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.
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]
A coupled KdV system [32]
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.
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
We first determine optimal system of onedimensional Lie subalgebras for (1.2). The adjoint representations are given by ref. [27]







0  0  0 


0  0  0 


0  0  0  0 



0  0 


























Using Tables 1 and 2, and following the method described in ref. [27], we obtain optimal system of Lie subalgebras as
We now use each element of the set (2.4) and reduce gHSKdVes (1.2) to ordinary differential equation (ODE) systems.
The symmetry
The symmetry generator
The symmetry generator
The symmetry generator
Finally, the symmetry
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
For the system of three ODEs (2.10), we consider its solutions in the form
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
As a result, we obtain a solution of (1.2) as
Solutions of (1.2) with the Riccati equation as simplest equation
As before, for this case, K = 2 and so
Solution (2.18) is sketched in Figure 1.
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
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.
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.
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.
[1] Gandarias ML , Rosa RDL , Rosa M. Conservation laws for a strongly damped wave equation. Open Phys. 2017;15:300–5. Search 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. Search in Google Scholar
[3] Wazwaz AM , Xu GQ. Bright, dark and Gaussons optical solutions for fourthorder Schrödinger equations with cubicquintic and logarithmic nonlinearities. Optik. 2020 ;202:163564. Search in Google Scholar
[4] Wazwaz AM , Xu GQ. KadomtsevPetviashvili hierarchy: two integrable equations with timedependent coefficients. Nonlinear Dyn. 2020 ;100:3711–6. Search in Google Scholar
[5] Wang G. A novel (3+1)dimensional sineGordon and a sineGordon equation: Derivation, symmetries and conservation laws. Appl Math Lett. 2021 ;113:106768 Search in Google Scholar
[6] Wang G , Liu Y , Wu Y , Su X. Symmetry analysis for a seventhorder generalized KdV equation and its fractional version in fluid mechanics. Fractals. 2020 ;28:2050044. Search 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. Search in Google Scholar
[8] Hu W , Wang Z , Zhao Y , Deng Z. Symmetry breaking of infinitedimensional dynamic system. Appl Math Lett. 2020 ;103:106207. Search in Google Scholar
[9] Yildirim Y , Yasar E. An extended Korteweg–de Vries equation: multisoliton solutions and conservation laws. Nonlinear Dyn. 2017 ;90:1571–9. Search 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. Search 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. Search in Google Scholar
[12] Korkmaz A , Hepson OE , Hosseini K , Rezazadeh H , Eslami M. SineGordon expansion method for exact solutions to conformable time fractional equations in RLWclass. J King Saud Univ Sci. 2020 ;32:567–74. Search in Google Scholar
[13] Gao XY , Guo YJ , Shan WR , Yuan YQ , Zhang CR , Chen SS. Magnetooptical/ferromagneticmaterial computation: Bäcklund transformations, bilinear forms and N solitons for a generalized (3+1)dimensional variablecoefficient modified KadomtsevPetviashvili system. Appl Math Lett. 2021 ;111:106627. Search in Google Scholar
[14] Wazwaz AM. The tanhcoth method for solitons and kink solutions for nonlinear parabolic equations. Appl Math Comput. 2007 ;188:1467–75. Search in Google Scholar
[15] Ablowitz MJ , Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge, UK; 1991. Search in Google Scholar
[16] Hirota R. The direct method in soliton theory. Cambridge University Press, Cambridge, 2004. Search 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. Search in Google Scholar
[18] Gu C , Hu H , Zhou Z. Darboux transformation in soliton theory and its geometric applications. Springer, The Netherlands; 2005. Search 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. Search in Google Scholar
[20] Vitanov NK. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact travelingwave solutions for a class of PDEs with polynomial nonlinearity. Commun Nonlinear Sci Numer Simul. 2010 ;15:2050–60. Search in Google Scholar
[21] Kudryashov NA , Loguinova NB. Extended simplest equation method for nonlinear differential equations. Appl Math Comput. 2008 ;205:396–402. Search 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. Search in Google Scholar
[23] Zhang L , Khalique CM. Classification and bifurcation of a class of secondorder ODEs and its application to nonlinear PDEs. Discrete Contin Dyn Syst Ser B. 2018 ;11:777–90. Search 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. Search in Google Scholar
[26] Bluman GW , Kumei S. Symmetries and differential equations. SpringerVerlag, New York; 1989. Search in Google Scholar
[27] Olver PJ. Applications of Lie groups to differential equations. second ed., SpringerVerlag, Berlin, 1993. Search 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 KleinGordonSchrödinger equations. Phys Lett A. 2003 ;318:84–92. Search 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. Search in Google Scholar
[32] Hirota R , Satsuma J. Soliton solutions of a coupled Korteweg–de Vries equation. Phys Lett A. 1981 ;85:407–8. Search 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. Search 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. Search in Google Scholar
[35] Ali AHA. The modified extended tanhfunction method for solving coupled MKdV and coupled Hirota–Satsuma coupled KdV equations. Phys Lett A. 2007 ;363:420–5. Search 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. Search 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. Search 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. Search in Google Scholar
[40] Bluman GW , Cheviakov AF , Anco SC. Applications of symmetry methods to partial differential equations. Springer, New York, 2010. Search in Google Scholar
[41] Khalique CM , Abdallah SA. Coupled Burgers equations governing polydispersive sedimentation; a Lie symmetry approach. Results Phys. 2020 ;16:102967. Search in Google Scholar
[42] Khalique CM , Moleleki LD. A (3 + 1)dimensional generalized BKPBoussinesq equation: Lie group approach. Results Phys. 2019 ;13:102239. Search in Google Scholar
[43] Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007 ;333:311–28. Search 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. Search 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. Search in Google Scholar
[46] Kudryashov NA. Exact solitary waves of the Fisher equation. Phys Lett A. 2005 ;342:99–106. Search 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. Search in Google Scholar
© 2021 Chaudry Masood Khalique, published by De Gruyter
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