On the Nonlinear Mathematical Model Representing the Coriolis Effect

Nonlinear partial dierential equations are mathematical models that represent many natural phenomena. Such mathematical models also take an active role in various branches of science. For example, it is an eective model for analyzing events in many science subjects, such as plasma physics and geophysics. Researchers need to investigate the solutions of such nonlinear mathematical models. Because the obtained ndings allow the analysis of the event, for this reason, there are various methods related to numerical or exact solutions of partial dierential equations in the literature. Some of them are respectively the trial equation method [1], the extended trial equation method [2], the newfunction method [3–7], the improved Bernoulli subequation function method [8–10], Kudryashov method [11, 12], the sine-Gordon equation expansion method [13–16], generalized auxiliary equation method [17], rst integral method [18], new extended direct algebraic method [19], Hirota bilinear method and the tanh-coth method [20, 21], the modied exponential function method [22], and Chebyshev–Tau method [23]. e GpKdv equation, an important model in geophysics, is analyzed in this study.is equation is especially preferred because it contains a coecient representing a Coriolis eect, which is in the model and is very important for science. e Coriolis eect is the name given to the phenomenon that aects the scattering of ¥uids such as water or air in nature while moving on the Earth. For example, while storms move counterclockwise at the north pole, they move clockwise at the south pole. e direction of the circular air¥ow, which generally occurs in natural events such as storms, is from the high-pressure region to the low-pressure area. However, this orientation cannot move vertically because the factor that prevents this and causes the storm to be blown is the Coriolis eect. erefore, it can be said that it has an active role in the occurrence of the dierence in direction at the poles. In order to analyze the Coriolis eect, the GpKdv equation in this study is [24–26]


Introduction
Nonlinear partial di erential equations are mathematical models that represent many natural phenomena. Such mathematical models also take an active role in various branches of science. For example, it is an e ective model for analyzing events in many science subjects, such as plasma physics and geophysics. Researchers need to investigate the solutions of such nonlinear mathematical models. Because the obtained ndings allow the analysis of the event, for this reason, there are various methods related to numerical or exact solutions of partial di erential equations in the literature. Some of them are respectively the trial equation method [1], the extended trial equation method [2], the newfunction method [3][4][5][6][7], the improved Bernoulli subequation function method [8][9][10], Kudryashov method [11,12], the sine-Gordon equation expansion method [13][14][15][16], generalized auxiliary equation method [17], rst integral method [18], new extended direct algebraic method [19], Hirota bilinear method and the tanh-coth method [20,21], the modi ed exponential function method [22], and Chebyshev-Tau method [23].
e GpKdv equation, an important model in geophysics, is analyzed in this study. is equation is especially preferred because it contains a coe cient representing a Coriolis e ect, which is in the model and is very important for science. e Coriolis e ect is the name given to the phenomenon that a ects the scattering of uids such as water or air in nature while moving on the Earth. For example, while storms move counterclockwise at the north pole, they move clockwise at the south pole. e direction of the circular air ow, which generally occurs in natural events such as storms, is from the high-pressure region to the low-pressure area. However, this orientation cannot move vertically because the factor that prevents this and causes the storm to be blown is the Coriolis e ect. erefore, it can be said that it has an active role in the occurrence of the di erence in direction at the poles.
In order to analyze the Coriolis e ect, the GpKdv equation in this study is [24][25][26] where u is a function representing the independent surface feed and c is the Coriolis coe cient. e Coriolis constant c may di er from region to region depending on the depth of the water. Also, it is of great importance to include the Coriolis term c u x in the KdV equation in order to be able to observe the e ect of the Earth's rotation on the ows in tsunami waves.

Description of the Method
In this section, the modified exponential function method that is well known as an effective technique for obtaining many solution functions such as traveling wave, soliton, and periodic wave of nonlinear mathematical models will be introduced [27,28]. Let us take the general form according to the function derivative variables used in the GpKdV equation as follows: P u, u x , u t , u xx , u xxx , . . . � 0.
(2) e wave transforms according to derivative variables in equation (2):   2 Mathematical Problems in Engineering where k represents the height of the wave and c represents the frequency of the wave. e derivative terms in equation (2) are reduced to the following nonlinear ordinary differential equation form with a single derivative variable using the wave transform in equation (3): e solution function sought according to the method in which the nonlinear partial differential equation and the nonlinear ordinary differential equation obtained by applying the wave transform to this partial equation are planned to be provided are as follows, respectively: where In the solution function assumed as equation (5), there are parameters that must be determined, respectively. e first of these is the omega function that is a solution of the nonlinear differential equation While integrating equation (6) according to ξ, various Ω functions are obtained according to the states of λ and μ in the omega function as follows [29]: where E is an integration constant.
Family 4: when μ ≠ 0, λ ≠ 0, and λ 2 − 4μ � 0, Family 5: when μ � 0, λ � 0, and λ 2 − 4μ � 0, After obtaining the omega functions as above, a relation between m and n is found by applying the balancing principle to equation (4) according to the second operation of the method. en, by determining a parameter suitable for m in this resulting relationship, the n parameter is found. In this way, the boundaries of the total symbols are determined. en, the coefficients A 0 , A 1 , . . . , A n , B 0 , B 1 , . . . , B m in equation (5) will be determined. For this, when the necessary derivative terms of equation (4)

Application
When the derivative terms in the nonlinear mathematical model (1) are substituted using the wave transform in equation (3), we obtain the following equation: If the balance procedure is applied to the term u ″ with the highest order derivative in equation (12) and u 2 of the highest order nonlinear term, the balance equation is obtained as follows: Accordingly, if m � 1 in equation (13), it is obtained as n � 3. If these values are substituted in the solution function assumed to provide equation (1), the solution function and the derivative terms in the nonlinear ordinary differential equation are obtained as follows: After the terms in equation (14) are replaced in equation (12), the following coefficients are obtained when the system Mathematical Problems in Engineering of algebraic equations consisting of coefficients is solved by classifying them according to the powers of e Ω(ξ) .

Case 1.
ese coefficients are replaced in the solution functions and derivatives in equation (14). en, the omega functions in the following family cases are put into the solution function. Also, it is checked that the solution function satisfies the nonlinear ordinary differential equation, Figure 1 and then the nonlinear partial Figure 2 differential equation is performed with the help of the Mathematica.
Family 1: Family 2: Mathematical Problems in Engineering 5 Family 3: Family 4: In the case of Family 5, the solution function is determined as undefined since λ and μ are zero and λ 2 − 4μ � 0 is zero. For this reason, no graphic drawings related to the mathematical model could be made.   Mathematical Problems in Engineering 7

Conclusion
In this study, we successfully obtained the new traveling wave solutions of the GpKdv equation using the modified exponential function method. When the types of solution functions that provide the mathematical model are analyzed, it is seen that hyperbolic and trigonometric functions with periodic functions are singular solitons. We plotted the contour surfaces under appropriate constants, where all wave solutions are two and three-dimensional, and the Coriolis effect represented by the mathematical model is effectively seen. In Figures 1-14, the simulated graphs representing the mathematical model, the behavior of the solution function u representing the free surface progression, and the Coriolis effect accordingly are seen. e mentioned method is understood to be very effective in obtaining the wave solutions of such nonlinear differential equations. Because there is an exponential function in the solution function u, which is preferred as a hypothesis according to the method. In addition, the omega function, which is used as the power of the exponential function, is also a function that satisfies the Riccati equation. Considering all these situations, it allows obtaining solution functions with periodicity compared to other methods. is gives the researchers information about the behavior of the mathematical model used in a desired time interval. In addition, as far as we know, the obtained solution functions are included in the literature for the first time. When the literature is searched for this mathematical model, it is seen that various soliton solutions are also obtained using numerical methods, the homotopy perturbation method, finite element, and the Hirota bilinear method, which is expressed as an analytical method. Consequently, we believe the obtained solutions can effectively demonstrate the Coriolis effect in geophysics.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.