Recent Advances in Symmetry Groups and Conservation Laws for Partial Differential Equations and ApplicationsView this Special Issue
Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients
The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions of . Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.
The Lie group method is a powerful tool to perform Lie symmetry analysis, study conservation laws, and look for exact solutions of nonlinear partial differential equations (NLPDEs) [1–4]. The notion of conservation laws, which plays an important role in the study of nonlinear science, is used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence, uniqueness, and stability analysis [5, 6]. In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. On the other hand, seeking exact solutions of NLPDEs has become one central theme of perpetual interest in mathematical physics as explicit solutions will be helpful to better understand the phenomena described by the equations. To get exact solutions of NLPDEs, many effective methods have been presented such as inverse scattering method , Hirota’s bilinear method , and Painlevé expansion method . Among them the Lie group method offers a systematic algorithmic procedure to find the symmetry reductions and exact solutions of a partial differential equation. In this paper, we use the Lie group method to consider a time-dependent Kadomtsev-Petviashvili equation: with time-dependent coefficient functions , and .
The above equation was also called “a 2D KdV equation with time-dependent coefficients” by Hereman and Zhuang ; they performed Painlevé analysis for (1) and found that (1) was Painlevé integrable when . Equation (1) can be reduced to the KdV equation () or the KP equation (). Equation (1) can also be reduced to the cylindrical KdV equation when or the cylindrical KP equation when . The KdV and KP equations and their cylindrical generalizations (2a) and (2b) are all known to be completely integrable . Zhang et al.  performed Painlevé analysis for (1) and constructed bilinear auto-Bäcklund, analytic solutions in the Wronskian form. Soliton-like solutions, Jacobi elliptic function-like solutions, and other exact solutions have been obtained by the method of auxiliary equations [12–15]. Elwakil et al.  used the homogeneous balance method to study the exact solutions of (1). Based on the homogeneous balance method and Clarkson-Kruskal method, direct reduction and exact solutions have been obtained in  by Moussa and El-Shiekh. The bilinear formalism, bilinear Bäcklund transformation, and Lax pair of (1) have been obtained by the binary Bell polynomial approach in . As far as we know, conservation laws and symmetry reductions for (1) have not been studied.
The rest of the paper is organized as follows. In Section 2, the Lie group method is applied to the time-dependent Kadomtsev-Petviashvili equation (1) and thus Lie symmetries of (1) are obtained. In Section 3, using the obtained symmetries and the general theorem on conservation laws by Ibragimov, nontrivial and time-dependent conservation laws are derived. In Section 4, we use the symmetry to get symmetry reductions and new exact solutions of (1). The last section is a short summary and discussion.
2. Lie Symmetry Analysis of (1)
Generally speaking, Lie symmetry denotes a transformation that leaves the solution manifold of a system invariant; that is, it maps any solution of the system into a solution of the same system, so it is also called geometric symmetry. In this section, we will perform Lie symmetry analysis for (1) by the classical Lie group method. Suppose that Lie symmetry of (1) is expressed as follows: where , , , and are undetermined functions with respect to , , , and . According to the procedures of Lie group method, the vector field (3) can be determined by applying the fourth prolongation of to (1) and thus the undetermined functions , , , and must satisfy the following invariant condition: where Substituting (5) into (4) with being a solution of (1), that is, we obtain the determining equations of symmetry (3). Solving the determining equations with the aid of Maple, we can get the following cases.
Case 1. When and are arbitrary functions, where and are arbitrary functions. It shows that (1) admits an infinite-dimensional Lie algebra of symmetries where
Case 2. When , , , , and , where , , , and are constants and and are arbitrary functions. This shows that the symmetries of equation have the form of where is a one-dimensional Lie algebra of symmetries and and are two infinite-dimensional Lie algebra of symmetries as expressed by (9) with .
Case 3. When , ., and , where and are arbitrary functions. It shows that the KP equation admits an infinite-dimensional Lie algebra of symmetries where is a constant and ; and are expressed by (9) with .,
Case 4. When , , and , where and are arbitrary functions, is an integral constant, and and satisfy the following ordinary differential equation: This shows that, under the condition (19), the equation admits an infinite-dimensional Lie algebra of symmetries where and are expressed by (9):
3. Conservation Laws for (1)
3.1. A General Theorem on Conservation Laws
As expressed through the famous Noether theorem, for a given differential equation, there is a close connection between Lie symmetries and conservation laws. To derive conservation laws of (1), we use the following conclusion proved by Ibragimov in .
Theorem 1. Every Lie point, Lie-Bäcklund, and nonlocal symmetry of a system of equations with independent variables and dependent variables; provides a conservation law for system (24) and the corresponding adjoint system Then the elements of the conservation vector are defined by the following expression: with
3.2. Conservation Laws for (1)
To search for conservation laws of (1) by Theorem 1, adjoint equation and formal Lagrangian of (1) must be known. We first construct its adjoint equation. Following the idea in , the adjoint equation of (1) is where is a new dependent variable with respect to , , and .
By means of the symmetries of (1), conservation laws of the system consisting of (1) and (28) can be derived by Theorem 1. However, we are only interested in the conservation laws of (1). Therefore one has to eliminate the nonlocal variable which is introduced in the adjoint equation. To solve this problem, the concepts of self-adjointness, quasi-self-adjointness, and nonlinear self-adjointness are developed [20–24]. In the following, we will discuss the adjointness and nonlinear adjointness using these definitions.
Equation (1) is said to be self-adjoint if the equation obtained from the adjoint equation (28) by the substitution is identical with the original equation (1). It is easy to see that (28) is not identical with (1) when , so (1) is not a self-adjoint equation. According to the definition of nonlinear self-adjointness , (1) is said to be nonlinearly self-adjoint if its adjoint equation (28) is satisfied for all solutions of (1) upon a substitution In other words, (1) is nonlinearly self-adjoint if and only if where is an undetermined and smooth function.
From (31), we can get the following equation: Solving the above system with the aid of Maple, the final results read as where , , , and are arbitrary functions. In summary, we have the following statements.
Theorem 2. The time-dependent KP equation (1) is nonlinearly self-adjoint.
For the symmetry in Case 1, the corresponding components of the conservation laws are
For the symmetry in Case 2, the corresponding components of the conservation laws are Here we should note that the coefficient function in the expression of , , and satisfies , , , and are constants, and , .
For the symmetry in Case 3, the corresponding components of the conservation laws are
For the fourth symmetry, the two functions and are determined by the differential equation (19) and they have many explicit solutions. For simplicity, we take ; then and . When , the corresponding Lie symmetry is and the components of the conservation laws are
We should mention that in the above components of the conservation laws for (1) and (28), is a solution of (1) and is a solution of the adjoint equation (28). Making use of the explicit solutions of (28), local conservation laws for (1) can be obtained. For example, when and in (34), where and are arbitrary functions, is an exact solution of (28). Substituting (40) into the above four conservation laws, we can obtain time-dependent and local conservation laws for (1). Here we take as an illustrative example; when the components of the conservation laws become
These are local and explicit conservation laws of (1). Next we show that the above conservation laws () are nontrivial: Obviously, if , are not zero at the same time, . And we can easily check that
4. Symmetry Reductions and New Exact Solutions of (1)
In Section 2, we obtain the Lie symmetries of (1). In this section, we will investigate the symmetry reductions and exact solutions for the equation. Using the obtained symmetries (3), similarity variables and symmetry reductions can be found by solving the corresponding characteristic equation: For the four different cases, we determine the following symmetry reductions and exact solutions of (1).
4.1. For the Symmetry in Case 1, Where and Are Arbitrary Functions
(i) When , , we can obtain and is a solution of the following reduction equation: From the above equation, we can obtain an algebraically explicit analytical solution for (1): where and are arbitrary functions of .
(ii) When , , the corresponding symmetry is By the characteristic equations of the symmetry, we have , . Substituting it into (1), we get a symmetry reduction of (1): If the coefficient functions , ., the obtained symmetry reduction can be simplified to Integrating (50) with respect to and taking the constant of integration to zero, we get the following equation: Equation (51) is the (1 + 1)-dimensional generalized KdV equation with variable coefficients. To the best of our knowledge, exact solutions of (51) have not been studied up to now. Solving (51) by the method in , we can get the following solutions for (1): where , , and are arbitrary constants and the function satisfies where , , and are constants; solutions of (53) have been given in . By means of the solutions of (53), plenty of solutions for (1) can be obtained; for example, where denotes the modulus of the Jacobi elliptic function.
(iii) When , , , , , and , we can get And satisfies the following reduction equation: The above equation can be integrated by and, when we take the constant of integration to zero, we get a reduced reduction equation: Equation (57) is variable coefficient KdV equation and soliton-like solutions have been obtained in . By means of the known solutions, many explicit solutions of (1) can be obtained. For example, where , , and are constants.
(iv) When and