Exact Combined Solutions for the ()-Dimensional Dispersive Long Water-Wave Equations

Abstract

The homogeneous balance of undetermined coefficient (HBUC) method is presented to obtain not only the linear, bilinear, or homogeneous forms but also the exact traveling wave solutions of nonlinear partial differential equations. Linear equation is obtained by applying the proposed method to the ()-dimensional dispersive long water-wave equations. Accordingly, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the ()-dimensional dispersive long water-wave equations are obtained directly. The HBUC method, which can be used to handle some nonlinear partial differential equations, is a standard, computable, and powerful method.

1. Introduction

Nonlinear partial differential equations (NLPDEs) are used to describe a variety of phenomena not only in physics [1, 2], thermodynamics [3], fluid dynamics [4, 5], and practical engineering [6, 7] but also in several other fields [8]. How to obtain the traveling wave solutions for NLPDEs is very important in the nonlinear phenomena [1, 9, 10]. In recent decades, there are many excellent methods, such as the -expansion method [11, 12], the homotopy perturbation method [13, 14], the Riccati-Bernoulli sub-ODE method [15, 16], the three-wave method [17, 18], the inverse scattering method [19, 20], the first integral method [21, 22], Hirota’s bilinear method [23, 24], the homogeneous balance method [25, 26], the iteration method [27, 28], the tanh-sech method [29, 30], and the extended homoclinic test method [31, 32], which are applied to obtain the exact traveling wave solutions of some NLPDEs.

The above traditional methods can be used to handle some well-known NLPDEs. However, there is no unified approach, which can be dealt with all NLPDEs. To obtain the traveling wave solutions of NLPDEs, Hirota’s bilinear method, the three-wave method, and the -expansion method are employed to investigate the traveling wave solutions of many NLPDEs. Unfortunately, some exact solutions are omitted by using Hirota’s bilinear method, the three-wave method, and the -expansion method if the NLPDEs can be linearized. To solve this problem, the HBUC method is proposed to derive the linear forms of NLPDEs.

In this paper, the ()-dimensional dispersive long water-wave equations (DLWEs) [33, 34] are investigated as follows: where represents the surface velocity of water along the -direction and gives the surface velocity of water along the -direction.

The DLWEs can also be derived from the well-known Kadomtsev-Petviashvili equation using the symmetry constraint. The DLWEs were used to model nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The DLWEs also appear in many scientific applications such as nonlinear fiber optics, plasma physics, fluid dynamics, and coastal engineering. Moreover, the solutions of the DLWEs are very helpful for coastal and engineers to apply the nonlinear water model to coastal and harbor design [33].

The DLWEs were investigated where different approaches were exploited. Wu et al. reported that the DLWEs exist of many nonpropagating hydrodynamical solitons both in theory and in experiment, and the DLWEs have no Painleve property, though the system is Lax or inverse scattering transformation integrable [35]. Paquin and Winternitz investigated the DLWEs by the Lie group method [36]. The extended mapping approach [37], the extended projective approach [38], and the tanh-sech method [33] are among many other methods that were used to handle the DLWEs. Much effort has been focused on the existence of propagating solitons [36], multiple soliton solutions, and rational solutions [33].

In this paper, the linear equation of the DLWEs is derived by the HBUC method. Then, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the DLWEs are obtained directly.

The remainder of this paper is organized as follows: the HBUC method is presented in Section 2. In Section 3, the HBUC method is used to obtain -multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of Equations (1) and (2) directly. In Section 4, some important conclusions are given.

2. Description of the HBUC Method

In this section, the following general NLPDE in two variables is considered: where is a polynomial function of its arguments; the subscripts and denote the partial derivatives of , respectively. The HBUC method consists of three steps as follows:

Step 1. Assume that the Equation (3) has a solution of the following form: where , , and , and (balance numbers) and (balance coefficients) are constants to be determined later.
The balance numbers can be determined by balancing the highest nonlinear terms and the highest order partial derivative terms. A set of algebraic equations for the balance coefficients is obtained by substituting Equation (4) into Equation (3) and balancing the terms with .

Step 2. If the NLPDEs can be linearized, the linear equation can be obtained by solving the set of algebraic equations and simplifying Equation (3) directly or after integrating some time (generally, integrating times equal to the orders of the lowest partial derivative of Equation (3)) with respect to and .

Step 3. Based on Step 1 and Step 2, by using traveling wave transformations and Equation (3) can be reduced to a linear partial differential equation where , , and are constants. Then, solving the linear partial differential equation (7) yields the exact combined solutions of Equation (3). Next, Equations (1) and (2) are chosen to obtain the combined solutions by applying the HBUC method.

3. Application to the ()-Dimensional DLWEs

Assume that the solutions of Equations (1) and (2) are of the forms where , , and , and (balance numbers) and and (balance coefficients) are constants to be determined later.

Balancing , , and in Equation (1) and and in Equation (2), it is required that

Solving the above algebraic equations, we get . Then, Equations (8) and (9) can be written as where and are constants to be determined later.

Substituting Equation (11) into Equations (1) and (2) and equating the coefficients of on the left-hand side of Equations (1) and (2) to zero yield a set of algebraic equations for and as follows:

Solving the above algebraic equations and noticing , we get . Substituting and back into Equation (11), we get

Substituting Equation (13) back into Equations (1) and (2), Equation (1) minus Equation (2) is where

Obviously, setting , we find that Equation (1) coincides with Equation (2). According to the above analysis, suppose that the solutions of Equations (1) and (2) are of the forms where is an arbitrary constant and is a function of that will be determined later.

Substituting Equation (16) into Equations (1) and (2) yields a single NLPDE where

Simplifying Equation (17) and integrating with respect to once, we get

Equation (19) is identical to where is an arbitrary function of .

Particularly, taking in Equation (20), the bilinear equation of Equations (1) and (2) is obtained as follows:

Equation (21) can be written concisely in terms of -operator as where

By using the property of -operator, Equation (22) is identical to where is an arbitrary function of .

Particularly, taking in Equation (24), we get a linear equation

Remark 1. We note that Equation (25) does not depend on the variable , instead it depends only on the variables .
Using the transformation Equation (25) is reduced to where is an arbitrary function of and the prime denotes the derivation with respect to .
There are three types of traveling wave solutions of Equation (27) as follows.
When , where ; , and are arbitrary constants; and is an arbitrary function of .
Generally, noticing the linear property of Equation (25), we can get the exact solution of Equation (25) as follows: where and ; , and are arbitrary constants; and are arbitrary functions of .
When where ; , and are arbitrary constants; and is an arbitrary function of .
Generally, noticing the linear property of Equation (25), we can get the exact solution as follows: where and ; , and are arbitrary constants; and are arbitrary functions of .
When where ; , and are arbitrary constants; and is an arbitrary function of .
Generally, noticing the linear property of Equation (25), we can get the exact solution as follows: where and ; , and are arbitrary constants; and are arbitrary functions of .
Generally, we can get combined solutions of Equation (25) as follows: where are given by Equations (29), (31), and (33), respectively.
Accordingly, we can get combined solutions of Equations (1) and (2) Choosing the appropriate parameters in Equation (35) can obtain all solutions of DLWEs in Ref. [15]. For example, setting , , , and , we get the -kink solutions and the -soliton solutions where and are arbitrary constants.
Setting , , , and , we get the singular solution (periodic solutions) where , and are arbitrary constants.
Setting , ,, , and , we get the rational solutions where , and are arbitrary constants.

Remark 2. We can deal with Equation (25) by using some assumptions. For example, suppose that , , and , we get where , and are arbitrary functions of , and is an arbitrary constant.
Suppose that and , we get where , and are arbitrary functions of .
Similarly, when , Equation (25) is reduced to . We can get the exact solution where and are arbitrary functions of .

Similarly, we can assume that ; then, a new solution of Equation (25) can be obtained. Being similar to the above process, we omit it.

4. Conclusions

The ()-dimensional dispersive long water-wave equations can be linearized by the HBUC method. Then, the -multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of Equations (1) and (2) can be obtained. Many well-known NLPDEs, such as the Whitham-Broer-Kaup equations, the Broer-Kaup equations, and the variant Boussinesq equations, can be handled by the HBUC method. The performance of the HBUC method is found to be simple and efficient. The HBUC method is also a standard, computable, and powerful method, which allows us to solve complicated and tedious algebraic calculations by the availability of computer systems like Maple.

Data Availability

The authors confirm that the data supporting the findings of this article are available within the article and are available on request from the corresponding author.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgments

The research is supported by a Project of Shandong Province Higher Educational Science and Technology Program (J18KB100, J18KA217), NSFC cultivation project of Jining Medical University (JYP2018KJ15), the Doctoral Research Foundation of Jining Medical University (2017JYQD22), the Fundamental Research Funds for the Central Universities (2452017373), and the Doctoral Research Foundation of Northwest A&F University (2452017007).