Abstract
The Konopelchenko-Dubrovsky (KD) system is presented by the application of the improved Bernoulli subequation function method (IBSEFM). First, The KD system being Nonlinear partial differential equations system is transformed into nonlinear ordinary differential equation by using a wave transformation. Last, the resulting equation is successfully explored for new explicit exact solutions including singular soliton, kink, and periodic wave solutions. All the obtained solutions in this study satisfy the Konopelchenko-Dubrovsky model. Under suitable choice of the parameter values, interesting two- and three-dimensional graphs of all the obtained solutions are plotted.
1. Introduction
Various complex nonlinear phenomena in different fields of nonlinear sciences such as fluid mechanic, plasma physics, and optical fibers can be expressed in the form of nonlinear partial differential equations (NPDEs). Many analytical methods for solving such type of equations have been developed and used by different researchers from all over the world. Zayed et al. [1] used the generalized Kudryashov in addressing some NPDEs arising in mathematical physics. Wang [2] investigated the Sharma-Tosso-Olver by using the extended hamoclinic test function method. Nofal [3] solved some nonlinear partial differential equation by using the simple equation method. Shang [4] obtained the exact solutions of the long-short wave resonance equation by using the extended hyperbolic function method. Wazzan [5] used the modified tanh-coth method in solving the generalized Burgers-fisher and Kuramoto-Sivashinsky equations. Ren et al. [6] applied the generalized algebra method to the (2+1)-dimensional Boiti-Leon-Pempinelli system. In general various studies in this context have been submitted to the literature [7–23].
In this paper, the Konopelchenko-Dubrovsky (KD) equations [24] are investigated by using the improved Bernoulli subequation function method [25, 26].
The Konopelchenko-Dubrovsky (KD) equations are given by [24]where and are constants.
Various analytical approaches have been used in obtaining the exact solutions to the Konopelchenko-Dubrovsky equations. Sheng [27] used the improved F-expansion method in addressing (1), Wazwaz [28] employed the tanh-sech method, the cosh-sinh method, and the exponential functions method for obtaining the analytical solutions to (1), and Kumar et al. [29] solved the KD equations by traveling wave hypothesis and lie symmetry approach. Song and Zhang [30] utilized the extended Riccati equation rational expansion method and Feng and Lin [31] applied the improved Riccati mapping approach. Bekir [32] used the extended tanh method. Xia et al. [33] employed the new modified extended tanh function method.
2. The IBSEFM
In this section, we present Improved Bernoulli subequation function method (IBSEFM) formed by modifying the Bernoulli subequation function method. Therefore, we consider the following four steps.
Step 1. Let us consider the nonlinear partial differential equation given asand the wave transformationSubstituting (3) into (2) gives the following nonlinear ordinary differential equation:
Step 2. The trial solution of (4) is assumed to be According to the Bernoulli theory, we have the general form of the Bernoulli differential equation aswhere is Bernoulli differential polynomial. Substituting (6) into (4) gives equations in degrees of as The values of , , and are to be determined by using the homogeneous balance principle.
Step 3. Setting the summation of the coefficients of and equating each summation to zero yields an algebraic system of equation.Solving this system of equation, we reach the values of and .
Step 4. Equation (6) has the following solutions depending on the values of and :Using a complete discrimination system for the polynomial of , we obtain the analytical solutions to (4) with the help of Wolfram Mathematica. For a better interpretation of results obtained in this way, we can plot two- and three-dimensional figures of analytical solutions by considering suitable values of parameters.
3. Application
In this section, we present the application of IBSEFM method to the Konopelchenko-Dubrovsky (KD) equations. Using the wave transformation on (1)we get the following system of nonlinear ordinary differential equations:Integrating the second equation in the system (12), we getInserting (13) into the first equation of (12), we get the following single nonlinear ordinary differential equation:Finally, integrating (14), we haveBalancing (15) by considering the highest derivative and the highest power, we obtain .
Choosing and , gives Thus, the trial solution to (1) takes the following form:where . Substituting (16), its second derivative along with into (15) yields a polynomial in . We collect a system of algebraic equations from the polynomial by equating each summation of the coefficients of which have the same power. Solving the system of the algebraic equations yields the values of the parameter involved. Substituting the obtained values of the parameters into (16), yields the solutions to (1).
Case 1.
Case 2.
Case 3.
Case 4.
Case 5. Substituting (17) into (16), givesSubstituting (18) into (16), givesSubstituting (19) into (16), givesSubstituting (20) into (16), givesSubstituting (21) into (16), gives
4. Results and Discussion
This study uses the IBSEFM to obtain some travelling wave solutions to the Konopelchenko-Dubrovsky equations such as the singular soliton, kink-type soliton, and the periodic wave solutions. The results are presented in exponential function structure in Section 3. When the basic relation is used, solutions (22)-(26) becomewhere for valid solution,where for valid solution.
It has been reported in the literature that some computational approaches have been utilized to obtain the solutions of the Konopelchenko-Dubrovsky equations. Sheng [27] reported that some Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions improved F-expansion method. Wazwaz [28] obtained some solitary wave, periodic wave, and kink solutions to this equation by using the tanh–sech method and the cosh–sinh scheme. The traveling wave hypothesis and the lie group approach were used on the Konopelchenko-Dubrovsky equations by Kumar et al. [29] and some solitary wave, periodic singular waves, and cnoidal and snoidal solutions were reported. Song and Zhang [30] employed the extended Riccati equation rational expansion method to (1) and obtained some rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions. Via the improved mapping approach and variable separation method, Feng and Lin [31] constructed some solitary wave solutions, periodic wave solutions, and rational function solutions. Bekir [32] constructed some kink and singular solitons by using the extended tanh method. Xia et al. [33] employed the modified extended tanh function method to the Konopelchenko-Dubrovsky equation, and soliton-like solutions, periodic formal solution, and rational function solutions were successfully constructed. We observed that the analytical method use in this study revealed the wave solutions in exponential function structure that can be converted to solutions with hyperbolic and trigonometric function structure.
Remark 1. Solutions (22), (23), and (24) are singular soliton solutions. Solution (25) is a kink-type soliton and solution (26) is a periodic wave solution.
5. Conclusions
In this paper, the IBSEFM is applied to the KD equations. We successfully obtained some new traveling wave solutions to the studied model such as complex and exponential function solutions. We presented the 2D and 3D graphs to each of the obtained solutions with the help of some powerful software program.
It has been observed that these solutions have verified nonlinear KD equations. These traveling wave solutions, Figures 1, 2, 3, 4, 5, and 6, have been formed. Moreover, if we chose , , and , we could obtain some other new traveling wave solutions to the nonlinear KD equations. When we compare our results with some of the existing results in the literature, we observe that the results reported in this study especially (25) and (26) are new complex and exponential solutions. All the reported solutions in this study verify the Konopelchenko-Dubrovsky equation. Our results might be useful in explaining the physical meaning of various nonlinear models arising in the field of nonlinear sciences. IBSEFM is powerful and efficient mathematical tool that can be used to handle various nonlinear mathematical models.
The performance of this method is effective and reliable. In our research, these traveling wave solutions have not submitted to the literature in advance.
Data Availability
The data used to support the findings of this study are included within the article.
Disclosure
Part of this article is presented in 3rd International Conference on Computational Mathematics and Engineering Sciences (CMES 2018).
Conflicts of Interest
The author declares that he has no conflicts of interest.
Authors’ Contributions
The author read and approved the final manuscript.
Acknowledgments
This work is supported by MAÜ.BAP.18.MMF.020.