Abstract
Through assuming that nonlinear superposition principles (NLSPs) are embedded in a Lie group, a class of 3rd-order PDEs is derived from a general determining equation that determine the invariant group. The corresponding NLSPs and transformation to linearize the nonlinear PDE are found, hence the governing PDE is proved C-integrable. In the end, some applications of the PDEs are explained, which shows that the result has very subtle relations with linearization of partial differential equation.
1. Introduction
Construction of new solutions by superposition of known ones is a familiar tool in nonlinear partial differential equations. The idea of superpositions for nonlinear differential operators originated in 1893 by Vessiot [1]. It was immediately generalized by Guldberg [2]. Marius Sophus Lee pointed out that these are special cases of his own theory of “Fundamental Solutions of Differential Equations” [3, 4]. The problem was reconsidered in 1960 by Temple [5] who showed the existence of nonlinear d.e. and the general solution cannot be obtained as a finite number of particular solutions. In 1965, Inselberg showed that there exist classes of equations, involving nonlinear operators, where it is possible to “compose” two solutions to obtain a different solution [6]. This is different than the above works where the general solution is sought. In 2000, Ibragimov gives specific examples of the Vessiot-Guldberg-Lie algebra applied to some partial differential equations [7]. Jones and Ames introduce the definition “connecting function” to describe nonlinear superposition in [8].
Generally, these methods for finding NLSPs can be classified into two categories: one is based on ad hoc methods [9–12]; the other is Lie’s classical symmetry algorithm [7, 13–15]. In contrast with the ad hoc methods, Lie’s classical symmetry algorithm is systematic and automatic.
Following the definition of nonlinear superposition introduced by in [8, 15], formally, the simplest form of a superposition principle is an operation (where and are function spaces) which preserves some governing equations, so that if then where . Of course, it is a simple matter to extend this concept to any -ary operation that constructs new solutions from old solutions.
Goard and Broadbridge have investigated how to use Lie symmetry algebras to find general superposition principles for nonlinear PDEs [15] and they have given the NLSPs of some first-order and second-order PDEs in two independent variables and linearized the PDEs. However, for higher-order nonlinear PDEs, its linearization and NLSPs have not been fully investigated. It is the purpose of the paper to demonstrate how to find the superposition principles for a class higher-order nonlinear PDEs.
We assume that for any pair of solutions and to the governing PDE, there exists a parameter so that we have a one-parameter solution In further, suppose that the NLSP is surjective in the sense that, for any two solutions and , there exists a solution such that From (1.4) then From (1.5), if , the implicit function theorem allows us to regard as a function of and . Hence we write which we find by solving the classical linear determining equations of the PDE. Then solving we find the NLSP for the PDE.
Goard and others have shown in their work that for first- and second-order PDEs of two independent variables, this will actually imply that the PDE will have a generator of the form where is the general solution of a linear homogeneous PDE. Hence the PDE satisfies the necessary condition for the existence of a transformation to a linear PDE.
In the paper, we will start from a general third-order PDE and find the third-order PDE class with NLSPs. Furthermore, a discussion for higher-order PDEs is given.
2. A Class of Third-Order PDEs with NLSPs
Firstly, consider a general third-order determining equation: Substitution of into (2.1) gives To obtain a solution of (2.2) where depends on alone, the coefficients of (2.2) need to be in and alone. Hence we consider the possible way in which (2.2) can be written as an equation in and . It is shown that the determining equation admits separable solutions or, more generally, In the case, at least one of is not zero. (We note that at least one of these coefficients must be nonzero for a third-order governing PDE.) In addition, we suppose with , then from (2.2) we let for some two functions of , and (2.2) becomes Then if we let
(2.2) becomes where and notice that, from (2.6), where .
We consider the following possibilities for (2.7).(i), , not all , . From (2.9) we have Equation (2.12) admits the separable solution (2.3), where satisfies
The linear PDE which satisfies is(ii), , , . We then require in (2.9).
Equation (2.9) admits separable solutions where and satisfies for some nonzero constant .
From (2.15), we have satisfies
So, we obtain (2.8) which has symmetries with generator The PDEs are linearizable with (by [14, Theorem 6.4, 2-2]). Equation (2.8) represents a class of third-order PDEs with various choices of , and . A class of general third-order PDEs in two independent variables with NLSPs of (1.1) embedded in a Lie group are identified in (2.8).
3. Symmetry Analysis of (2.8)
We rewrite (2.8) in the following formulation: where for , , is constant, and for , is arbitrary.
Equation (3.1) has classical symmetry with generator with its third-order prolongation where satisfies We require where . Let then, It is easy to verify other equations if (3.6) is satisfied. Then, (3.4) degenerates into Letting and substituting into (3.8) give where needs to satisfy where if . Hence we choose as in (2.11), and we can find that make (3.10) satisfied by substituting it into (3.10).
Then solving (1.8), that is, gives the NLSP for (3.4) as where
If we let and substitute it into (3.8), then and satifies where again when . By solving we obtain where is given in (3.13). By using (2.20) and (3.15), the transformation to linearize (3.1) is obtained as which transforms (3.1) into the linear equation for some constant and where if . It is indicated that the governing PDE (3.1) is C-integrable [16].
It is noticed that there is no explicit general solution for the linearized equation except few special cases. Thus, it is generally hard to obtain explicit general solution of the original nonlinear PDE, although the NLSP and the transformation to linearise the system are explicit. Equation (3.1) can also be used to determine whether some equations are C-integrable. For example, the kdv equation is only S-integrable [16], but not C-integrable, since it is not included in the class. Many numerical methods (such as Finite Difference Method, Finite Element Method) can be used to solve the linearized equation, and the solution of (3.1) can be obtained through the corresponding variable transformation.
4. Some Explanations of the Result
By explicitly constructing the full class of PDEs with a Lie group of NLSPs, we arrive at a single expression (3.1) for a class of third-order variable-coefficient equations.
4.1. Relations with ODE
First of all, we should point out that when letting , (3.1) is second-order PDE, which is (2.44) in [15]
Second, letting , , be only related to variable , an ODE is obtained which is a special case of the equation [17] The special situation of equation has the NLSPs and can be transformed into linear equation.
Furthermore, we can get a class of third-order ODEs that have NLSPs
4.2. Relations with PDEs Whose Potential Equations Are in the Form (3.1)
Letting , , , , and differentiate it with variable , we get Making the replacement , we get a class nonlinear PDEs whose potential equations can be linearized. Here we only give out the form with that coefficients are constants for simplicity The transformation to linearize the PDE is In fact, it is the Cole-Hopf transform [18, 19] It is easy to calculate that the NLSP is
When the potential Burgers equation is included in (3.1) [15]. And at the same time, we can also get the Burgers equation from(4.6)
4.3. Some Examples of NLSPs and Transformations to Linearize Nonlinear PDEs
Some more concrete examples are given below. Letting , , in (3.1), then We discuss the two following cases.
Case 1. In the case of , let , then , and In this case, , hence the NLSP is The transformation to linearize the PDE is The corresponding linearized PDE is
Case 2. In the case of , let , then In the case , the NLSP is The transformation to linearize the PDE is The corresponding linearized PDE is
Let In the case , the NLSP is The transformation to linearize the PDE is The corresponding linearized PDE is
5. Conclusion
We have given out a class of nonlinear PDEs with NLSPs by using the idea suggested by Goard and Broadbridge in [15]. By explicitly constructing the full class of PDEs with a Lie group of NLSPs, we arrived at a single expression (3.1) for a class of third-order variable-coefficient equations that includes previously second-order PDEs obtained in [15]. The class of PDEs includes the type equations that can be linearized. We derive Cole-Hopf transformation of (4.3) whose potential equations are included in (3.1). PDE (3.1) is C-integrable since it can be linearized.
It is should be pointed out that higher-order nonlinear PDEs with NLSPs can be obtained in the same way, although much effort for symbolic computation is needed. For example, a fourth-order PDE with NLSP is in the following: Here , , , , is constant. When , is arbitrary. Here the C-integrable fourth-order ODE (5.1) can be obtained by the proposed Lie algorithm method relative easily on the computer.
Acknowledgment
The authors gratefully acknowledge fruitful discussions with Dr. Shen Shoufeng.