Exact and Approximate Solutions for Nonlinear PDEsView this Special Issue
Exact Solutions and Conservation Laws of the Drinfel’d-Sokolov-Wilson System
We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the -expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.
The classical Drinfel’d-Sokolov-Wilson (DSW) system given by where , , , and are nonzero constants, has been studied by . The authors obtained various types of explicit solutions for (1) by using the bifurcation method and qualitative theory of dynamical systems. Also, Yao and Li  and C. Liu and X. Liu  obtained some exact solutions for the DSW system (1) by using a direct algebra method. A special case of the classical DSW system, namely, was studied by several authors [4–8]. Hirota et al.  investigated the soliton structure of (2) and by employing an algebraic method, Fan  constructed some exact solutions. By using the improved generalized Jacobi elliptic function method, Yao  obtained some traveling wave solutions of (2), whereas by applying the Adomian decomposition method, Inc  obtained approximate doubly periodic wave solutions of (2). Zhao and Zhi  constructed exact doubly periodic solutions of (2) by using an improved -expansion method.
In this paper, we study the Drinfel’d-Sokolov-Wilson (DSW) system given bywhich can be derived from (1) by taking and . We obtain exact solutions and construct conservation laws of the DSW systems (3a) and (3b).
Nonlinear partial differential equations (PDEs) model diverse nonlinear phenomena in natural and applied sciences such as mechanics, fluid dynamics, biology, plasma physics, and mathematical finance. Therefore finding exact solutions of nonlinear PDEs is very important. Unfortunately, this is a very difficult task and there are no systematic methods that can be used to find exact solutions of the nonlinear PDEs. However, in the past few decades a number of new methods have been developed to obtain exact solutions to nonlinear PDEs. Some of these methods include the exp-function method, the homogeneous balance method, the sine-cosine method, the hyperbolic tangent function expansion method, and the -expansion function method [9–15].
We recall that conservation laws are mathematical expressions of the physical laws, such as conservation of energy, mass, and momentum. They are of great significance in the solution process and reduction of PDEs. In the literature, one finds that the conservation laws have been widely used in studying the existence, uniqueness, and stability of solutions of nonlinear partial differential equations [16–18], as well as in the development and use of numerical methods [19, 20]. Also, conserved vectors associated with Lie point symmetries have been employed to find exact solutions of partial differential equations [21–23]. There are various methods of constructing conservation laws. One of the methods for variational problems is by means of Noether’s theorem . In order to apply Noether’s theorem, the knowledge of a suitable Lagrangian is necessary. For nonlinear differential equations that do not have a Lagrangian, several methods have been developed (see, e.g., [25–30]).
This paper is structured as follows. In Section 2, exact solutions of (3a) and (3b) are obtained using the -expansion function method. In Section 3, we construct Noether’s symmetries and the conserved vectors for the DSW system (3a) and (3b). Concluding remarks are presented in Section 4.
In this section, we obtain exact solutions of the DSW system (3a) and (3b). We first transform the system (3a) and (3b) into a system of ordinary differential equations by using the substitutions where . Substituting (4) into the system (3a) and (3b) and integrating with respect to , we obtain the following ordinary differential equations (ODEs): where integration constants are taken to be zero. From the first equation, we obtain . Substituting this value of in the second equation of the system, we obtain Now multiplying the above equation by and integrating while taking the constant of integration to be zero, we arrive at a first-order variables separable equation. Integrating this equation and reverting back to our original variables, we obtain where is an arbitrary constant of integration. Since , we have Thus, we have obtained one exact solution of the DSW system (3a) and (3b).
To obtain more exact solutions of the DSW system (3a) and (3b), we employ the -expansion function method . We assume that the solutions of the ODE (6) can be expressed as a polynomial in by where is the balancing number to be determined and the function satisfies the second-order linear ODE given by with and being arbitrary constants. In our case, the balancing procedure gives . Thus Substituting (11) into (6) and making use of (10) and then equating all the terms with the same powers of to zero yield the following system of algebraic equations: Solving the above equations, with the aid of Mathematica, we obtain Consequently, we obtain the following two types of travelling wave solutions of the DSW system.
For , we obtain the hyperbolic functions travelling wave solutions For , we obtain the trigonometric function travelling solutions
In this section, we construct the conservation laws of the DSW system (3a) and (3b). Since the third-order DSW system (3a) and (3b) does not have a Lagrangian, we cannot apply the Noether theorem. However, if we transform the third-order DSW system (3a) and (3b) to a fourth-order with the aid of the transformation , , we obtainThis system has a Lagrangian given by and it satisfies the Euler-Lagrange equations where and are defined by respectively.
Let us now consider the vector field
The second prolongation operator, of , is given by where We recall that , given by (20), is a Noether symmetry of (16a) and (16b), if it satisfies where and are the gauge functions. Expanding the above equation gives This leads to an overdetermined system of PDEs for the functions , and . Solving the system of PDEs gives We may choose and as they contribute to the trivial part of the conserved vector. The conserved vector for the second-order Lagrangian is given by [24, 31] where and are the characteristic functions. Now using (26) in conjunction with (25), , and , we obtain the following independent conserved vectors for system (3a) and (3b): We note that (27) is a nonlocal conserved vector, whereas (28) is a local conserved vectors. Also, for the arbitrary functions and , we obtain the following conserved vectors: which gives us infinitely many nonlocal conservation laws.
The third-order DSW system (3a) and (3b) was studied. Exact solutions of the DSW system were obtained using direct integration and the -expansion function method. The solutions obtained were hyperbolic and trigonometric solutions. In addition conservation laws were also derived. This system does not have a Lagrangian. In order to invoke Noether’s theorem we used the transformations and to convert the DSW system to a fourth-order system, which has a Lagrangian. The conservation laws were then obtained and consisted of a local and infinite number of nonlocal conserved vectors.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
All authors would like to thank NRF and North-West University, Mafikeng Campus, for financial support.
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