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On the solutions and conservation laws of the coupled Drinfeld-Sokolov-Satsuma-Hirota system
Boundary Value Problems volume 2014, Article number: 248 (2014)
In this paper we study the coupled Drinfeld-Sokolov-Satsuma-Hirota system, which was developed as one example of nonlinear equations possessing Lax pairs of a special form. Also this system was found as a special case of the four-reduction of the Kadomtsev-Petviashivilli hierarchy. We obtain exact solutions of the system by using Lie symmetry analysis along with the simplest equation and Jacobi elliptic equation methods. Also, symmetry reductions are obtained based on the optimal system of one-dimensional subalgebras. In addition, the conservation laws are derived using two approaches: the new conservation theorem due to Ibragimov and the multiplier method.
In recent years many nonlinear evolution equations (NLEEs) have been used to model many real world problems in various fields of science and engineering. Thus, finding exact explicit solutions of NLEEs is a very important endeavor. It is also true that finding solutions of NLEEs is a difficult task, and only in few special cases one can write down the explicit solutions. However, despite of this fact, various methods of solving NLEEs have been proposed in the literature recently. Some of the most important methods found in the literature include the ansatz method , , the Weierstrass elliptic function expansion method , the Darboux transformation , Hirota’s bilinear method , the -expansion method , the Jacobi elliptic function expansion method , , the inverse scattering transform method , the homogeneous balance method , the Bäcklund transformation , the F-expansion method , the exp-function method , the multiple exp-function method , the variable separation approach , the sine-cosine method , the tri-function method , , and the Lie symmetry method –.
In this paper we study the coupled Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system
This system was introduced independently by Drinfeld and Sokolov , and by Satsuma and Hirota . The coupled DSSH system  was given as one of numerous examples of nonlinear equations possessing Lax pairs of a special form. Also, the coupled DSSH system  was found as a special case of the four-reduction of the KP hierarchy, and its explicit one-soliton solution was constructed. Gürses and Karasu  found a recursion operator and a bi-Hamiltonian structure for (1). Wazwaz  used three distinct methods, namely the Cole-Hopf transformation, Hirota’s bilinear and the exp-function methods, and obtained solitons, multiple soliton solutions, multiple singular soliton solutions, and plane periodic solutions. Zheng  used the -expansion method and obtained traveling wave solutions of (1).
In this paper we firstly perform symmetry reductions of (1) using Lie group analysis –, which are based on the optimal systems of one-dimensional subalgebras. The simplest equation method  and the Jacobi elliptic function method  are later employed to obtain some exact solutions of (1). In addition to this, conservation laws are derived for (1) using the new conservation theorem  and the multiplier method .
It is well known that the conservation laws play a very important role in the solution process of differential equations. Also, one can safely say that the existence of a large number of conservation laws of a system of partial differential equations is a strong indication of its integrability . Recently, conservation laws have been used to find exact solutions of certain partial differential equations , .
2 Symmetry analysis of (1)
The symmetry group of the coupled DSSH system (1) will be generated by the vector field of the form
The application of the third prolongation to (1) results in an overdetermined system of linear partial differential equations. The general solution of these equations with the aid of Maple is given by
where , , are arbitrary constants. The above general solution contains four arbitrary constants, and hence the infinitesimal symmetries of (1) form the four-dimensional Lie algebra spanned by the following linearly independent operators:
2.1 Optimal system of one-dimensional subalgebras
In this subsection we present the optimal system of one-dimensional subalgebras for equation (1) to obtain the optimal system of group-invariant solutions. The method which we use here for obtaining the optimal system of one-dimensional subalgebras is given in . The adjoint transformations are given by
The commutator table of the Lie point symmetries of equation (1) and the adjoint representations of the symmetry group of (1) on its Lie algebra are given in Table 1 and Table 2, respectively. Table 1 and Table 2 are then used to construct the optimal system of one-dimensional subalgebras for equation (1).
2.2 Symmetry reductions of (1)
In this subsection we use the optimal system of one-dimensional subalgebras calculated above to obtain symmetry reductions.
The operator gives rise to the group-invariant solution
Case 2. ;
The symmetry gives rise to the group-invariant solution
3 Exact solutions of (1) using the simplest equation method
Taking the linear combination of the translation symmetries, viz., and solving the corresponding Lagrange system for the symmetry , one obtains an invariant and the group-invariant solution of the form
where the functions F and G satisfy
Now we use the simplest equation method ,  to solve system (5a)-(5b); and as a result, we obtain the exact solutions of our coupled DSSH system (1). Bernoulli and Riccati equations will be used as the simplest equations.
where satisfies the Bernoulli or Riccati equation, M and N are positive integers that can be determined by a balancing procedure and ’s and ’s are parameters to be determined.
3.1 Solutions of (1) using the Bernoulli equation as the simplest equation
Substituting (7a)-(7b) into (5a)-(5b) and making use of the Bernoulli equation  and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations in terms of () and (). Solving the resultant system of algebraic equations with the aid of Maple, one possible set of values of and is as follows:
As a result, a solution of (1) is
3.2 Solutions of (1) using the Riccati equation as the simplest equation
Substituting (9a)-(9b) into (5a)-(5b) and making use of the Riccati equation , we obtain an algebraic system of equations in terms of and . Solving the resultant system, one possible set of values is as follows:
Hence solutions of (1) are
3.3 Solutions of (1) in terms of Jacobi elliptic functions
We now present exact solutions of the coupled DSSH system (1) that are expressed in Jacobi elliptic functions. The cosine-amplitude function and the sine-amplitude function satisfy the first-order differential equations
Treating the above first-order ODEs as our simplest equations and then proceeding as before, we obtain the cnoidal and snoidal wave solutions that are given by
4 Conservation laws
In this section we construct conservation laws for the coupled Drinfeld-Sokolov-Satsuma-Hirota system (1). The new conservation theorem due to Ibragimov  and the multiplier method  will be used. For the notations used in this section, the reader is referred to .
4.1 Construction of conservation laws using the new conservation theorem
The coupled DSSH system together with its adjoint equation is given by
We recall that the coupled DSSH system admits the following four Lie point symmetries:
Thus we have the following four cases:
For the Lie point symmetry , the corresponding Lie characteristic functions are and . Thus, by using Ibragimov’s theorem , the components of the conserved vector are given by
The Lie point symmetry has the Lie characteristic functions that are given by and . Hence, by the application of Ibragimov’s theorem , the conserved vector is given by
The symmetry generator has the Lie characteristic functions given by and , and hence in this case one can obtain the conserved vector whose components are
Finally, we consider the symmetry generator , which has the Lie characteristic functions and . By invoking Ibragimov’s theorem , the components of the conserved vector are given by
4.2 Construction of conservation laws using the multiplier method
where , , are arbitrary constants. Corresponding to the above multipliers, we obtain the following three local conserved vectors of (1):
It should be noted that higher-order conservation laws of (1) can be computed by increasing the order of the multipliers.
5 Concluding remarks
In this paper firstly we obtained the solutions of the Drinfeld-Sokolov-Satsuma-Hirota equation by employing Lie group analysis together with the simplest and Jacobi elliptic equation methods. Also symmetry reductions were obtained based on the optimal systems of one-dimensional subalgebras. The exact solutions obtained were traveling wave solutions, cnoidal and snoidal wave solutions. Furthermore, the conservation laws for the underlying equation were derived by using two different approaches, namely the new conservation theorem and the multiplier method. The importance of the conservation laws was explained in the introduction.
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KRA is grateful to the National Research Foundation of South Africa for the generous financial support.
The authors declare that they have no competing interests.
KRA and CMK worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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Adem, K.R., Khalique, C.M. On the solutions and conservation laws of the coupled Drinfeld-Sokolov-Satsuma-Hirota system. Bound Value Probl 2014, 248 (2014). https://doi.org/10.1186/s13661-014-0248-6
- coupled Drinfeld-Sokolov-Satsuma-Hirota system
- Lie symmetry methods
- simplest equation method
- Jacobi elliptic function method
- conservation laws