ( ρ ) F ′ ( ρ ) + 6 F ( ρ ) G ′ ( ρ ) = 0 ,

(2.10b) 3 E ( ρ ) F ′ ( ρ ) − F ‴ ( ρ ) + ν F ′ ( ρ ) = 0 ,

(2.10c) 3 E ( ρ ) G ′ ( ρ ) − G ‴ ( ρ ) + ν G ′ ( ρ ) = 0 .

2.3 Exact solutions via the simplest equation method

The simplest equation technique is an effective and robust technique, which can be used to construct closed-form solutions of DEs. It was introduced and used by the Russian mathematician Kudryashov [45,46]. Here we invoke this technique and use it on the reduced ODE system (2.10). The technique involves the use of a well-known ODE whose solution exists in the closed form. In our work, we shall make use of two famous ODEs, namely Bernoulli and Riccati equations.

We consider Bernoulli’s equation

(2.11) ℋ ′ ( ρ ) = r ℋ ( ρ ) + s ℋ 2 ( ρ )

with r and s constants, whose solution is

ℋ ( ρ ) = r cosh [ ( ρ + C ) r ] + sinh [ ( ρ + C ) r ] 1 − s cosh [ ( ρ + C ) r ] − s sinh [ ( ρ + C ) r ] ,

where C is an arbitrary integration constant. For Riccati’s equation

(2.12) ℋ ′ ( ρ ) = r ℋ 2 ( ρ ) + s ℋ ( ρ ) + c

(r, s, c constants) the two solutions we use are

ℋ ( ρ ) = − s 2 r − θ 2 r tanh 1 2 ϑ ( ρ + C )

and

ℋ ( ρ ) = − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 ,

with ϑ ² = s ² − 4rc > 0 and C an integration constant.

For the system of three ODEs (2.10), we consider its solutions in the form

(2.13) E ( ρ ) = ∑ μ = 0 K A μ ( ℋ ( ρ ) ) μ , F ( ρ ) = ∑ μ = 0 K ℬ μ ( ℋ ( ρ ) ) μ , G ( ρ ) = ∑ μ = 0 K C μ ( ℋ ( ρ ) ) μ ,

where ℋ ( ρ ) solves Bernoulli or Riccati equation, K > 0 is an integer and A μ , ℬ μ and C μ ( μ = 0 , 1 , … , K ) are unknown constants.

Solutions of (1.2) with Bernoulli’s equation as simplest equation

From (2.10) by balancing the highest order derivatives with the nonlinear terms [45] yields K = 2, so (2.13) takes the form

(2.14a) E ( ρ ) = A 0 + A 1 ℋ + A 2 ℋ 2 ,

(2.14b) F ( ρ ) = ℬ 0 + ℬ 1 ℋ + ℬ 2 ℋ 2 ,

(2.14c) G ( ρ ) = C 0 + C 1 ℋ + C 2 ℋ 2 .

We substitute (2.14) into (2.10) and invoke (2.11) and then equate the coefficients of like powers of ℋ to zero. This gives a system of algebraic equations in A i , ℬ i , C i ( i = 0 , 1 , 2 ) . Using Mathematica to solve the above system, one obtains

A 0 = 1 3 r 2 − ν , A 1 = 4 r s , A 2 = 4 s 2 , ℬ 2 = s ℬ 1 r , C 0 = 2 3 ℬ 1 2 − 6 r 2 s 2 ℬ 0 + r 3 s ℬ 1 − 4 r s ν ℬ 1 , C 1 = 4 r s 3 ℬ 2 , C 2 = s C 1 r .

As a result, we obtain a solution of (1.2) as

(2.15a) u ( t , x ) = A 0 + A 1 r cosh ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r + A 2 r 2 cosh ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r 2 ,

(2.15b) v ( t , x ) = ℬ 0 + ℬ 1 r × cosh ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r + ℬ 2 r 2 cosh r ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r 2 ,

(2.15c) w ( t , x ) = C 0 + C 1 r × cosh ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r + C 2 r 2 cosh ( ρ + C ) r + sinh ( ρ + C ) r 1 − s cosh ( ρ + C ) r − s sinh ( ρ + C ) r 2 ,

with ρ = x − νt and C an integration constant.

Solutions of (1.2) with the Riccati equation as simplest equation

As before, for this case, K = 2 and so

(2.16a) E ( ρ ) = A 0 + A 1 ℋ + A 2 ℋ 2 ,

(2.16b) F ( ρ ) = ℬ 0 + ℬ 1 ℋ + ℬ 2 ℋ 2 ,

(2.16c) G ( ρ ) = C 0 + C 1 ℋ + C 2 ℋ 2 .

Following the same procedure as above, but using (2.12), we obtain

A 0 = 1 3 s 2 + 8 r c − ν , A 1 = 4 r s , A 2 = 4 r 2 , ℬ 2 = r ℬ 1 s , C 0 = 2 3 ℬ 2 2 − 6 r 4 ℬ 0 + r 3 b ℬ 1 + 8 r 3 c ℬ 2 − 4 r 2 ν ℬ 2 , C 1 = 4 r 3 s ℬ 2 , C 2 = r C 1 s .

Thus, the two solutions of (1.2) are

(2.17a) u 1 ( t , x ) = A 0 + A 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) + A 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) 2 ,

(2.17b) v 1 ( t , x ) = ℬ 0 + ℬ 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) + ℬ 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) 2 ,

(2.17c) w 1 ( t , x ) = C 0 + C 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) + C 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ( ρ + C ) 2

and

(2.18a) u 2 ( t , x ) = A 0 + A 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 + A 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 2 ,

(2.18b) v 2 ( t , x ) = ℬ 0 + ℬ 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 + ℬ 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 2 ,

(2.18c) w 2 ( t , x ) = C 0 + C 1 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 + C 2 − s 2 r − ϑ 2 r tanh 1 2 ϑ ρ + sech ϑ ρ 2 C cosh ϑ ρ 2 − 2 r ϑ sinh ϑ ρ 2 2 ,

with ρ = x − νt and C an integration constant.

Solution (2.18) is sketched in Figure 1.

Figure 1

Profile of solitary wave solution (2.18).

3 Conservation laws

We now determine conservation laws for gHS-KdVes (1.2) by employing the general multiplier technique [27,40, 41,42]. Here we seek second-order multipliers Λ¹, Λ² and Λ³ that depend on (t, x, u, v, w, u _x, v _x, w _x, u _xx, v _xx, w _xx). The multipliers are obtained from the equations

(3.19) δ δ u Λ 1 Q 1 + Λ 2 Q 2 + Λ 3 Q 3 = 0 , δ δ v Λ 1 Q 1 + Λ 2 Q 2 + Λ 3 Q 3 = 0 , δ δ w Λ 1 Q 1 + Λ 2 Q 2 + Λ 3 Q 3 = 0 ,

where δ/δu ^l is the Euler operator

δ δ u l = ∂ ∂ u l − D j ∂ ∂ u j l + D j D k ∂ ∂ u j k l − ⋯ , l = 1 , 2 , 3 ,

with u ¹ = u, u ² = v, u ³ = w and

D j = ∂ ∂ x j + u j l ∂ ∂ u l + u j k l ∂ ∂ u k l + ⋯ , j = 1 , 2 ,

is the total differentiation operator. Expanding (3.19) and after some calculations, we obtain five conservation law multipliers

Λ 1 = − 3 2 C 1 u 2 + C 1 v w + ( C 3 t + C 4 ) u + 1 2 C 1 u x x − C 3 x 3 + C 5 , Λ 2 = C 1 u w − ( C 3 t + C 4 ) w − C 1 w x x − C 2 w x , Λ 3 = C 1 u v − ( C 3 t + C 4 ) v − C 1 v x x + C 2 v x ,

where C _i, i = 1, 2,…, 5 are constants. Thus, the corresponding five local conserved vectors of (1.2) are as follows [27,44,47]

T 1 t = 1 4 u x x u − 2 v x x w − 2 w x x v + 4 u v w − 2 u 3 , T 1 x = 1 8 16 v x w x u + 8 v x x u w + 8 w x x u v − 8 u x v x w − 8 u x w x v − 4 u x x v w + 6 u x x u 2 − 2 u u t x + 4 w v t x + 4 v w t x + 12 u 2 v w − 9 u 4 − 12 v 2 w 2 + 2 u t u x − 4 v t w x − 4 w t v x − u x x 2 − 8 v x x w x x ; T 2 t = 1 2 v x w − w x v , T 2 x = 1 2 − v t w + w t v − 2 v x x w x + 2 v x w x x ; T 3 t = 1 6 3 t u 2 − 2 x u − 6 t v w , T 3 x = 1 12 − 6 t u x x u − 12 t v x x w − 12 t w x x v + 12 t u 3 − 6 x u 2 + 12 x v w + 3 t u x 2 + 12 t v x w x − 2 u x + 2 x u x x ; T 4 t = 1 2 u 2 − 2 v w , T 4 x = 1 4 − 2 u x x u − 4 v x x w − 4 w x x v + 4 u 3 + u x 2 + 4 v x w x ; T 5 t = u , T 5 x = 1 2 3 u 2 − 6 v w − u x x .

Remark

As far as the physical meaning of the conservation laws derived above are concerned, we observe that the first four of them are purely mathematical, whereas the fifth describes mass density and the flux representing conserved currents for the mass. Also, note that by raising the order of multipliers, higher-order conserved vectors of gHS-KdVes (1.2) can be derived.

4 Concluding remarks

In this article, we investigated the generalized system of Hirota–Satsuma coupled KdV equations (1.2) from the group standpoint. The symmetries of the system were found and then used to build optimal system of one-dimensional Lie subalgebras. With the assistance of these subalgebras, system (1.2) was reduced to systems of ODEs and thereafter the simplest equation technique was invoked to manufacture closed-form solutions of (1.2). Moreover, conserved vectors were derived for system (1.2) using the general multiplier method. Five multipliers were computed and accordingly five conservation laws were constructed. The advantages of conservation laws were mentioned in Section 1.

Acknowledgement

The author would like to thank the North-West University, Mafikeng Campus, for its continued support and the reviewers for their positive suggestions, which helped to improve the paper enormously.

References

[1] Gandarias ML , Rosa RDL , Rosa M. Conservation laws for a strongly damped wave equation. Open Phys. 2017;15:300–5.10.1515/phys-2017-0033Search in Google Scholar

[2] Qurashi MMA. Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications. Open Phys. 2020;18:164–9.10.1515/phys-2020-0127Search in Google Scholar

[3] Wazwaz AM , Xu GQ. Bright, dark and Gaussons optical solutions for fourth-order Schrödinger equations with cubic-quintic and logarithmic nonlinearities. Optik. 2020 ;202:163564.10.1016/j.ijleo.2019.163564Search in Google Scholar

[4] Wazwaz AM , Xu GQ. Kadomtsev-Petviashvili hierarchy: two integrable equations with time-dependent coefficients. Nonlinear Dyn. 2020 ;100:3711–6.10.1007/s11071-020-05708-1Search in Google Scholar

[5] Wang G. A novel (3+1)-dimensional sine-Gordon and a sine-Gordon equation: Derivation, symmetries and conservation laws. Appl Math Lett. 2021 ;113:106768 10.1016/j.aml.2020.106768Search in Google Scholar

[6] Wang G , Liu Y , Wu Y , Su X. Symmetry analysis for a seventh-order generalized KdV equation and its fractional version in fluid mechanics. Fractals. 2020 ;28:2050044.10.1142/S0218348X20500449Search in Google Scholar

[7] Wang G. Symmetry analysis and rogue wave solutions for the(2+1)-dimensional nonlinear Schrödinger equation with variable coefficients. Appl Math Lett. 2016 ;56:56–64.10.1016/j.aml.2015.12.011Search in Google Scholar

[8] Hu W , Wang Z , Zhao Y , Deng Z. Symmetry breaking of infinite-dimensional dynamic system. Appl Math Lett. 2020 ;103:106207.10.1016/j.aml.2019.106207Search in Google Scholar

[9] Yildirim Y , Yasar E. An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws. Nonlinear Dyn. 2017 ;90:1571–9.10.1007/s11071-017-3749-xSearch in Google Scholar

[10] Chulián S , Rosa M , Gandarias ML Symmetries and solutions for a Fisher equation with a proliferation term involving tumor development. Math Meth Appl Sci. 2020 ;43:2076–84.10.1002/mma.6105Search in Google Scholar

[11] Rosa M , Chulián S , Gandarias ML , Traciná R. Application of Lie point symmetries to the resolution of an interface problem in a generalized Fisher equation. Physica D. 2020 ;405:132411.10.1016/j.physd.2020.132411Search in Google Scholar

[12] Korkmaz A , Hepson OE , Hosseini K , Rezazadeh H , Eslami M. Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. J King Saud Univ Sci. 2020 ;32:567–74.10.1016/j.jksus.2018.08.013Search in Google Scholar

[13] Gao XY , Guo YJ , Shan WR , Yuan YQ , Zhang CR , Chen SS. Magneto-optical/ferromagnetic-material computation: Bäcklund transformations, bilinear forms and N solitons for a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili system. Appl Math Lett. 2021 ;111:106627.10.1016/j.aml.2020.106627Search in Google Scholar

[14] Wazwaz AM. The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl Math Comput. 2007 ;188:1467–75.10.1016/j.amc.2006.11.013Search in Google Scholar

[15] Ablowitz MJ , Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge, UK; 1991.10.1017/CBO9780511623998Search in Google Scholar

[16] Hirota R. The direct method in soliton theory. Cambridge University Press, Cambridge, 2004.10.1017/CBO9780511543043Search in Google Scholar

[17] Wei Y , He X , Yang X. The homogeneous balance of undetermined coefficients method and its application. Open Math. 2016 ;14:816–26.10.1515/math-2016-0078Search in Google Scholar

[18] Gu C , Hu H , Zhou Z. Darboux transformation in soliton theory and its geometric applications. Springer, The Netherlands; 2005.10.1007/1-4020-3088-6_5Search in Google Scholar

[19] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals. 2005 ;24:1217–31.10.1016/j.chaos.2004.09.109Search in Google Scholar

[20] Vitanov NK. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Commun Nonlinear Sci Numer Simul. 2010 ;15:2050–60.10.1016/j.cnsns.2009.08.011Search in Google Scholar

[21] Kudryashov NA , Loguinova NB. Extended simplest equation method for nonlinear differential equations. Appl Math Comput. 2008 ;205:396–402.10.1016/j.amc.2008.08.019Search in Google Scholar

[22] Kudryashov NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat. 2012 ;17:2248–53.10.1016/j.cnsns.2011.10.016Search in Google Scholar

[23] Zhang L , Khalique CM. Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs. Discrete Contin Dyn Syst Ser B. 2018 ;11:777–90.10.3934/dcdss.2018048Search in Google Scholar

[24] Taghizadeh N , Mirzazadeh M , Paghaleh AS. The first integral method to nonlinear partial differential equations. Appl Appl Math. 2012 ;7:117–32.Search in Google Scholar

[25] Ovsiannikov LV. Group analysis of differential equations. Academic Press, New York; 1982.10.1016/B978-0-12-531680-4.50012-5Search in Google Scholar

[26] Bluman GW , Kumei S. Symmetries and differential equations. Springer-Verlag, New York; 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[27] Olver PJ. Applications of Lie groups to differential equations. second ed., Springer-Verlag, Berlin, 1993.10.1007/978-1-4612-4350-2Search in Google Scholar

[28] Ibragimov NH. CRC handbook of Lie group analysis of differential equations. Vols 1–3, CRC Press, Boca Raton, Florida, 1994–1996.Search in Google Scholar

[29] Ibragimov NH. Elementary Lie Group Analysis and Ordinary Differential Equations. John Wiley & Sons, Chichester, NY; 1999.Search in Google Scholar

[30] Wang ML , Zhou YB. The periodic wave solutions for the Klein-Gordon-Schrödinger equations. Phys Lett A. 2003 ;318:84–92.10.1016/j.physleta.2003.07.026Search in Google Scholar

[31] Korteweg DJ , de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 1985 ;39:422–43.10.1080/14786449508620739Search in Google Scholar

[32] Hirota R , Satsuma J. Soliton solutions of a coupled Korteweg–de Vries equation. Phys Lett A. 1981 ;85:407–8.10.1016/0375-9601(81)90423-0Search in Google Scholar

[33] Wu YT , Geng XG , Hu XB , Zhu SM. A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and miura transformations. Phys Lett A. 1999 ;255:259–64.10.1016/S0375-9601(99)00163-2Search in Google Scholar

[34] Raslan KR. The decomposition method for a Hirota–Satsuma coupled KdV equation and a coupled MKdV equation. Int J Comput Math. 2004 ;81:1497–505.10.1080/0020716042000261405Search in Google Scholar

[35] Ali AHA. The modified extended tanh-function method for solving coupled MKdV and coupled Hirota–Satsuma coupled KdV equations. Phys Lett A. 2007 ;363:420–5.10.1016/j.physleta.2006.11.076Search in Google Scholar

[36] Abbasbandy S. The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys Lett A. 2007 ;361:478–83.10.1016/j.physleta.2006.09.105Search in Google Scholar

[37] Zuo JM , Zhang YM. A new method for a generalized Hirota–Satsuma coupled KdV equation. Appl Math Comput 2011 ;217:7117–25.10.1016/j.amc.2011.01.048Search in Google Scholar

[38] Zuo JM , Zhang YM. Application of the (G′∕G) -expansion method to solve coupled MKdV equations and coupled Hirota–Satsuma coupled KdV equations. Appl Math Comput. 2011 ;217:5936–41.Search in Google Scholar

[39] Noether E. Invariante variationsprobleme. Nachr. v. d. Ges. d. Wiss. zu Göttingen. 1918 ;2:235–57.10.1007/978-3-642-39990-9_13Search in Google Scholar

[40] Bluman GW , Cheviakov AF , Anco SC. Applications of symmetry methods to partial differential equations. Springer, New York, 2010.10.1007/978-0-387-68028-6Search in Google Scholar

[41] Khalique CM , Abdallah SA. Coupled Burgers equations governing polydispersive sedimentation; a Lie symmetry approach. Results Phys. 2020 ;16:102967.10.1016/j.rinp.2020.102967Search in Google Scholar

[42] Khalique CM , Moleleki LD. A (3 + 1)-dimensional generalized BKP-Boussinesq equation: Lie group approach. Results Phys. 2019 ;13:102239.10.1016/j.rinp.2019.102239Search in Google Scholar

[43] Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007 ;333:311–28.10.1016/j.jmaa.2006.10.078Search in Google Scholar

[44] Benoudina N , Zhang Y , Khalique CM. Lie symmetry analysis, optimal system, new solitary wave solutions and conservation laws of the Pavlov equation. Commun Nonlinear Sci Numer Simulat. 2021 ;94:105560.10.1016/j.cnsns.2020.105560Search in Google Scholar

[45] Kudryashov NA. Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals. 2005 ;24:1217–31.10.1016/j.chaos.2004.09.109Search in Google Scholar

[46] Kudryashov NA. Exact solitary waves of the Fisher equation. Phys Lett A. 2005 ;342:99–106.10.1016/j.physleta.2005.05.025Search in Google Scholar

[47] Cheviakov AF. Symbolic computation of local symmetries of nonlinear and linear partial andordinary differential equations. Math Comp Sci. 2010 ;4:203–22.10.1007/s11786-010-0051-4Search in Google Scholar

Received: 2020-10-31

Revised: 2020-12-12

Accepted: 2020-12-21

Published Online: 2021-02-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Abstract

1 Introduction

2 Symmetry reductions and exact solutions

2.1 Optimal system of Lie subalgebras

Table 1

Table 2

2.2 Symmetry reductions of (1.2)

Case 1

Case 2

Case 3

Case 4

Case 5

2.3 Exact solutions via the simplest equation method

Figure 1

3 Conservation laws

Remark

4 Concluding remarks

Acknowledgement

References

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