Abstract
In this article, we study the generalized ()-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ()-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.
1. Introduction
Nonlinear issues are widespread in some natural disciplines, and many difficult problems in some disciplines can be reduced to solving a certain PDE or investigating some properties of a PDE [1, 2]. With the rapid development of research fields like hydrodynamics and quantum physics, it has become increasingly important to investigate the exact solutions and certain properties of nonlinear evolution equations [3–5]. Compared with the PDEs with constant coefficients, the PDEs with variable coefficients can describe richer natural phenomena and construct more detailed and complex physical models [6–8].
In this paper, we focus on the generalized ()-dimensional Boiti-Leon-Pempinelli equation with time-part variable coefficients as where , , , , and are any functions with respect to time . It represents the development of the components in the horizontal velocity in the and directions when the water wave propagates in a channel of unchanging depth and infinitely small width [9]. The vcBLP is conditionally integrable, and the necessary conditions for it to be Painlevé integrable are and . Some exact solutions of vcBLP were obtained in [10] by extended tanh-function method. We can find some periodic solutions and soliton solutions of vcBLP obtained with the homogeneous balance method in [11], and the conservation laws for the constant coefficients BLP were discussed in [9]. By reviewing the relevant literatures, no one has studied vcBLP using the Lie group method.
The outline of this article is as follows. In Section 2, we construct the equivalence transformations and differential invariants of vcBLP. Based on these, we give an explicit transformation to its constant coefficient form. In Section 3, the infinitesimal generators of vcBLP are obtained using the Lie group method, and then, the optimal system for the one-dimensional subalgebras is constructed. In Section 4, we obtain six sets of ()-dimensional reduced PDEs by similarity reductions to vcBLP. In Section 5, some exact solutions of vcBLP are shown using the -expansion method on the basis of the reduced PDEs in above section. In Section 6, we use the multiplier method to calculate the conservation laws of vcBLP. We can find the conclusions of this paper in Section 7.
2. Equivalence Transformations and Differential Invariants of vcBLP
In this part, we construct the equivalence transformations [12] of Equation (1). The equivalence transformation of Equation (1) is a nondegenerate point transformation which from to [13]. It has the same differential structure but different coefficient functions than the original equation. First, we assume that the auxiliary conditions are with the one-parameter group of equivalence transformations is determined on the basis of and is the group parameter. The vector field or generators of Equation (1) which corresponds to transformations (3) as
Since Equation (1) is invariant under the above transformations (3) and there exists the 3rd derivatives, we have to use the 3rd prolongation . We define that
On the basis of the above Equation (5) and , the 3rd prolongation can be written as where with
Under transformations (3), the invariance of Equation (1) requires that to satisfy the conditions
We can obtain a determining equation by bringing Equation (6), Equation (7), and Equation (8) into Equation (9). Subsequently, solving this determining equation, we get where , and are arbitrary smooth functions. The corresponding infinite dimensional equivalence group is generated by the following operators:
In the following, our task is to derive the zero-order differential invariants and the first-order differential invariants. We assume that the form of the zero-order differential invariant is
Applying the invariant test to the operators (11), we can obtain and the above equations can be reduced to solving this system, we get that the zero-order differential invariants are with invariant equations are
Next, we derive the interesting first-order differential invariants. Similar to the above case, we suppose that the first-order differential invariant is the form
In order to get the first-order differential invariants, we need to make the first prolongation of the operators using , and Equation (11) can be rewritten as
The invariance test associated with Equation (18) is solving this system yields with invariant equations are
Based on the above facts, we use differential invariants to give the transformation of vcBLP to its constant coefficient form. We take the constant coefficient form of Equation (1) as where are arbitrary nonzero constants. To obtain the transformation from Equation (1) to Equation (22), we need a necessary condition that the coefficient functions of Equation (1) must satisfy the following equations
Under this condition, the more general form of the coefficient functions are therefore, Equation (1) becomes and there exists transformation where are arbitrary constants and is an arbitrary function. It is easy to verify that the above transformation maps Equation (1) into so the constant coefficient form in other literature [9] is a special case of Equation (1), and it is easy to find that Equation (24) satisfies the conditions for integrability.
We take then the above Equation (27) and Equation (28) become subsequently, we integrate Equation (29) once with respect to yields therefore, when , the above Equation (27) and Equation (28) can be converted into the well-known constant coefficient Burgers equation.
3. Lie Classical Symmetry Analysis of vcBLP
Lie group method is an effective way to find invariant solutions and to explore certain properties by reducing the dimensionality of the equations [14]. It has been described in sufficient detail in many literatures [15–18]. To begin with, we suppose that the one-parameter Lie group in Equation (1) is and Equation (1) remains invariant under transformations (32).
The vector field or infinitesimal generator of Equation (1) which corresponds to transformations (32) is its 3rd prolongation is written as . Equation (1) remains invariant under transformations (32), which requires that
Expanding Equation (34), the invariant conditions are redefined as where with , , and are the total differentiation of , , and , respectively.
Solving Equation (35) yields the following results as where are any constants. Furthermore, the coefficient functions , , , , and in Equation (1), which depend on time and have to satisfy the conditions
Thus, the infinitesimal generators of Equation (1) are expanded by the below vector field
Depending on the , we have the following four groups where are one-parameter Lie point symmetry groups. It is not difficult to find that and are space translations and is a dependent variable translation. For , it is a time translation when is an arbitrary constant. For the combination of and can be understood as a translation along a certain direction, the group invariant solutions are the traveling wave solutions. The most important application of the traveling wave solutions is to construct soliton solutions of the PDE. The soliton reflects a rather common nonlinear phenomenon in nature, which is mainly characterized by its superstability, i.e., the wave shape remains stable after the collision of two solitary waves with different velocities. We can also understand solitons as local traveling wave solutions of nonlinear development equations. Also, symmetry has a great connection with conservation laws in physics; for example, space translation corresponds to momentum conservation, and time translation corresponds to energy conservation.
Generally speaking, it is possible to construct it group-invariant solutions for arbitrary subgroup or subalgebra. However, the Lie group has infinitely many subgroups with the same dimension, and it is impossible to compute the group-invariant solutions of all subgroups [19]. We have to sort them into some mutual equivalence, which requires the optimal system of one-dimensional subalgebras to be computed.
To get the optimal system, we start by constructing the commutator table as Table 1 with the help of Lie algebra [15, 20].
With reference to Table 1, the adjoint relationship table can be acquired as Table 2, where with is an infinitesimal real number.
Through Tables 1 and 2, it is quite simple to obtain the optimal system for the one-dimensional subalgebras of Equation (1), which is given by the following forms
4. Similarity Reductions of vcBLP
In the first step, we reduce Equation (1) to the ()-dimensional PDEs based on the above optimal system using the similarly reductions. The similarity variables and the ()-dimensional reduced PDEs can be found in Table 3, and the expressions for the coefficient functions depending on time can be found in Table 4. We only show the process of calculation with the example of case ; the results of other cases are in Tables 3 and 4.
For this Lie vector, its corresponding characteristic equation is solving this equation to obtain the relative similarity variables are
Through Equation (44), Equation (1) is reduced to the following forms: here to ensure that there are only two independent variables and in Equation (45), and the coefficient functions satisfy conditions (38), so the expressions for the coefficient functions are where are arbitrary constants. Substituting Equation (46) to Equation (45) yields the reduced PDEs as
5. Exact Solutions of vcBLP
In this section, the exact solutions of corresponding reduced PDEs are found for some cases in Table 3 using the -expansion method [21, 22]. For computational simplicity, we let for Tables 3 and 4.
Case 1.
First, we assume that the traveling wave variables are [23].
where is the traveling wave speed to be determined.
Next, using Equation (48), the ()-dimensional reduced PDEs are transformed into ordinary differential equations (ODEs)
where represents the derivative of . Through homogeneous balance, we assume that the solutions of the ODEs (49) can be expressed as
where are coefficients to be determined and satisfies
with and are arbitrary constants.
Substituting Equation (50) and Equation (51) into Equation (49), then collecting the coefficients of the same order of and making them equal to zero yields
Solving Equation (52), we get
with are any constants.
By applying Equation (53), Equation (50) can be rewritten as
where . Substituting the solutions of the Equation (51) into Equation (54), we obtain that the three types of exact solutions of reduced PDEs are as follows [24]:
When where , , and are any constants.
When ,
where , , and are any constants.
When ,
where , , and are any constants.
By substituting the corresponding similarity variables in Table 3 into the above solutions, the exact solutions of vcBLP are obtained as follows:
When ,
where , , and are any constants.
When ,
where , , and are any constants.
When ,
where , , and are any constants.
For the other cases, we also obtained three types of exact solutions of the reduced PDEs by the method of -expansion, and we will not repeat the calculation process, just list the results of the calculation.
Case 2.
When ,
where , , , and are any constants. We choose parameters , , , , , , , and ; the images of (61) and (62) are, respectively, in Figures 1 and 2.
When ,
where , , , and are any constants. We choose parameters , , , , , , , and ; the images of (63) and (64) are, respectively, in Figures 3 and 4.
When where , , , and are any constants. We choose parameters , , , , , and ; the images of (65) and (66) are, respectively, in Figures 5 and 6.
Case 3. .
When ,
where , , , and are any constants.
When ,
where , , , and are any constants.
When ,
where , , , and are any constants.
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(a)
(b)
(c)
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(b)
(c)
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We can easily find that all the above solutions are traveling wave solutions when takes any constant, and all the above solutions are group-invariant solutions when is an arbitrary function of .
6. Conservation Laws of vcBLP
The conservation law is extremely valuable for studying the integrability and exploring the exact solutions of PDE [25–27]. We can use it to explain many physical phenomena described by PDE [28–30]. In this section, we use the multiplier method [31–33] to calculate the conservation laws of vcBLP. The first order multipliers and of vcBLP can be obtained by the following equations with where and are Euler operators.
By expanding Equation (70) and decomposing them according to the derivatives of , we can obtain a system and solve this system can obtain where and are arbitrary constants, and are arbitrary functions about , and is an arbitrary function about .
Therefore, we can obtain the following low-order conservation laws and the corresponding multipliers. The details are discussed as below.
Case 1. .
Case 2. .
Case 3. .
Case 4. .
The results obtained above have been verified using Maple software to ensure that holds.
7. Conclusions of This Article
In this paper, the zero-order and first-order differential invariants were determined by using the equivalence group of vcBLP. With the help of these, the explicit transformation to its constant coefficient form was given. Subsequently, we have successfully performed the Lie symmetry analysis of vcBLP, obtained some exact solutions, and plotted the corresponding 3-dimensional figures to describe the evolution of the solutions. Moreover, the four conservation laws of vcBLP were obtained using the multiplier method.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 11505090), the Research Award Foundation for Outstanding Young Scientists of Shandong Province (No. BS2015SF009), the doctoral foundation of Liaocheng University under Grant No. 318051413, and the Liaocheng University level science and technology research fund No. 318012018.