Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations

and Applied Analysis 3 to construct the solution of (1). Nevertheless it is easy to get the solution of the equation


Introduction
It is well known that the classical Lie symmetry method of point transformations is often used for reducing the number of independent variables in partial differential equation to obtain ordinary differential equations.After integration of reduced differential equations one can obtain partial solutions of the equation under study [1][2][3].The main problem is that the maximal invariance group of point transformations of differential equations used in applications is not sufficiently wide and thus the group approach can not be successfully applied to these equations.The concept of generalized conditional symmetry has been introduced in [4,5] to extend the applicability of the symmetry method to the construction of solutions of evolution equations.The relationship of generalized conditional symmetries of evolution equations to compatibility of system of differential equations is studied in [6].The method for construction of nonlocally related partial differential equation systems for a given partial differential equation has been proposed in [7].The starting point for the method is the existence of operator of point symmetry of the equation under study.Through nonlocally related systems one can construct operators of nonlocal symmetry and nonlocal conservation laws of initial equation.
We use operators of nonpoint classical and conditional symmetries to extend the class of differential equations to which the symmetry method is applicable.In this paper we study the symmetry reduction of partial differential equations with two independent variables by using the operators of nonpoint symmetry because the prolongated operators of classical point symmetry lead to the classical invariant solutions.The method can be naturally generalized to the multidimensional case.We construct the ansatz for dependent variable  or its derivatives which reduces the scalar partial differential equation to a system of ordinary differential equations.We use the operators of the classical point symmetry [1,2] of the corresponding system which are not the prolongated operators of point symmetries admitted by the original equation to construct the ansatz for derivatives.We construct the ansatz for  by using ordinary differential equation admitting the operators of Lie-Bäcklund symmetry (in the classical sense [2,4]).We consider nonlinear evolution and wave type equations and present the operator of conditional symmetry for the corresponding system which generates the Bäcklund transformations for nonlinear wave equation.
Recall, that the well-known integrable nonlinear differential equations such as Korteweg-de-Vries, sine-Gordon, and cubic Schroedinger equations admit an infinite number of Lie-Bäcklund symmetry operators [1,2].Another goal of this paper is to show that such important properties of nonlinear partial differential equations as existence of Bäcklund transformations, linearization, and existence of the class of solutions depending on arbitrary function can be related to their invariance under the finite number of nonpoint symmetry operators.

Nonpoint Symmetry and Reduction of Nonlinear Wave Type and Evolution Equations with Two Independent Variables
The concept of differential invariant solutions based on infinite Lie group  is introduced in [3].This group is a classical symmetry group of point transformations of dependent and independent variables for the equation under study.Generally speaking, analysis similar to that in constructing differential invariant solutions enables us to obtain the ansätze for derivatives   1 ,   2 by virtue of operators of nonpoint symmetry [3,8].Let us consider nonlinear differential equation We search for the ansatz for the derivatives of such form where  = ( 1 ,  2 , ).Operators of classical and conditional symmetry of the corresponding system can be used to find  1 ,  2 .The corresponding system has the form where To construct ansatz of type (2) we use the symmetry operator of system (3).It is obvious that operator  generates nonpoint group transformations for variables  1 ,  2 , .It is easy to find the invariants of one-parameter Lie group with generator By using these invariants one can construct the ansatz for From ( 6) we have where   2 =  2 /.Substituting ( 6) and ( 7) into the equation Thus we get the first reduced ordinary differential equation The second one we obtain from the compatibility condition We take the particular solution of reduced system of ordinary differential equations ( 10) and (11) in the form where  1 = const.Thus one has to integrate overdetermined compatible system of differential equations to construct the solution of (1).Nevertheless it is easy to get the solution of the equation in such form . ( The tangent transformations groups are also used in the framework of this approach.Let us consider the nonlinear evolution equation One can construct operator of tangent transformations of the form admitted by (16).The first order functionally independent differential invariants of the corresponding one-parameter Lie group of tangent transformations can be chosen in the form In order to construct ansatz of type ( 2 ) .
From ( 19) and ( 16) we have and first ordinary differential equation From the condition   =   it follows that ,  satisfy the second ordinary differential equation Thus the reduced system consists of ( 21) and (22).From (21), ( 22), and (19) it follows that the solutions of ( 16) can be constructed by integrating overdetermined compatible system where  1 ,  2 are arbitrary real constants.
Next we emphasize that the operators of conditional symmetry of corresponding system can be used for construction the Bäcklund transformations for nonlinear wave equation Indeed we showed that is the operator of conditional symmetry of the corresponding system where  ≡  3 .Using operator  we can write the ansatz in the following form: where  1 ,  2 are unknown functions on  1 ,  2 and hence the Bäcklund transforms relating ( 24) and sine-Gordon equation   1  2 = sin .These Bäcklund transforms (28) have been obtained for the first time in [9] by another technique.
Note that this approach is also applicable for linearization of nonlinear partial differential equations with two independent variables.Indeed, consider the second-order differential equation where  is a smooth function.Using the invariance of (29) under Lie group of transformations with corresponding fivedimensional Lie algebra given by basic elements   0 ,   1   ,  0   , and  1   we write the corresponding system in the form where , and   1  1 ≡ V 3 ( 0 ,  1 ).One can prove that (30) possesses infinite Lie classical symmetry and can be linearized by hodograph transformations.Thus we obtained the method of linearization of the second-order partial differential equation of the form (29) for arbitrary function .
Let us note that the symmetry group of corresponding system written in the general form contains the symmetry group of point transformations of initial equation as a subgroup and generators of point transformations can be used to construct ansatz (2).However these operators lead to invariant solutions in the classical Lie sense.We shall illustrate this property by the following example.Let us consider the wave equation where  is a smooth function.It is invariant with respect to the three-parameter Lie group.The basis of Lie algebra is given by {  1 ,   2 , which reduces (31) to the system Let () = 0. Then we obtain two cases (35) By integrating system one obtains the solution where  3 is arbitrary real constant and  1 ̸ = 0, of (31) with  = 0.In the second case we have and solution has the form where ℎ( 1 ) is arbitrary differentiable function.Let us consider the operator where ,  are arbitrary real constants.One can verify that in the following form: where  1 ( 2 ) is arbitrary smooth function.So in the framework of this approach we have constructed solution with arbitrary function  1 ( 2 ) to nonlinear wave type partial differential equation (60) for arbitrary functions  and ℎ.

Conclusions
We have constructed ansätze (6) and ansätze (19) which reduce nonlinear evolution equations ( 1) and ( 16) to ordinary differential equations and can not be obtained by using classical Lie method.We have found the solution of nonlinear heat equation ( 14).It turns out that some of these ansätze result in the classical invariant solutions.Obviously, one can construct such ansätze by prolongated operators of point symmetry admitted by the initial equation but they lead to the invariant solutions too.It is necessary that operators of nonpoint and conditional symmetry should be applied to obtain new results.As was noted above the linearization of class of nonlinear partial differential equations (29) is possible in the framework of this approach.
Finally we show that the existence of even at least one operator of Lie-Bäcklund symmetry to ordinary differential equations (47) gives the possibility of constructing solutions (61) defined by arbitrary functions to (60).To our knowledge the inverse scattering tranformation method is not applicable in this case.