AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2015/181275 181275 Research Article Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations http://orcid.org/0000-0001-6665-3934 Tsyfra Ivan 1 http://orcid.org/0000-0002-5024-7883 Czyżycki Tomasz 2 Naz R. 1 AGH University of Science and Technology, Faculty of Applied Mathematics 30 Mickiewicza Avenue 30-059 Krakow Poland agh.edu.pl 2 Institute of Mathematics University of Białystok Ciołkowskiego 1M 15-245 Białystok Poland uwb.edu.pl 2015 7102015 2015 20 02 2015 25 05 2015 7102015 2015 Copyright © 2015 Ivan Tsyfra and Tomasz Czyżycki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

1. 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 . 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 . The method for construction of nonlocally related partial differential equation systems for a given partial differential equation has been proposed in . 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 u 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 u 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.

2. 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 G is introduced in . 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 ux1, ux2 by virtue of operators of nonpoint symmetry [3, 8]. Let us consider nonlinear differential equation(1)ux2x2=11-ux1r,r0,±1.

We search for the ansatz for the derivatives of such form(2)ux1=R1x1,x2,u,φ1ω,φ2ω,ux2=R2x1,x2,u,φ1ω,φ2ω,where ω=ω(x1,x2,u). Operators of classical and conditional symmetry of the corresponding system can be used to find R1, R2. The corresponding system has the form(3)v21=v12,v22=11-v1r,where v1=ux1, v2=ux2, and vki=vxki, i,k=1,2. To construct ansatz of type (2) we use the symmetry operator (4)Q=r+1x1x1+rv2x2-v1v1+rv2v2of system (3). It is obvious that operator Q generates nonpoint group transformations for variables x1,x2,u. It is easy to find the invariants of one-parameter Lie group with generator Q(5)ω=x2-v2,ω1=v1x11/r+1,ω2=v2x1-r/r+1.By using these invariants one can construct the ansatz for v1, v2:(6)v1=x1-1/r+1φ1ω,v2=x1r/r+1φ2ω.From (6) we have(7)v22=x1r/r+1φ21+x1r/r+1φ2,where φ2=dφ2/dω. Substituting (6) and (7) into the equation (8)v22=11-v1ryields(9)x1r/r+1φ2-φ2φ1r=1+x1r/r+1φ2.Thus we get the first reduced ordinary differential equation(10)φ2φ1r=-1.The second one we obtain from the compatibility condition v21=v12. It has the form(11)rr+1φ2=φ1.We take the particular solution of reduced system of ordinary differential equations (10) and (11) in the form(12)φ1=rr+12r-1ω+C12/r+1,φ2=2r+1rr-1rr+12r-1ω+C11-r/r+1,where C1=const. Thus one has to integrate overdetermined compatible system of differential equations(13)ux1=x1-1/r+1rr+12r-1x2-ux2+C12/r+1,ux2=x1r/r+12r+1rr-1rr+12r-1x2-ux2+C11-r/r+1to construct the solution of (1). Nevertheless it is easy to get the solution of the equation(14)wt-1rw-11-r/rwxw1+r/rx=0in such form(15)1-1w1/r=t-1/r+1·rr+12r-1x-θ+C12/r+1,θ=tr/r+1·2r+1rr-1rr+12r-1x-θ+C11-r/r+1.

The tangent transformations groups are also used in the framework of this approach. Let us consider the nonlinear evolution equation(16)ut=e1/uxx.One can construct operator of tangent transformations of the form(17)K=-tt+uxx+ux22u+ututadmitted 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(18)ω=xux-2u,ω1=lnut-xux,ω2=ux,ω3=tut.In order to construct ansatz of type (2) reducing (16) to system of ordinary differential equations we consider two-dimensional Lie algebra with basic operators {K,Pt=/t}. The operators satisfy the commutation relation [K,Pt]=Pt. The invariants of two-parameter Lie group with generators K, Pt are ω, ω1, and ω2. Then we construct the ansatz by using these invariants in the form(19)ux=fω,ut=expφω+xfω.From (19) and (16) we have(20)uxx=-ff1-xfand first ordinary differential equation(21)ffφ=-1.From the condition uxt=utx it follows that f, φ satisfy the second ordinary differential equation(22)f-φf3f2=-2f.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(23)ut=exp2C1-4xux-2u+x2C2-C1-4xux-2u,ux=2C2-C1-4xux-2u2,where C1, C2 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(24)ux1x2=1-k2ux221/2sinu.Indeed we showed that(25)Q=x3+kcosx3v1+k-11-k2v22v2is the operator of conditional symmetry of the corresponding system (26)v21+v31v2=v12+v32v1,v12+v32v1=1-k2v22sinx3,where ux3. Using operator Q we can write the ansatz in the following form:(27)ux1=φ2+ksinu,ux2=k-1sinu-φ1,where φ1, φ2 are unknown functions on x1, x2 and hence the Bäcklund transforms(28)ux2=k-1sinu-w,ux1=wx1+ksinurelating (24) and sine-Gordon equation wx1x2=sinw. These Bäcklund transforms (28) have been obtained for the first time in  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 (29)ux0x0=Fux0x1,ux1x1,where F is a smooth function. Using the invariance of (29) under Lie group of transformations with corresponding five-dimensional Lie algebra given by basic elements x0, x1u, x0u, and x1u we write the corresponding system in the form(30)Fv2v2x1+Fv3v3x1=v2x0,v3x0=v2x1,v1=Fv2,v3,where ux0x0v1(x0,x1), ux0x1v2(x0,x1), and ux1x1v3(x0,x1). 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 F.

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(31)ux1x2=Fu,where F is a smooth function. It is invariant with respect to the three-parameter Lie group. The basis of Lie algebra is given by {x1,x2,x1x1-x2x2}. Consider two-dimensional subalgebra with basic elements {x2,x1x1-x2x2}. By using the differential invariants u, x1ux1, and ux2/x1 of the corresponding two-parameter Lie group we construct ansatz of the form(32)ux1=fux1,ux2=x1φuwhich reduces (31) to the system(33)fφ=φ+φf=Fu.Let F(u)=0. Then we obtain two cases(34)1  f=0,φ+φf=0and solution of reduced system has the form(35)f=C1=const,φ=C2exp-uC1,C2=const.By integrating system(36)ux1=C1x1,ux2=C2x1exp-uC1one obtains the solution(37)u=C1lnC2C1x1x2+C3x1,where C3 is arbitrary real constant and C10, of (31) with F=0. In the second case we have(38)2  φ=0,ux1=1x1fu,ux2=0and solution has the form(39)u=hx1,where h(x1) is arbitrary differentiable function. Let us consider the operator(40)Q=αx2+βx1x1-x2x2,where α, β are arbitrary real constants. One can verify that(41)Qu-C1lnC2C1x1x2+C3x1=0if and only if (42)αC2C1+βC3=0.It means that solution (37) is invariant with respect to one-dimensional subgroup of symmetry group of (31) with generator Q where α, β satisfy condition (42). It is obvious that the solution (39) is invariant with respect to one-parameter group with generator Q=αx2 (β=0). Thus we conclude that any solution of (31) when F=0 constructed by this method with the help of two-dimensional Lie algebra with basic elements {x2,x1x1-x2x2} is an invariant one in the classical Lie sense.

Further we show how the operators of Lie-Bäcklund symmetry [2, 4] are used for reducing partial differential equations. Let us consider equation(43)Ux,u,u1,u2,,uk=0,where x=(x1,x2,,xn), u=u(x)Ck(Rn,R1), and uk denotes all partial derivatives of kth order and the mth order ordinary differential equation of the form(44)Hx1,x2,,xn,u,ux1,,mux1m=0.Let(45)u=Fx,C1,,Cm,where F is a smooth function on variables x,C1,,Cm and C1,,Cm are arbitrary functions on variables x2,x3,,xn, be a general solution of (44).

We use Theorem  1 from  which implies that if (44) is invariant with respect to the Lie–Bäcklund operator X=U(x,u,u1,u2,,uk)u then the ansatz (46)u=Fx,φ1,φ2,,φm,where φ1,φ2,,φm depend on n-1 variables x2,x3,,xn, reduces partial differential equation (43) to the system of k1 equations for unknown functions φ1,φ2,,φm with n-1 independent variables and k1m. We show the application of the theorem to nonlinear partial differential equation. Consider linear ordinary differential equation(47)ux1x1+α2ux1=0,where α=const. Recall that the concepts of local theory of differential equations such as symmetry, conditional symmetry, conservation laws, and Lax representations are defined by differential equalities which must be satisfied only for solutions of the equations under study. One can prove that (47) admits the following Lie–Bäcklund operator:(48)X=ux1x2-ux1Fux1+α2uu,where FC2(R1,R1). It means that the following criterium of invariance(49)X2ux1x1+α2ux1=0whenever  ux1x1+α2ux1=0,where X(2) is the prolongated operator of the second order , is fulfilled. We have proved that (47) admits operator of nonpoint (tangent) symmetry(50)X1=fu,uxuif f(u,ux) satisfies the following equation:(51)fuu-2α2fuux+α4fuxux=0.The general solution of this equation has the form(52)f=Aux1+α2uu+Bux1+α2u,where A, B are arbitrary smooth functions of one variable. One can verify that (47) also admits operator(53)X2=e-α2x1hux1+α2uu,where h is arbitrary function on variable ux1+α2u. Then the ansatz(54)u=φ1x2+e-α2x1φ2x2obtained from the general solution of (47) reduces wave type partial differential equations(55)ux1x2=ux1Fux1+α2u+Aux1+α2uu+Bux1+α2u+kux2+e-α2x1hux1+α2u,where k is a real constant. In general, the x1 dependent coefficients in partial differential equations enable us to study the effects of field gradients.

Substituting (54) into (55) we obtain the reduced system of two ordinary differential equations(56)-α2φ2=-α2φ2Fα2φ1+Aα2φ1φ2+kφ2+hα2φ1,Aα2φ1φ1+Bα2φ1+kφ1=0for unknown functions φ1(x2), φ2(x2). One can obtain partial solutions of (55) from solutions of system (56). In particular, if A=B=0 and k=0 then system (56) is reduced to one ordinary differential equation of the form(57)φ2=φ2Fα2φ1-1α2hα2φ1.This equation is integrable by quadratures for arbitrary φ1(x2). Its general solution has the form(58)φ2=C1-1α2hα2φ1x2Hx2dx2·expFα2φ1x2dx2,where C1=const,(59)Hx2=exp-Fα2φ1x2dx2.Using (54) one can construct the solution of nonlinear wave equation(60)ux1x2=ux1Fux1+α2u+e-α2x1hux1+α2uin the following form:(61)u=φ1x2+C1-1α2hα2φ1x2Hx2dx2·expFα2φ1x2dx2-α2x1,where φ1(x2) is arbitrary smooth function. So in the framework of this approach we have constructed solution with arbitrary function φ1(x2) to nonlinear wave type partial differential equation (60) for arbitrary functions F and h.

3. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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