Reduction operators and exact solutions of variable coefficient nonlinear wave equations with power nonlinearities

Reduction operators, i.e. the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities is investigated within the framework of singular reduction operator. A classification of regular reduction operators is performedwith respect to generalized extended equivalence groups. Exact solutions of some nonlinear wave model which are invariant under certain reduction operators are also constructed.


Introduction
The investigation of Lie-point symmetry of differential equations is a well established and powerful tool for constructing exact solutions of the equations under consideration [6,19,27,29]. In fact, classical Lie-point symmetry group analysis is the only systematic method we know for deducing exact solutions (in particular, invariant, partially invariant and separation of variables solutions) of nonlinear partial differential equations [6,19,22,26,27,29]. However, for a lot of important applications differential equations because their classical Lie symmetry groups are rather trivial including at most space and time translations and scale transformations, the obtained solutions are not sufficient for our understanding the mechanics and mathematics of mathematical models. Thus, this stimulates the efforts devoted to generalization of Lie's original concept of symmetry in order to find more other types exact solutions.
The first approach to these symmetries is conditional ones (called also nonclassical symmetries or Q-conditional symmetries), which was introduced by Bluman and Cole [5] in 1969. It consists in augmenting the original partial differential equations with invariant surface conditions, a system of first order differential equations satisfied by all functions invariant under a certain vector field. The latter is chosen as a classical infinitesimal symmetry for the augmented system. The basic equations for conditional symmetries are similar to Lie's determining equations except that they are nonlinear and less overdetermined. That is why one rarely succeeds in obtaining all possible solutions to the determining equations for conditional symmetries, especially in the case of second order equations.
In contrast to classical Lie symmetry, another difference in the research of such two kinds of symmetries is the procedure of deriving the determining equations. Namely, deriving systems of determining equations for conditional symmetries crucially depends on the interplay between the operators and the equations under consideration. For example, for the linear heat equation u t = u xx , the determining equation are derived usually based on the condition that the general form of conditional symmetry operators Q = τ (t, x, u)∂ t +ξ(t, x, u)∂ x +η(t, x, u)∂ u , where (τ, ξ) = (0, 0), were divided into two essentially different cases, i.e., the regular case τ = 0 and the singular case τ = 0. However, such partition of sets of conditional symmetries operators with the conditions of vanishing and nonvanishing coefficients of operators is not appropriate for all of differential equations. Therefore, investigation of algorithms for deriving the determining equations for condition symmetries are essential importance, which will provide some useful guidelines for simplifying the obtained determining equation and make its complete solving become possible. Recently, Popovych et.al give a detail investigation on this question and present a novel framework, namely singular reduction operators or singular reduction modules [8,24], for finding an optimal way of obtaining the determining equation of conditional symmetries. As examples, properties of singular reduction operators of (1+1)-dimensional evolution equations and a specific wave equations have been studied [24, 30-32, 39, 41]. However, for more general nonlinear wave equation, there exist no general results.
In this paper we extend this new framework of singular reduction operators [24] to study a class of variable coefficient nonlinear wave equations which possesses three arbitrary functions f = f (x), g = g(x) and h = h(x) and two power nonlinearities u n and u m of the form where f (x)g(x) = 0, n and m are arbitrary constants, t is the time coordinate and x is the onespace coordinate. The linear case is excluded from consideration because it was well-investigated. We also assume the variable wave speed coefficient u n to be nonlinear, i.e. n = 0. The case n = 0 is quite singular and will be investigated separately. Many specific nonlinear wave models describing a wide variety of phenomena in Mechanics and Engineering such as the flow of one-dimensional gas, shallow water waves theory, longitudinal wave propagation on a moving threadline, dynamics of a finite nonlinear string, elastic-plastic materials and electromagnetic transmission line and so on, can be reduced to equation (1) (see [1] p.50-52 and [2]). Classical Lie symmetries of various kinds of quasi-linear wave equations in two independent variables that intersect class (1) have been investigated in [3, 4, 6, 7, 9, 10, 12-14, 17-20, 28, 35-37, 42]. Recently, we have present a complete local symmetry and conservation law classification of class (1) in [15,16]. Classical Lie symmetry reduction and invariant solutions of some variable coefficients wave models which are singled out from the classification results are also investigated. However, the nonclassical symmetries of class (1) remains open. In this paper, we will make a detailed investigation on this subjects so that we can use it to obtain some new non-Lie exact solutions. Below, following [24] we use the shorter and more natural term 'reduction operators' instead of 'operators of conditional symmetry' or 'operators of nonclassical symmetry'.
The rest of paper is organized as follows. In Section 2, known results on equivalence transformations and classical group analysis of the wave equations (1) are adduced in a form which is suitable for purposes of our investigations. Then in section 3, singular reduction operators, and in particular regular reduction operators classification for the class under consideration are investigated after some brief review of the notation of singular and regular reduction operator. Section 4 contains nonclassical symmetry reduction of some nonlinear wave models including some truly 'variable coefficient' ones which are singled out from the class of (1+1)-dimensional wave equations. New non-Lie exact solutions of the models are constructed by means the reduction. Conclusions and discussion are given in section 5.

Equivalence transformations and Lie symmetries
The exhaustive result on classical group classification of class (1) is presented by the statements adduced below [15]. Using the direct method, we construct different kinds of equivalence groups including usual and generalized extended ones and discuss their structure (for the details about equivalence group can be seen in [21,29,33,34,38]). Theorem 1. The usual equivalence group G ∼ for the class (1) consists of the transformations where ǫ j (j = 1, . . . , 4) are arbitrary constants, ǫ 1 ǫ 2 ǫ 4 = 0, X is an arbitrary smooth function of It is shown that class (1) admits other equivalence transformations which do not belong to G ∼ and form, together with usual equivalence transformations, an generalized extended equivalence group. Using the direct method [23], we can construct the following complete generalized extended equivalence groupĜ ∼ of class (1).
Theorem 2. The generalized extended equivalence groupĜ ∼ of the class (1) is formed by the transformations where ǫ j (j = 1, 2, 3) are arbitrary constants, ǫ 1 ǫ 2 = 0, X is an arbitrary smooth function of x, Using the transformatioñ from theorem 1, we can reduce equation (1) (Similarly, any equation of form (1) can be reduced to the same form withf (x) = 1.) Thus, without loss of generality we can restrict ourselves to investigation of the equation All results on symmetries, solutions and conservation laws of class (3) can be extended to class (1) with transformations (2).
The above representation of the transformations from G ∼ 1 is very clumsily, but it shows that the family of transformations is continuously parameterized with the parameter n, including the value n = −1. In fact, we have lim n→−1 X(x) = x δ + ǫ 4 δǫ 3 , lim n→−1 Y (x) = e ǫ 3 x+ǫ 4 . Therefore, we can reformulate the above transformations as a more simple and clear formulae than in theorem 3.
Since the parameters n and m are invariants of all the above equivalence transformations, class (1) (or class (3)) can also be presented as the union of disjoint subclasses where each from the subclasses corresponds to fixed values of n and m. Thus the generalized equivalence groupĜ ∼ of class (1) can be considered as a family of usual equivalence groups of the subclasses parameterized with n and m. Motivated by this, we can obtain wider equivalence groups for some of the subclasses apriori assuming the parameters n and m satisfy a condition. In particular, for the case m = n + 1, we have admits the equivalence transformation group G ∼ m=n+1 consisting of the transformations: where X and Y are arbitrary functions of x, ǫ j (j = 1, 2, 3) are arbitrary constants, ǫ 1 ǫ 2 X x Y = 0.
If we further gauge g with the condition g = 1 and impose the restrictions ǫ 1 X x = Y 2n+2 on parameters of G ∼ m=n+1 , we can obtain the following equivalence group G ∼ 1,m=n+1 for the class of equations Corollary 1. The equivalence group G ∼ 1,m=n+1 of class (5) is formed by the transformations: where X and Y are arbitrary functions satisfying the condition ǫ 1 X x = Y 2n+2 , ǫ j (j = 1, 2, 3) are arbitrary constants, ǫ 1 ǫ 2 Y = 0.
Using the above-mentioned results, we obtained the classical symmetry classification for the class (3) in [15].
In table 1, we list tuples of parameter-functions f (x) and h(x), the constant parameters m and n and bases of the invariance algebras in all possible inequivalent cases of Lie symmetry extension. The operators from table 1 form bases of the maximal Lie invariance algebras iff the corresponding values of the parameters are inequivalent to ones with more abundant Lie invariance algebras. In cases 2 and 3 we do not try to use equivalence transformations as much as possible since otherwise a number of similar simplified cases would be derived, see footnote after the table.
Here α is arbitrary constant, ε = ±1, and it can be assumed up to equivalence with respect to G ∼ 1 that the parameter tuple (a, b, c, d) takes only the following non-equivalent values: where d ′ is arbitrary constant. In all the cases we put g(x) = 1. In case 10 the parameter-functions f and h can be additionally gauged with equivalence transformations from G ∼ 1,m=n+1 . For example, we can put f = 1 if n = −4, −4/3 and f = e x otherwise. Remark 1. It should be emphasized that we adduce only the cases of extensions of maximal Lie invariance algebra, which are inequivalent with respect to G ∼ 1 if m = n + 1 and with respect to G ∼ 1,m=n+1 if m = n + 1 or h = 0 in table 1. What's more, right choice of a gauge of the parameter tuple (f, g, h) is another crucial point of our investigations. In fact, the major choice have been made from the very outset of classification when the parameter-function g was put equal to 1. It is the gauge that leads to maximal simplification of both the whole solving and the final results. Although all gauges are equivalent from an abstract point of view, only the gauge g = 1 allows one to exhaustively solve the problem of group classification of class (1) with reasonable quantity of calculations. Even after the other simplest gauge f = 1 chosen, calculations become too cumbersome and sophisticated.

Nonclassical symmetries
In this section, we first review some necessary definitions and statements on nonclassical symmetries [11,25,40,43] and singular reduction operator [24]. Then perform a detailed investigation on the reduction operators of class (3).
Consider an rth order differential equation L of the form L(t, x, u (r) ) = 0 for the unknown function u of the two independent variables t and x, where L = L[u] = L(t, x, u (r) ) is a fixed differential function of order r and u (r) denotes the set of all the derivatives of the function u with respect to t and x of order not greater than r, including u as the derivative of order zero.
In order to discuss the conditional symmetries of equation L, we will first treat equation L from a geometric point of view as an algebraic equation in the jet space J r of order r and is identified with the manifold of its solutions in J r [27]: Let Q denote the set of vector fields of the general form which is a first-order differential operator on the space R 2 × R 1 with coordinates t, x, and u. Then all functions invariant under Q and only such functions satisfy a first order differential equation called the the characteristic equation (also known as invariant surface condition). Denote the manifold defined by the set of all the differential consequences of the characteristic equation Q[u] = 0 in J r by Q (r) , i.e., are the operators of total differentiation with respect to the variables t and x, and the variable u α,β of the jet space J r corresponds to the derivative ∂ α+β u ∂t α ∂x β . Denote also by Q (r) the standard rth prolongation of Q to the space J r : Definition 1. The differential equation L is called conditionally invariant with respect to the operator Q if the relation holds, which is called the conditional invariance criterion. Then Q is called conditional symmetry (or nonclassical symmetry, Q-conditional symmetries or reduction operator) of the equation L.
We denote the set of reduction operators of the equation L by Q(L) which is a subset of Q. Any Lie symmetry operator of L belongs to Q(L). Sometimes Q(L) is exhausted by the operators equivalent to Lie symmetry ones in the sense of the following definition.
Definition 2. Two differential operators Q andQ in Q are called equivalent (Q ∼Q) if they differ by a multiplier which is a non-vanishing function of t, x and u :Q = λQ, where λ = λ(t, x, u), λ = 0.
Consider a vector field Q in the form (6) and a differential function L = L[u] of order ord L = r (i.e., a smooth function of variables t, x, u and derivatives of u of orders up to r).
Definition 3. The vector field Q is called singular for the differential function L if there exists a differential functionL =L[u] of an order less than r such that L| Q (r) =L| Q (r) . Otherwise Q is called a regular vector field for the differential function L. If the minimal order of differential functions whose restrictions on Q (r) coincide with L| Q (r) equals k (k < r) then the vector field Q is said to be of singularity co-order k for the differential function L. The vector field Q is called ultra-singular for the differential function L if L| Q (r) ≡ 0.

Definition 4.
A vector field Q is called weakly singular for the differential equation L : L[u] = 0 if there exists a differential functionL =L[u] of an order less than r and a nonvanishing differential function λ = λ[u] of an order not greater than r such that L| Q (r) = λL| Q (r) . Otherwise Q is called a weakly regular vector field for the differential equation L. If the minimal order of differential functions whose restrictions on Q (r) coincide, up to nonvanishing functional multipliers, with L| Q (r) is equal to k(k < r) then the vector field Q is said to be weakly singular of co-order k for the differential equation L.
Definition 5. A vector field Q is called a singular reduction operator of a differential equation L if Q is both a reduction operator of L and a weakly singular vector field of L.
After the factorization of the reduction operator under the usual equivalence relation of reduction operators in Definition 2 to singular and regular cases, the classification of reduction operators can be considerably enhanced and simplified by considering Lie symmetry and equivalence transformations of (classes of) equations. Lemma 1. Any point transformation of t, x and u induces a one-to-one mapping of Q into itself. Namely, the transformation g :t = T (t, x, u),x = X(t, x, u),ũ = U (t, x, u) generates the mapping g * : Q → Q such that the operator Q is mapped to the operator g * Q =τ ∂t +ξ∂x +η∂ũ, Therefore, the corresponding factorized mapping g f : Q f → Q f also is well defined and bijective.
Lemma 1 results in appearing equivalence relation between operators, which differs from usual one described in Definition 2.
Definition 6. Two differential operators Q andQ in Q are called equivalent with respect to a group G of point transformations if there exists g ∈ G for which the operators Q and g * Q are equivalent. We denote this equivalence by Q ∼Q mod G.
The problem of finding reduction operators is more complicated than the similar problem for Lie symmetries because the first problem is reduced to the integration of an overdetermined system of nonlinear PDEs, whereas in the case of Lie symmetries one deals with a more overdetermined system of linear PDEs. The question occurs: could we use equivalence and gauging transformations in investigation of reduction operators as we do for finding Lie symmetries?
The following statements give the positive answer. Lemma 2. Given any point transformation g of an equation L to an equationL, g * maps Q(L) to Q(L) bijectively. The same is true for the factorized mapping g f from Q f (L) to Q f (L).

Corollary 2.
Let G be the point symmetry group of an equation L. Then the equivalence of operators with respect to the group G generates equivalence relations in Q(L) and in Q f (L).
Consider the class L| S of equations L θ : L(t, x, u (r) , θ) = 0 parameterized with the parameterfunctions θ = θ(t, x, u (r) ). Here L is a fixed function of t, x, u (r) and θ. The symbol θ denotes the tuple of arbitrary (parametric) differential functions θ(t, x, u (r) ) = (θ 1 (t, x, u (r) ), ..., θ k (t, x, u (r) )) running through the set S of solutions of the system S(t, x, u (r) , θ (q) (t, x, u (r) )) = 0. This system consists of differential equations on θ, where t, x and u (r) play the role of independent variables and θ (q) stands for the set of all the derivatives of θ of order not greater than q. In what follows we call the functions θ arbitrary elements. Denote the point transformation group preserving the form of the equations from L| S by G ∼ .
Let P denote the set of the pairs consisting of an equation L θ from L| S and an operator Q from Q(L θ ). In view of Lemma 2, the action of transformations from the equivalence group G ∼ on L| S and {Q(L θ )|θ ∈ S} together with the pure equivalence relation of differential operators naturally generates an equivalence relation on P .
The classification of reduction operators with respect to G ∼ will be understood as the classification in P with respect to this equivalence relation, a problem which can be investigated similar to the usual group classification in classes of differential equations. Namely, we construct firstly the reduction operators that are defined for all values of θ. Then we classify, with respect to G ∼ , the values of θ for which the equation L θ admits additional reduction operators.
In what follows we use above-mentioned method to investigate reduction operators of the (1+1)-dimensional variable coefficient nonlinear wave equations (3). For convenient, we rewrite it as the form

Singular reduction operators
Using the procedure given by Popovych et al. in [24], we can obtain the following assertion.
Proof. Suppose that τ = 0. According to the characteristic equation τ u t + ξu x − η = 0, we can get Substituting the formulas of u tt from above formulaes into L, we obtain a differential functioñ According to the definition 3 of singular vector field, we have ordL < 2 if and only if f (x)( ξ τ ) 2 − u n = 0.
Therefore, for any f, h, n and n with f u n > 0 the differential function L = f (x)u tt −(u n u x ) x − h(x)u m possesses exactly two set of singular vector fields in the reduced form, namely, S = {∂ t + u n /f ∂ x +η∂ u } and S * = {∂ t − u n /f ∂ x +η∂ u }, whereη = η τ . Any singular vector field of L is equivalent to one of the above fields. The singular sets are mapped to each other by alternating the sign of x and hence one of them can be excluded from the consideration. Thus taking into accountant the conditional invariance criterion for an equation from class (9) and the operator ∂ t + u n /f ∂ x + η∂ u , we can get Theorem 6. Every singular reduction operator of an equation from class (9) is equivalent to an operator of the form where the real-valued function η(t, x, u) satisfies the determining equations

Regular reduction operators
The above investigation of singular reduction operators of nonlinear wave equation of the form (9) shows that for these equations the regular case of the natural partition of the corresponding sets of reduction operators is singled out by the conditions ξ = ± u n /f τ . After factorization with respect to the equivalence relation of vector fields, we obtain the defining conditions of regular subset of reduction operator: τ = 1, ξ = ± u n /f . Hence we have Proposition 3. For any variable coefficient nonlinear wave equations in the form (9) the differential function L = f (x)u tt − (u n u x ) x − h(x)u m possesses exactly one set of regular vector fields in the reduced form, namely, S = {∂ t +ξ∂ x +η∂ u } withξ = ± u n /f . Consider the conditional invariance criterion for an equation from class (9) and the operator ∂ t + ξ(t, x, u)∂ x + η(t, x, u)∂ u with ξ(t, x, u) = ± u n /f , we can get the following determining equations for the coefficients ξ and η: From the first two equations of system (11), we have Substituting these expression into the last three equations of system (11), we have the following assertion.
Theorem 7. Every regular reduction operator of an equation from class (9) is equivalent to an operator of the form where the real-valued function ξ(t, x) satisfies the overdetermined system of partial differential equations Solving the above system with respect to the coefficient functions ξ, f and h under the equivalence group G ∼ 1 , we can get a classification of regular reduction operator for the class (9). It is easy to know that some of the regular reduction operator are equivalent to Lie symmetry operators, while some of are nontrivial. Below, we give a detailed investigations for these cases.
In fact, the first three equations of system (13) implies there are two cases should be considered: n = 1 or not. (It should be noted that ξ = 0 should be exclude from the consideration because it leads to η = 0). Case 1: n = 1. In this case, we have ξ t = 0. Thus system (13) can be reduced to Thus, there are two cases should be considered: m = n + 1 or not.  (14) we obtain The first equation of (14) suggests that (3n + 4)ξ x + 2(n + 1)ξ fx f is independent of the variable x, so there exists a constant r such that (3n + 4)ξ x + 2(n + 1)ξ fx f = nr. The second equation of (15) suggests that there exist two constants a and b such that 2ξ x + ξ fx f = nax + nb. By solving the last two equations we obtain which together with the first equation of (15) imply ξ = a(n + 1)x 2 + [2b(n + 1) − r]x + s, where a, b, r, s are arbitrary constants. Thus, the corresponding regular reduction operator has the form which is equivalent to Lie symmetry operator.
Case 1.2: m = n + 1. In this case, system (14) can be rewritten as Integrating these two equations with respect to functions f (x) and g(x), we can obtain where p, r are arbitrary constants, ξ is an arbitrary smooth function and n = −1. In addition, η = 1 n (2ξ x + ξ fx f )u = ( r n + ξx 2n+2 )u. Thus, we have a nontrivial regular reduction operator It should be noted that for n = −1 the reduction operator is also equivalent to Lie symmetry operator.
Case 2: n = 1. In this case, we have η = 2f ξ t + (2ξ x + ξ fx f )u. Thus, system (13) can be reduced to After some brief analysis, we find that there are five cases should be considered.
Case 2.1: m = 0. In this case, the third equation of (19) implies From the last equation of system (20) we can know that there exist two functions a(t) and b(t) such that 2ξ x + ξ fx f = a(t)x + b(t). On the other hand, the first equation of (19) implies there exists a function c(t) such that 7ξ x + 4ξ fx f = c(t). Solving the last two equations gives from which we can get where d(t) is an arbitrary functions. Since fx f is independent of t, we see that Now, we multiply both sides of the second equation of (19) by ξ and substitute (21) into it, then simplify the equation and compare the coefficient of x i (i = 0, 1, . . . , 5) to obtain Note that ξ is assumed not to be identical with zero, after some simple but lengthy computations, we find that systems (22) and (23) can be reduced to: or where q is an arbitrary constant.
Case 2.1a: If system (24) is satisfied, then ξ t = 0, the second equation of (20) is an identity. The expression (21) can be rewritten as where a, b, c and d are arbitrary constants. The first equation of (20) is reduced to Substitute the expression of ξ and f (x) into it and integrate both sides to obtain In addition, η = 2f ξ t + (2ξ x + ξ fx f )u = (ax + b)u. Therefore, we have where a, b, c, d are arbitrary constants. Thus, the corresponding regular reduction operator has the form which is equivalent to Lie symmetry operator.
Case 2.1b: If system (25) is satisfied, then the expression (21) can be rewritten as Hence, ξ x = 0, k = b(t). Substituting these formulaes into the second equation of (20) we obtain Combine it with the fourth equation of (25) to get b ′ (t) = 0. Hence a(t), b(t), c(t), d(t) satisfy system (24), and the solution is included in the case 2.1a.
Case 2.1c: If system (26) is satisfied, then the expression (21) can be rewritten as Substitute it into the second equation of (20) to obtain 2b(t)b ′ (t) + b ′′ (t) = 0. Combine it with the third equation of (26) to get b ′ (t) = 0. Substitute it into the fourth equation of (26) to obtain bd ′ (t) + d ′′ (t) = 0, which implies d(t) = γ 1 e −bt + γ 0 , where γ 1 and γ 0 are arbitrary constants. Therefore ξ = b 2 x + γ 1 e −bt + γ 0 . Substitute it into the first equation of (20) to obtain Since h, γ 1 , γ 0 are independent of t, the preceding equation suggests that which leads to b = 0. Therefore, we have where d 1 , d 0 , h 0 are constants. Thus, the corresponding regular reduction operator has the form which is equivalent to Lie symmetry operator.
Since h and f are independent of t, there is a constant r such that b ′′ (t) = r. It follows that there exist constants s and w such that b(t) = rt 2 /2 + st + w. Substitute it into the fourth equation of (27) to obtain r 2 t 3 + 3rst 2 + 2(wr + s 2 )t + 2ws + r = 0.
Then r = 0 and s = 0. Hence b(t) = w, ξ t = 0, the solution is included in the case 2.1a.
Case 2.2: When m = 1, the third equation of (19) implies Similar to the case of m = 0, from the third equation of (33) and the first two equations of (19) we get the expression of ξ and f (x) as stated in (21), where a(t), b(t), c(t), d(t) satisfy the condition (24) or (25) or (26) or (27).
Case 2.2a: If system (24) is satisfied, then ξ t = 0, the second equation of (33) is an identity. The expression (21) can be rewritten as (28). The first equation of (33) is reduced to h where a, b, c, d are arbitrary constants and ǫ = ±1. Thus, the corresponding regular reduction operator has the form which is equivalent to Lie symmetry operator.

Case 2.3:
When m = 2, system (19) implies From the first and the last equation of system (39), we can get where α(t) is an arbitrary function, q is a constant. Substituting these expressions into the rest equations of system (39), we can see that ξ(t, x) and α(t) satisfy the overdetermined system of partial differential equations In addition, we have Thus, we have a nontrivial regular reduction operator where ξ(t, x) and α(t) satisfy the overdetermined system of partial differential equations (40).
In particular, if ξ t = 0, from system (39) we can obtain where a, q are arbitrary constants. Thus, we have a nontrivial regular reduction operator Notice that the second equation of (46) indicates that ξ is independent of t, therefore the second equation of (19) is satisfied automatically. Similar to the case of m = 0, from the third equation of (46) and the first equation of (19) we get the expression of ξ and f (x) as stated in (28), where a, b, c, d are constants. Substituting the expression of ξ and f into the first equation of (46) we obtain In addition, η = 2f ξ t + (2ξ x + ξ fx f )u = (ax + b)u. Therefore, we have where a, b, c, d are arbitrary constants. Thus, the corresponding regular reduction operator has the form which is equivalent to Lie symmetry operator.
From the above discussion, we can arrival at the following two theorems.
Theorem 9. Any reduction operator of an equations from class (3) having the form (12) with ξ t = 0, ξ xxx = 0 is equivalent to a Lie symmetry operator of this equation.

Exact solutions
In this section, we construct nonclassical reduction and exact solutions for the classification models in table 2 by using the corresponding regular reduction operator. Lie reduction and exact solutions of equation from class (3) have been investigated in reference [15]. We choose case 1 in table 2 as an example to implement the reduction, the other cases can be considered in a similar way. For the first case in table 2, the corresponding equation is which admit the regular reduction operator Q = ∂ t + ξ(x)∂ x + ( r n + ξ x 2n + 2 )u∂ u .
Thus we obtain an explicit solution u(t, x) = [1 ± 1 + 2(c 1 ω + c 2 )]| ξ| 1 4 , If we take different functions for ξ, then we can obtain a series of solutions for the corresponding equations. In order to avoid tediousness, we do not make a further discussion here.

Conclusion and Discussion
In this paper we have given a detailed investigation of the reduction operators of the variable coefficient nonlinear wave equations (1) (equivalently to (3)) by using the singular reduction operator and the equivalence transformation theory. A classification of regular reduction operators is performed with respect to generalized extended equivalence groups. The main results on classification for the equation (3) are collected in table 2 where we list three inequivalent cases with the corresponding regular reduction operators. Nonclassical symmetry reduction of a class nonlinear wave model (48) which are singled out the classification models are also performed. This enabled to obtain some non-Lie exact solutions which are invariant under certain conditional symmetries for the corresponding model. The present paper is a preliminary nonclassical symmetry analysis of the class of hyperbolic type nonlinear partial differential equations (1). Therefore, further investigations of different properties such as nonclassical potential symmetries, and nonclassical potential exact solutions as well as physical application of this class of equations would be extremely interesting. These results will be reported in subsequent publications.