Abstract
A modified tanh-coth method with Riccati equation is used to construct several explicit solutions of (3+1)-dimensional Kudryashov-Sinelshchikov equations in bubble gas liquid flow. The solutions include solitons and periodic solutions. The method applied can be used in further works to obtain entirely new solutions to many other nonlinear evolution equations.
1. Introduction
Many complex phenomena in nature are describe in various scientific and industrial fields and especially in areas of physics such as fluid mechanics [1], optical fibres [2], plasma physics [3], and so on. Nonlinear evolution equations are used to describe these nonlinear phenomena, and that has led to the development of methods to look for exact solutions of nonlinear partial differential equations [4].
Recently, Kudryashov and Sinelshchikov [5] developed a (3+1)-dimensional nonlinear evolution equation in a model of wave propagation in bubbly fluid flow. The model equation has gained a lot of attention where Bäcklund transformation and conservation laws [6], bifurcation [7], and density-fluctuation [8] analysis were carried out the important evolution equation.
Many methods have been developed to find the explicit solutions of nonlinear evolution equations; example of such methods are the first integral method [9], Jacobi elliptic function method [10], Hirota bilinear method [11], Wronskian determinant technique [12], F-expansion method [13], Darboux Transformations [14], Backlund transformation method [6], Miura transformation [15], homotopy perturbation method [16], and Adomian decomposition method [17]. Many algebraic methods were proposed so far, such as tanh method which was proposed by Malflie [18]. Fan [19] extended the tanh method and obtained new exact solution that cannot be obtained by using the conventional tanh method. Further extension called tanh-coth was proposed by Wazwaz [4] which provides a wider applicability for solving nonlinear evolution equations. A modification was also proposed by El-Wakil [20] and Soliman [21, 22].
In this paper, we present two equations (3+1)-dimensional Kudryashov and Sinelshchikov equation written aswhere , , and represent the nonlinearity, dissipation dispersion terms, while and stand for transverse variation of wave in and directions; we assume all the coefficient to be constant parameters. We shall use modified tanh-coth to obtain many explicit exact solutions for (3+1)-dimensional Kudryashov-Sinelshchikov equations. The travelling wave solution of a special case of (1) is given in [7]. Note that when , (1) reduces to their two-dimensional counterpartswhich were widely studied in [16, 17, 23, 24]. Furthermore when , (1) reduces to one-dimensional Korteweg-de-Vries equations (KdV) and one-dimensional Korteweg-de-Vries-Burgers equations (KdVB). These equations have many applications in fluid dynamics [25], plasma physics [26], cold atomic gases [27], and so on. It is well known that the KdV and KdVB equations take the formThese two equations have been extensively investigated by many researcher in recent years [28–32].
In the following section, we briefly describe the modified tanh-coth method in three variables and the application of the modified tanh-coth to (3+1)-dimensional Kudryashov and Sinelshchikov equations and their explicit solutions will be given in Section 3. Conclusions will follow in Section 4.
2. Description of the Method
Let us consider a three-dimensional nonlinear partial differential equation in the formUpon using the transformation , where , (4) can be reduced to an ordinary differential equation (ODE)where the independent variable composed of the new variable , is the wave number, is wave speed, are the spatial coordinates, and is time. The ODE (5) is then integrated as long as all terms contain derivatives, where integration constants are considered zero. The simplified ODE is then solved via the tanh-coth method [28], which admits the use of finite expansionwith the Riccati equationBy changing of variablewhere , , and shall be given, while , are constants to be determined later. The positive integer can be determined by considering the homogeneous balance [33] between the highest order derivatives and the most nonlinear terms appearing in (5). If is not an integer, then a transformation formula should be used to overcome the difficulty. Substituting (6) into (5) and make use of (7) and (8) yield an equation in terms of . Equating the coefficients of all the power to zero, we obtain a set of algebraic equations for , and .
For the aforementioned method, expansion (8) reduces to the standard tanh-coth method. In this work, we shall use the following solutions of Riccati equation: and , then , , then .
3. Application of the Method
In this section, we will apply the method to the (3+1)-dimensional Kudryashov-Sinelshchikov equation for both dispersive and dissipative cases.
3.1. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion
The generalised equation is given asBy using the transformation(9) is reduced to an ODE of the formIntegrating (11) twice and assuming the integration constant to be zero, we obtainBalancing terms and in (12), we get , which enables us to make the following ansatz:Substituting (13) into (12) and making use of (7) and (8), we obtain a system of equation in terms of , , , and .
Case I. If are substituted into the system of (14) and solving the equations, we have the following solutions:Substituting (15a) and (15b) into (13) using , with , we get the solutions in simple form as
Case II. If we set into the system of (14) and solve the equations, we have the following solutions:
Substituting into (13), making use of (18a), we obtain a soliton solutionand periodic solutionswhere . Using (18b), we have a soliton solution, written in simple form asand periodic solutionsEquation (18c) gives a soliton solutionand periodic solutionswhere . Equation (18d) gives a soliton solutionand periodic solutionsEquation (18e) gives soliton solutionsand periodic solutionsFinally (18f) gives soliton solutionsand periodic solutions
The graphical representation of the solitary wave profile to the solution of (9) at is given in Figure 1, while for is given in Figure 2.
3.2. The (3+1)-Dimensional Kudryashov-Sinelshchikov Equation with Dispersion and Dissipation
The equation is given asBy using the transformation(39) is reduced to an ODE of the formIntegrating (41) twice and assuming the integration constant to be zero, we obtainBalancing and in (42), we get , which enables us to make the following ansatz:Substituting (43) into (42) and making use of (7) and (8), we obtained a system of equations in , , and .
Case I. If are used in the system (44) and solving the equations, we have obtained the following solutions:Substituting (45a) and (45b) into (43) and using , we getand
Case II. If we set , , and into system (44) and solve the equation, we have
It can be shown that if , the set of values in (48a), (48c), and (48e) give the same solution asThe set values in (48b), (48d), and (48f) give exactly the same solution as
We use to represent the solution to (39) graphically; Figure 3 gives a kink wave profile at , while at it is given in Figure 4.
4. Conclusion
In this paper, the modified tanh-coth method has been applied to nonlinear evolution equations. The explicit travelling wave solutions of the generalised (3+1)-dimensional Kudryashov-Sinelshchikov equation are derived. Some of the solutions obtained are in good agreement with the known exact solutions [7] using specific parameter values. The results show that the modified tanh-coth method is an efficient approach for the seeking solutions of such type of nonlinear evolution equations.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.