Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg-de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations

We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg-de Vries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation.


Introduction
Almost all processes occurring in human life and in nature can be considered to be complex systems. Examples of such complex systems are stock markets, research groups, traffic networks, etc. [1][2][3][4][5][6]. Moreover, most complex systems are characterized by their non-linearity. Examples of non-linear complex systems can be found in many scientific areas, from fluid mechanics and solid-state physics to biology and medicine [7][8][9][10][11]. Usually, the non-linear behavior of the complex systems is described by differential or difference equations [12][13][14][15]. In this direction, finding analytical and numerical solutions of non-linear differential equations is a great challenge for researchers from various scientific fields.
Research related to finding exact analytical solutions of non-linear partial differential equations (NPDFs) has a long history. At the beginning, to remove the non-linearity of the solved equation, an appropriate transformation is introduced. An example can be given the by so-called Hopf-Cole transformation [16,17], by which the non-linear Burger's equation is reduced to the linear heat equation. Later, the transformation, which reduces the standard KdV equation to the famous linear equation of Schrdinger, leads to the appearance of the Method of Inverse Scattering Transform [18][19][20]. Other popular methods using appropriate transformations are the method of Hirota [21][22][23] and the method including the Painleve expansions [24][25][26].
In this study, we shall use the SEsM (Simple Equations Method) for obtaining exact solutions of non-linear differential equations. The idea for development of this method comes from the Method of Simplest Equation (MSE), proposed by Kudryashov [27]. MSE is based on searching for particular solutions of NPDEs as a series containing powers of integrals of the modal function for the linear long-wave theory. In addition, the fifth-order KdV equation was used to describe internal waves of moderate amplitude in densitystratified fluids [94]. All these references are only a part of the possibilities that the studied equation gives in a purely physical sense. This emphasizes the importance of finding its exact analytical solutions.
The paper is structured as follows. In Section 2, we formulate the problem studied. The methodology of SEsM is presented in the same section. In Section 3, we present various types of exact travelling-wave solution of the extended fifth-order KdV equation depending on the simple equations used. Numerical examples of the obtained analytical solutions are shown in the same section. Some concluding remarks are made in Section 4.

Problem Formulation and Methodology
In this study, we discuss the extended KdV equation, presented in the form [86][87][88][89][90][91][92][93][94]: where u(x, t) is a displacement of surface at any varied natural instances, x is the spatial coordinate, and t is time. In more detail, Equation (1) is a hydrodynamic model of an incompressible, inviscid fluid and its irrotational motion is governed by gravitational forces. In addition to the standard non-linear term with a coefficient = a/h << 1, and the standard linear dispersion term with a coefficient δ = (h/l) 2 << 1 (a denotes the wave amplitude, h the average depth of the fluid container, and l is the average wavelength), involved in the standard KdV equation, Equation (1) involves a cubic non-linear term (with a coefficient γ), a linear dispersion term of 5th order (with a coefficient of 1), and also higher-order non-linear dispersion terms with coefficients α and β.
Here, we shall search for analytical solutions of Equation (1) applying the SEsM. The SEsM can be used for obtaining analytical solutions of NPDEs: where the left-hand side of Equation (2) is a relationship containing the function u(x, t) and some of its derivatives. The algorithm of SEsM includes the following four steps [71,72]: (1). The transformation is made, where Tr(F 1 (x, . . . , t), F 2 (x, . . . , t), . . . F N (x, . . . , t)) is a composite function of other functions . . , t) are functions of several spatial variables, as well as of time. The transformations Tr(F i ) have two goals: (1) They can remove some non-linearities if possible (an example is the Hopf-Cole transformation, which leads to the linearization of the Burger's equation); (2) They can transform the non-linearity of the solved differential equations to a more treatable kind of non-linearity (e.g., to polynomial non-linearity). In many particular cases one may skip this step (then we have just u(x, . . . , t) = F(x, . . . , t)), but in numerous cases this step is necessary for obtaining a solution of the studied NPDE. The substitution of Equation (3) in Equation (2) leads to a non-linear PDE for the function F(x, . . . , t). In many cases, the general form of the transformation Tr(F) is not known.
(2). This step is based on the use of composite functions. In this step, the functions F 1 (x, . . . , t), F 2 (x, . . . , t), . . . are chosen as composite functions of the functions f i1 , . . . , f iN , . . . , which are solutions of simpler differential equations. There are two possibilities: (1) The construction relationship for the composite function is not fixed. Then, the Faa di Bruno relationship for the derivatives of a composite function is used; (2) The construction relationship for the composite function is fixed. For example, for the case of one solved equation and one function F, the construction relationship can be given as: Then, one can directly calculate the corresponding derivatives from the solved differential equation.
(3). In this step, the simple equations for the functions f i 1 , . . . , f i N must be selected. In addition, in accordance with the hypothesis of Point (1) of Step 2, the relationship between the composite functions F 1 (x, . . . , t), . . . , F N (x, . . . , t) and the functions f i 1 , . . . , f i N must be fixed. The fixation of the simple equations and the fixation of the relationships for the composite functions are connected. The fixations transform the left-hand sides of Equation (2). The result of this transformation can be functions that are the sum of terms. Each of these terms contains some function multiplied by a coefficient. This coefficient is a relationship containing some of the parameters of the solved equations and some of the parameters of the solutions of the simple equations used. The fixation mentioned above is performed by a balance procedure that ensures that the relationships for the coefficients contain more than one term. This balance procedure leads to one or more additional relationships among the parameters of the solved equation and parameters of the solutions of the simple equations used. These additional relationships are known as balance equations.
(4). A non-trivial solution of Equation (2) is obtained if all coefficients mentioned in Step 3 are set to zero. This condition usually leads to a system of non-linear algebraic equations. The unknown variables in these equations are the coefficients of the solved non-linear differential equation and the coefficients of the solutions of the simple equations. Any non-trivial solution of this algebraic system leads to a solution of the studied nonlinear PDE.
Below, we shall apply the methodology above given to obtain exact solutions of Equation (1). We shall consider u as a composite function of two functions of two variables, i.e., where as where ζ i 1 , i 1 = 0, . . . , n 1 and η i 2 , i 2 = 0, . . . , n 2 are parameters, and n 1 and n 2 shall be determined by means of balance procedure. Let us present the solutions of functions f 1 and f 2 by the special functions V µ 0 ,µ 1 ,...,µ m 1 (ξ 1 ; k 1 , l 1 , m 1 ) and V ν 0 ,ν 1 ,...,ν m 2 (ξ 2 ; k 2 , l 2 , m 2 ), which are solutions of the simple equations of the following kind: where k 1,2 are the orders of derivatives of f 1 and f 2 , l 1,2 are the degrees of derivatives in the defining ODEs and m 1,2 are the highest degrees of the polynomials of f 1 and f 2 in the defining ODE. The special functions V µ 0 ,µ 1 ,...,µ m 1 (ξ 1 ; k 1 , l 1 , m 1 ) and V ν 0 ,ν 1 ,...,ν m 2 (ξ 2 ; k 2 , l 2 , m 2 ) have interesting properties. These functions can be hyperbolic, trigonometric, elliptic functions of Jacobi, etc. For our study, we choose one specific case of the functions V. We shall assume that k 1 = k 2 = 1 and l 1 = l 2 = 1. Then, the functions V µ 0 ,µ 1 ,...,µ m 1 (ξ 1 ; 1, 1, m 1 ) and V ν 0 ,ν 1 ,...,ν m 2 (ξ 2 ; 1, 1, m 2 ) are solutions of the simple equations: In the study, we shall present various examples of application of the special functions V depending on the numerical value of m 1 and m 2 . We shall use the following general types of simple equations: • The Bernoulli equation, whose general form is: The general solution of this equation is: for the case a > 0, b < 0 and for the case a < 0, b > 0, as ξ 0 is a constant of integration. • The Abel equation of first kind, whose general form is: For the special case a = c 3d (b − 2c 2 9d ), this equation has the following solution: where C * and ξ 0 are constants of integration. • The Riccati equation, whose general form is: The general solutions of this equation are: and where θ 2 = b 2 − 4ac > 0 and C * and ξ 0 are constants of integration. In this study, we shall use only the extended variant of the Riccati equation (Equation (17)). In addition, as a particular case of the use of equations of Riccati as simple equations, we shall consider also the so-called extended tanh-function equation: Equation (18) is obtained from Equation (15) when b = 0, a = −ā 2 , c =c 2 and its solution is: The linear ODE, which has the following form: and its solution is: where C * and ξ 0 are constants.

Exact Solutions of the Extended KdV Equation
Following the above given algorithm, we skip Step 1 of the SEsM (no additional transformation of non-linearity). In Step 2, we consider u as a composite function of two functions of two variables (see Equation (5)). Substitution of Equations (5)- (7) in Equation (1) leads to the following ODE: In Step 3 of the SEsM, we must select the equation for u( ) (the relationship for the composite function) and the equations for f 1 (ξ 1 ) and f 1 (ξ 2 ) (the simple equations). We assume that the expression for u is of kind (5). In addition, the simple equations are assumed to be of kind (9). The substitution of Equations (5), (7) and (9) in Equation (22) leads to polynomials of the functions f 1 and f 2 . To obtain the system of non-linear algebraic equations, we must balance the largest degrees of these polynomials. This procedure leads to the balance equations Then Equation (1) may have solutions of the kind (24) and the functions f 1 (ξ 1 ) and f 2 (ξ 2 ) are solutions of the simple Equation (9).
First, we shall search for analytical solutions of Equation (1) for m 1 = 3, m 2 = 3. According to the balance Equation (23), n 1 = 4, n 2 = 4. The general solution of Equation (1) can be written as where Substitution of Equations (25) and (26) in Equation (22) leads to the following system of non-linear algebraic equations: 5 µ 3 Thanks to the computer algebra system MAPLE, solutions of the system (27) can be derived. We note that this is performed step by step starting from the simplest equation of the system of algebraic equations and then moving in the direction of solving of the more complicated equations. There are many variants of solutions, which can be obtaining by the computational software, but our goal is to express the coefficients of the solved equation and the coefficients of its solution by the coefficients of the simple equations and the coefficients in the solutions of the simple equations, so far as it is possible.

Conclusions
In this paper, we have shown the effectiveness of the SEsM for obtaining exact solutions of a famous evolution equation in mathematical physics. We have presented various types of the travelling-wave solution of the fifth-order KdV equation, using the special functions V, which are solutions of so-called simple equations in SEsM. The obtained results are only a part of the possible variety of exact solutions of the studied equation that can be derived using the special functions V. We believe that the presented results are new. Moreover, the use of composite functions in the methodology of the SEsM gives possibilities for obtaining other specific solutions of the physical phenomena, discussed in the paper. However, this will be the goal of further investigations.