Exact meromorphic solutions of cubic Ostrovsky equation: Kudryashov method

We analyze the cubic Ostrovsky equation which is a modification of Ostrovsky equation. This equation can be found in physical phenomena such as wave mechanism due to a rotating media. We use the Kudryashov method to solve this equation. Since this equation is one type of nonlinear partial differential equation, we initially transformed this equation into its nonlinear ordinary differential equation form. Next, we compute the Laurent series of this equation and we get two types of Laurent series with a second-order pole; the coefficients of these series are complex numbers so that the symmetrical form is complex conjugation. Based on these two Laurent series, we construct two types of solutions, each of which consists of elliptic solution (doubly periodic) in the form of Weierstrass-℘ functions, simply periodic solutions, and rational solution.


Introduction
One of the most interesting topics in mathematics and nonlinear physics is the problem of solving nonlinear partial differential equations. Nonlinear partial differential equations play a very important role as a language in explaining dynamics in nonlinear physics. Many important and current phenomena require nonlinear partial differential equations to explain them [1][2][3][4][5][6]. Because this equation generally contains differentiation terms that are spatially dispersed and nonlinear terms that have the opposite properties, the combination of these terms in an equation will have many implications, especially in one of the most important physical applications, namely the soliton phenomenon. Solitons are a type of solution to nonlinear partial differential equations that most people are interested in because of their localized and stable nature over a long time and spatial range. This is what is widely applied/ studied in various fields such as [7,8], [9].
By looking at the brief explanation above, it is important to have a good understanding of how to analyze the solutions to these equations. Nowadays, many methods have been developed for analyzing the solution of nonlinear partial differential equations such as the Exp method [10], Tanh method [11], and Tanh-Coth Method [12]. However, these methods generally produce specific solutions in the form of certain specific functions that correspond to the method's naming. For this reason, a more general method is needed so that more interpretations can be obtained and analyzed. The method in question is the Kudryashov method, which was developed in 2010 ( [13], [14], [15]). This method is based on the existence of Laurent series (which has more general properties than Taylor series) of a differential equation which makes it possible to produce negative term expansions ( [13,15,16]). Given this negative term, solutions with doubly periodic (which of course have more general properties) can be formed, such as elliptic functions.
The Ostrovksy equation is a type of nonlinear partial differential equation [17]. This equation already has several modifications ( [5,7,[17][18][19][20][20][21][22][23]), one of which is the cubic Ostrovsky equation (referring to the existence of a cubic nonlinear term). This equation can be found in physical problems such as the wave mechanism caused by the rotating [5]. In this paper, we analyze the meromorphic exact solution of the Ostrovsky cubic equation based on the existence of Laurent series. Therefore, this paper generally consists of several parts, namely first is an introduction, second on the Kudryashov method, third on the meromorphic solutions of the cubic Ostrovksy equation, and finally the conclusion.

Methods
In this study, we use the method introduced by Kudryahov to solve nonlinear differential equations ( [13,15]). This method is very powerful because it allows us to obtain more general solutions of a nonlinear differential equation in three forms, namely simply periodic solutions, doubly periodic solutions, and rational solutions. These general solutions provide the possibility for us to obtain certain specific solutions based on the selection of certain particular circumstances. The basis on which this method is able to produce general solutions is because before constructing an exact meromorphic solution of a nonlinear differential equation, what must be done first is to find the existence of Laurent series of the equation. It is well known that Laurent series provides a complex expansion from a negative expansion to an infinite positive expansion term. Thus, it is clear that we can construct doubly periodic solutions such as the Weierstrass functions ℘ and . This is not found in other methods. Thus, this method can be an excellent reference in finding meromorphic exact solutions of a nonlinear differential equation.
This method can be broadly explained as follows: first, if what is being studied is a nonlinear partial differential equation, then the equation must first be transformed into a nonlinear ordinary differential equation, of course by using the appropriate transformation variables. Next, we find the Laurent series of this equation. If the equation has Laurent series, then proceed by applying the theorems in [15]. It can be seen from [15] that there are necessary conditions that must be met by an equation in order to find an exact meromorphic solution, namely for Theorem 1 is = 0, while for Theorem 2 it is ∑ ( ) ∈ + ( ) = 0, where is the series coefficient. What distinguishes the application of Theorems 1 and 2 is how many types of Laurent series a nonlinear differential equation has. If an equation has only one type of Laurent series, then we use Theorem 1, but if the studied equation has more than one type of Laurent series, then we use Theorem 2. The further detailed and explicit steps of this method have been shown very clearly on paper [15].

Meromorphic Solutions of the Cubic Ostrovksy Equation
The Kudryashov method used in this paper refers to paper [15]. As previously explained, this paper examines the Ostrovsky Cubic equation, which is one type of modification of the Ostrovsky equation. The equation reviewed in this paper is where ≡ ( , ). It is clear that this equation is a nonlinear partial differential equation. Then, equation (1) is transformed into a nonlinear ordinary differential equation using the form where # = − $ . Using equation (2), equation (1) (3) is further expanded using equation (6) in [15], and two types of Laurent series are obtained, namely and From equations (4) to (11) it can be seen that the solution of equation (3)  with the values : and : as shown in equation (17). Next, we compute the simply periodic solution of the first type, namely where ( ) is equal to equation (14), and Laurent series of equation (20) around # = 0 is Next, we compare this series with the coefficients of the series " ( ) (#) (see equation (4) .Now we continue the analysis on calculating the meromorphic solution of the second type taking advantage of the symmetry in equation (12). These solutions possess poles of two types at the same type, and without losing the generality, we suppose that the point # = 0 is a pole of the first type.
where ( ) can be seen in equation (14), ( ) in equation (28), and √D in equation (21). Then, equation Thus, a meromorphic solution of the second type of period takes the form