λ -Symmetry and µ -Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations

: On one hand, we construct λ -symmetries and their corresponding integrating factors and invariant solutions for two kinds of ordinary differential equations. On the other hand, we present µ -symmetries for a (2+1)-dimensional diffusion equation and derive group-reductions of a ﬁrst-order partial differential equation. A few speciﬁc group invariant solutions of those two partial differential equations are constructed.


Introduction
Lie symmetry method is a powerful technique which can be used to solve nonlinear differential equations algorithmically, and there are many such examples in mathematics, physics and engineering [1,2]. If an nth-order ordinary differential equation (ODE) that admits an n-dimensional solvable Lie algebra of symmetries, then a solution of the ODE, involving n arbitrary constants, can be constructed successfully by quadrature. If a partial differential equation (PDE) admits a Lie point symmetry, then its dimension can be reduced by one, and further its group invariant solution can be systematically constructed. However, there exist some kinds of differential equations which have trivial Lie point symmetries or have no symmetry, and Lie symmetry method cannot be applied directly. It is also known that the existence of nontrivial Lie point symmetries is not necessary for guaranteeing the integrability by quadrature for differential equations [3,4].
In 2001, a new kind of symmetries, called λ-symmetries, was introduced by Muriel and Romero [3]. Indeed, ODEs which have trivial Lie point symmetries or no symmetry but possess λ-symmetries can be integrated by means of the λ-symmetry approach. λ-symmetries can also be used to construct first integrals and integrating factors of such equations [5,6]. Gaeta and Morando considered the case of PDEs, and extended λ-symmetries to µ-symmetries [7,8]. It was proved that µ-symmetries are as useful as standard symmetries in respect to symmetry reductions, and the determination of invariant solutions by using µ-symmetries is completely similar to the standard one in the Lie symmetry method (see, for example, [9][10][11][12][13][14][15][16] for many other interesting applications and theoretical developments about λ-and µ-symmetries).
Both λ-symmetries and µ-symmetries are generalizations of Lie point symmetries, which could be viewed as Lie point symmetries of integrable couplings [17], and provide new insights into the development of the Lie symmetry theory. The determination of both symmetries depends on the prolongation formula that generalizes the standard Lie symmetry prolongation of vector fields. The most outstanding factor is that the determining equations are nonlinear, and so calculations are much more complicated. In this paper, we use the package of the differential characteristic set method [18,19] and symbolic computing systems to determine the existence of generalized symmetries and to simplify the corresponding determining equations. The differential characteristic set method, developed by Wentsun Wu [20] in the 1970s, is a fundamental algorithmic method, together with the Gröbner base algorithm. The method is very effective in calculating both classical and non-classical symmetries (for further applications, please refer to [21]). This paper is structured as follows. In Section 2, we calculate λ-symmetries of two kinds of second-order ODEs and construct their integrating factors and invariant solutions by using the obtained λ-symmetries. In Section 3, we generate µ-symmetries of two different PDEs and construct some invariant solutions of the equations through applying the obtained µ-symmetries. In Section 4, we are devoted to providing some concluding remarks.

The Basic Concept of λ-symmetries
Consider an n-th order ordinary differential equation (ODE) where (x, u (k) ) = (x, u, u 1 , · · · , u k ) and for i = 1, ..., k, u i denotes the derivative of order i of the dependent variable u with respect to the independent variable x. The canonical form of this equation reads as follows u (n) = Ψ(x, u (n−1) ). (2) where η [λ,(0)] = η(x, u) and we will say that a vector field v, defined on M, is a λ-symmetry of the Equation (1). Obviously, if λ = 0, the λ-prolongation of order n of v is exactly the classical nth prolongation of v [1].

λ-Symmetries Reductions and Integrating Factors without Using Lie Symmetries
Consider the following ordinary differential equation We can use the differential characteristic set method [18] to determine that this equation has no Lie point symmetries easily. Assume that λ-symmetry generator of Equation (6) is and the second prolongation formula is of the form From Equations (5), we know that v satisfies the following λ-symmetry condition: The determining equation of (6) is where Substituting the above η [λ,(1)] , η [λ, (2)] into the Equation (8), one can get a set of over-determined homogeneous differential equations for ξ, η It can be checked that these equations, whose unknowns are ξ, η and λ, admit the solution . Now, we use the prolongation formula (4) to construct invariant solutions. We can determine v [λ, (2) It can be checked that are two functionally independent invariants for v [λ, (1)] . Upon calculating an additional invariant by derivation [1] Equation (6) can be reduced to the equation of y, w, w y , Solving (9), one can get We recover the invariant solution of Equation (6) by solving the auxiliary first-order differential equation Letũ = u 3 . The equation of (10) turns intõ and by integrating this equation, we get the invariant solution of the Equation (6): where c 1 , c 2 are arbitrary constants. Now we calculate first integrals of the Equation (6) by using method given in [5]. According to [5], if the equation admits a λ-symmetry: v = ∂ ∂u , then we can construct an integrating factor. From (4) in Section 2, we have For the sake of simplicity, the solution of λ is assumed to be λ = λ 1 (x, u)u x + λ 2 (x, u), and then the Equation (12) turns into From the first equation of the system (13), we get a special solution of λ 1 (x, u) = 1 u , and the other equation becomes 3x + λ 2u u 4 = 0, From the first equation of (14), we get λ 2 (x, u) = x u 3 + c 1 (x), and substituting it to the second equation, we get Taking Then the corresponding characteristic equation of the Equation (15) is Then upon calculating function D[w] substituting (16) into (17), and simplifying, the result turns into Now we calculate the first-order partial differential equation Solving the corresponding characteristic equation of (18), we get a special solution Substituting (16) into (19), we get the first integral Therefore, from Theorem 1 in [5], the integrating factor of the Equation (6) is

λ-Symmetry Reductions and Integrating Factors Using Lie Symmetry
Consider the following ordinary differential equation where A is an arbitrary constant. The Lie symmetries of Equation (20) are Now we use the relationship between Lie point symmetries and λ-symmetres given in [3] to get λ-symmetries of Equation (20). Let us consider P 1 . Then we have and the characteristic function of P 1 and the total derivative operator The symmetry v 1 = ∂ ∂u is the λ-symmetry [5] when Similarly, we consider P 2 and obtain The above-mentioned (v 1 , λ 1 ) and (v 2 , λ 2 ) are not equivalent, owing to Integrating the characteristic equation of (22) we get a special solution Secondly, calculating function D[w], one can get Next, calculating the first-order partial differential equation and solving the corresponding characteristic equation, we get a special solution Finally, substituting (23) into (24), we get the first integral Similarly, we get a first integral from λ 2 In the following, we calculate an integrating factor from λ 1 . According to [5], we get The corresponding characteristic equation is So we get a special solution of the Equation (25) So the above formula provides an integrating factor of the Equation (20). Using the same procedure as above, we get another integrating factor from λ 2 By using both of the first integrals I 1 and I 2 , the invariant solution of the Equation (20) can be obtained. The resulting solution is

The Basic Concept of µ-Symmetries
Let us consider the kth-order partial differential equation (PDE) where u = u(x) = u(x 1 , x 2 , · · · , x p ) and u (k) represents all kth order derivatives of u with respect to x. We recall that M is vector space with the coordinates x and u, and M can be prolonged to the k-th jet bundle (J (k) M, π k , B), with J 0 M ≡ M. We equip (J (1) M, π, B) with a distinguished semi-basic one-form µ [16], We require that µ is compatible with the contact structure defined in J (k) M, for k ≥ 2, in the sense that where J(ε) is the Cartan ideal generated by ε. According to [16], condition (27) is equivalent to Lemma 1 [16]. Let Y be a vector field on the jet space J (k) M, written in coordinates as where X = ξ i ∂ ∂x i + ϕ ∂ ∂u is a vector field on M. Let ε be the standard contact structure in J (k) M, and µ = λ i dx i a semi-basic one-form on (J (1) M, π, B), compatible with ε. Then Y is the µ-prolongation of X if and only if its coefficients (with ψ 0 = ϕ) satisfy the µ-prolongation formula is the solution manifold for ∆. If Y leaves invariant each level manifold for ∆, we say that X is a strong µ-symmetry for ∆.

Conclusions
"λ-symmetries and µ-symmetries are both useful in establishing effective alternative methods to analyze nonlinear differential equations without using Lie point symmetries. In this paper, we presented four examples to illustrate the efficiency of λ-symmetries and µ-symmetries for analyzing nonlinear differential equations. The integrating factors and invariant solutions of two kinds of nonlinear ordinary differential equations were constructed by using λ-symmetries and different techniques. And using µ-symmetries, we found many satisfactory new invariant solutions of two types of nonlinear partial differential equations.
The main obstacle to determining λ-symmetries and µ-symmetries is to solve the nonlinear determining equations. At present, there is no general algorithm and package to solve this problem directly. Therefore, it is difficult to determine the general form of λ and µ. However, appropriate assumptions of λ and µ can simplify the difficult calculation, so that the existing algorithms and programs can be used and satisfactory results can be obtained. In this paper, we used the package of the differential characteristic set method and symbolic computing systems to determine the complicated work of existence of generalized symmetries and to reduce the corresponding determining equations. It is an open question to improve the efficiency of symmetry computations and any alternative advanced algorithm for computing µ-symmetry needs to be investigated. It is also interesting to see if µ-symmetries can be used to generate lump solutions, particularly with higher-order dispersion relations [22], or in the case of linear partial differential equations (see, e.g., [23])."