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 nonlinear 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.
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, multisoliton, 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 closedform solutions of such NEEs. Some of the predominant techniques that exist in the literature include the sineGordon 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 Fexpansion method [30].
It is common knowledge that the renowned Korteweg–de Vries (KdV) equation [31]
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]
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 (gHSKdVes) that are given by
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
In this study, an entirely distinct approach is used to investigate the gHSKdVes (1.2). Lie’s theory combined with the simplest equation technique is employed to find explicit closedform 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 closedform explicit solutions. Perhaps the first explicit solution was the travelling wave solution of the secondorder 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 closedform 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 gHSKdVes (1.2) using Lie’s theory and according to the optimal systems of onedimensional 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 gHSKdVes (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
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
holds. Here
2.1 Optimal system of Lie subalgebras
We first determine optimal system of onedimensional Lie subalgebras for (1.2). The adjoint representations are given by ref. [27]
where
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







0  0  0 


0  0  0 


0  0  0  0 



0  0 
Table 2


























Using Tables 1 and 2, and following the method described in ref. [27], we obtain optimal system of Lie subalgebras as
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 gHSKdVes (1.2) to ordinary differential equation (ODE) systems.
Case 1
The symmetry
with ρ = t
^{−1/3}
x as an invariant of
Case 2
The symmetry generator
where ρ = (a
_{2}
x − a
_{1}
t)/a
_{2} is an invariant of
Case 3
The symmetry generator
where invariant of
Case 4
The symmetry generator
with E, F, G satisfying
Case 5
Finally, the symmetry
where E, F, G solve
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 closedform 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 wellknown 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
with r and s constants, whose solution is
where C is an arbitrary integration constant. For Riccati’s equation
(r, s, c constants) the two solutions we use are
and
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
where
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
We substitute (2.14) into (2.10) and invoke (2.11) and then equate the coefficients of like powers of
As a result, we obtain a solution of (1.2) as
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
Following the same procedure as above, but using (2.12), we obtain
Thus, the two solutions of (1.2) are
and
with ρ = x − νt and C an integration constant.
Solution (2.18) is sketched in Figure 1.
Figure 1
3 Conservation laws
We now determine conservation laws for gHSKdVes (1.2) by employing the general multiplier technique [27,40, 41,42]. Here we seek secondorder multipliers Λ^{1}, Λ^{2} and Λ^{3} 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
where δ/δu ^{ l } is the Euler operator
with u ^{1} = u, u ^{2} = v, u ^{3} = w and
is the total differentiation operator. Expanding (3.19) and after some calculations, we obtain five conservation law multipliers
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]
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, higherorder conserved vectors of gHSKdVes (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 onedimensional 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 closedform 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 NorthWest University, Mafikeng Campus, for its continued support and the reviewers for their positive suggestions, which helped to improve the paper enormously.
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© 2021 Chaudry Masood Khalique, published by De Gruyter
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