the second - order evolution equation ( 1)l[u]ut+uux+uxx=0 was proposed by burgers   as a one - dimensional turbulence model . 
( 1 ) is also applied to model other phenomena in physics , chemistry , mathematical biology , etc .
a fairly complete review of properties of the burgers equation can be found in  [ 3 ,  chapter  4 ] .
( 1 ) is linearized to the heat equation  vt+vxx=0 using the so - called hopf  cole transformation u=2vx / v  [ 4 ,  p.  102
] . at the same time , due to the importance of the burgers equation for various applications , the exhaustive study of its properties in the framework of symmetry analysis is still topical . lie symmetries of the burgers equation and  some of its generalizations were studied since the 1960s . the maximal lie invariance algebra  gb of eq .
( 1 ) was first computed by katkov   in the course of group classification of differential equations of the general form ut+uux=(f(u)ux)x .
the algebra  gb is spanned by the vector fields pt=t , d=2tt+xxuu , k = t2t+txx+(xut)u , px=x , g = tx+u . the complete point symmetry group  gb of eq .
( 1 ) consists of the transformations t=t+t+,x=x+1t+0t+,u=(t+)ux+10 , where ( ,,,,,0,1 ) is an arbitrary set of constants defined up to a nonzero multiplier , and =2>0 .
up to composition with continuous point symmetries , the group  gb contains the single discrete symmetry ( t , x , u)(t,x,u ) .
generally , reductions of partial differential equations using their lie symmetries do not provide sufficiently large families of exact solutions of these equations .
the nonclassical method of reduction was proposed in   ( see also  ) in order to utilize a wider class of vector fields than lie symmetries .
later such vector fields were called nonclassical symmetries   or conditional symmetries  [ 911 ] or reduction operators  . the notion of nonclassical symmetries
can be extended in several directions , e.g. ,  to the concept of weak symmetry introduced in  , which is also closely related to compatibility theory of differential equations and the general method of differential constraints  [ 1418 ] .
generalized notions of ansatzes and reductions associated with weak symmetries were intensively discussed , e.g. ,  in  [ 13,1921 ] , see also references therein . a reduction operator of eq .
( 1 ) is a vector field of the general form ( 2)q=(t , x , u)t+(t , x , u)x+(t , x , u)u , where the coefficients   and   do not simultaneously vanish , which allows one to construct an ansatz reducing the initial eq .
see , e.g. ,   for the general definition of involutive families of reduction operators .
the multiplication by nonvanishing functions of  ( t , x , u ) generates an equivalence relation on the set of reduction operators .
the determining equations on coefficients of a reduction operator  q are derived from the conditional invariance criterion  ( 3)q(2)l[u]lq(2)=0 .
here q(2 ) is the second prolongation of the vector field  q , l is the manifold in the second - order jet space j(2 ) that corresponds to the burgers equation l[u]=0 , and q(2 ) is the manifold in the same jet space determined by the invariant surface condition q[u]=0 jointly with its differential consequences dtq[u]=0 and  dxq[u]=0 , q[u]=utux is the characteristic of the vector field  q , dt and  dx are the operators of total differentiation with respect to t and x , respectively . in view of the evolution
kind of the burgers equation it is natural to partition the set of its reduction operators into two subsets , singular and regular , depending on whether or not the coefficient   vanishes  .
up to the above equivalence relation , one can assume ( ,)=(0,1 ) and =1 for singular and regular reduction operators of the burgers equation , respectively .
it is the burgers equation that was first considered from the nonclassical symmetry point of view after the prominent paper  .
namely , in   the determining equations for regular reduction operators of  ( 1 ) were derived under the gauge =1 and a few of their particular solutions satisfying the additional constraint u=0 were constructed .
the determining equations for both regular and singular nonclassical symmetries of  ( 1 ) were presented in  . therein the regular case was studied in detail under the gauge =1 , for which the consideration was shown to be partitioned into three cases , u=0 , u=1 and u=12 . the case u=0 proved to result merely in nonclassical symmetries which are equivalent to lie symmetries .
( within the framework of the direct method , the same result was earlier obtained in   in terms of the corresponding ansatzes and reductions . )
the unique reduction operator t+ux satisfying the constraint u=1 was also found and used for reducing the burgers equation .
for the case u=12 some particular solutions of the determining equations jointly with the corresponding ansatzes and invariant solutions of the burgers equation were constructed .
the above consideration of regular nonclassical symmetries from   was extended in   with more particular solutions satisfying the constraint u=12 . still
wider families of particular solutions of the determining equations in this case were given in  .
in   an algorithmic procedure to derive determining equations for nonclassical symmetries was proposed , and the burgers equation was one of the illustrative examples for application of this procedure .
the system  sb of determining equations for the case u=12 was not well investigated for a surprisingly long time although the study of the analogous system  sh for regular reduction operators of the linear heat equation vt+vxx=0 , whose form is very similar to  sb , had already been completed in  .
see also   for preliminary results on  sh and   for further generalizations to ( 1 + 1)-dimensional second - order linear evolution equations .
namely , this system was reduced by a differential substitution to the uncoupled system of three copies of the linear heat equation .
as shown in  , the systems sb and  sh as well as the substitutions linearizing them can be interpreted in terms of the matrix burgers equation and the matrix hopf  cole transformation .
the above in fact means that the case  u=12 can be referred  to as  no - go  . in general , looking for reduction operators in a family of vector fields is said to result in a  no - go 
case if the corresponding system of determining equations for coefficients of reduction operators is reduced to a well - determined system whose general solution can not be represented in a closed form , and , moreover , solving this system is equivalent , in a certain sense , to solving the initial equation .
singular reduction operators of the burgers equation were in fact not studied until  , where no - go results of   on reduction operators with =0 for the linear heat equation were extended to general evolution equations of order greater than one .
these results were treated in   within the framework of singular reduction operators . in this paper
we intend to enhance and complete the above results on nonclassical symmetries and reductions of the burgers equation .
in particular , extending methods from   we present a new proof on the linearization of the system  sb and show that solutions of this system are expressed via triples of solutions of the burgers equation .
we first exhaustively describe all possible nonclassical reductions of the burgers equation to single ordinary differential equations including reductions associated with the no - go case =1 and u=12 .
a part of the description is the assertion stating that any lie reduction of the burgers equation is equivalent via the hopf  cole transformation to a parameterized family of lie reductions of the linear heat equation .
as the burgers equation  ( 1 ) is a ( 1 + 1)-dimensional second - order evolution equation , every reduction operator of the form  ( 2 ) with =0 is singular since the corresponding reduced equation is of a lower ( namely , the first ) order  .
all basic results on such reduction operators of  ( 1 ) follow from the general results on singular reduction operators of co - order singularity one . after setting =1 in  q due to the equivalence of reduction operators ,
the conditional invariance criterion  ( 3 ) implies a single determining equation on the single coefficient  =(t , x , u ) , ( 4)t+ux+2+xx+2xu+2uu=0 .
namely , the composition of the differential substitution =x/u with u0 , where  is a smooth function of ( t , x , u ) , and the hodograph transformation , where the new independent variables are t=t , x=x and = and the new dependent variable is u=u , reduces eq .
( 1 ) on the function u=u(t,x, ) with  playing the role of a parameter .
moreover , up to equivalences of reduction operators and solution families , there exists a bijection between one - parameter families of solutions of eq .
namely , each operator of the above kind corresponds to the family of solutions which are invariant with respect to this operator .
the problems of construction of all one - parameter solution families of  ( 1 ) and the exhaustive description of its reduction operators with zero coefficients of t are completely equivalent . given a family f={u = f(t , x, ) } of solutions of  ( 1 ) parameterized by a single essential parameter   , the corresponding singular reduction operator is q=x(x/u)u , where the function  is obtained by solving the equality u = f(t , x, ) with respect to   , =(t , x , u ) .
the ansatz u = f(t , x,( ) ) , where =t , associated with  q , reduces eq .
the simplicity of the reduced equation is explained by the specific choice of the ansatz based on knowing the one - parameter family  f of solutions .
now we look for regular reduction operators of the burgers equation  ( 1 ) , which are of the form  ( 2 ) with nonvanishing values of the coefficient  . up to equivalence of reduction operators , for any regular operator  q we can set =1 . in view of this gauge
we do not need to use the differential consequences in order to derive the determining equations , i.e.  it suffices to take into account only the equations ut+uux+uxx=0 and  utux=0 in the course of confining to the manifold lq(2 ) in the conditional invariance criterion  ( 3 ) . substituting the expressions for  ut and  uxx obtained from these equations into the differential function q(2)l[u ] and splitting the result with respect to ux , we get ( 5)uu=0,2xu2u+2uu+uu=0,2xu+2u+t+uxxx2x=0,t+ux+xx+2x=0 . integrating the first two equations
, we represent the functions  and   as polynomials of  u with the coefficients depending on t and x , ( 6)=1u+0,=131(11)u3+(x1+10)u2+1u+0 . then we split the third equation of the system  ( 5 ) with respect to  u and get a system of differential equations on the functions 1 , 0 , 1 and 0 , ( 7)1(11)(21 + 1)=0,10(21 + 1)+41x1=0,3xx1 + 2(x10+1x0)+(21 + 1)1t1+x0=0,t0 + 20x0+xx0(21 + 1)02x1=0 .
the further consideration depends on the choice among the three possible solutions of the first equation of the system  ( 7 ) .
we rewrite the last equation of  ( 5 ) in terms of 1 , 0 , 1 and  0 and split it with respect to u severally for each value of  1 .
the corresponding vector field q1=t+ux is a unique reduction operator ( up to the equivalence relation ) for the burgers equation in this case .
the set of q1-invariant solutions consists of two families , one of which is two - parameter and the other is one - parameter .
the two - parameter family is formed by the functions u=(x+c1)/(t+c0 ) , where  c1 and  c0 are arbitrary constants , and all these solutions are lie - invariant and equivalent to the scale - invariant solution u = x / t .
they are invariant with respect to shifts of both  t and  x. the optimal way for the construction of q1-invariant solutions of  ( 1 ) is to integrate at first the equation l[u]+q1[u]=uxx=0 , which gives the representation u=(t)x+(t ) with some smooth functions   and   of  t. the generalized vector field q=(l[u]+q1[u])u = uxxu , is equivalent to the evolutionary representative q1[u]u of  q1 on the set of solutions of  ( 1 ) and hence it is a generalized conditional symmetry of  ( 1 ) , cf . 
this is why the associated ansatz u=(t)x+(t ) reduces the burgers equation  ( 1 ) to a system of two ordinary differential equations with respect to the functions   and   , t+2=0,t+=0 . a complete set of functionally independent integrals of the equation q1[u]=0 consists of xut and  u. therefore ,
directly with  q1 we construct the implicit ansatz u=( ) , where =xut ( resp . 
the case 1=0 is discussed in   in detail ( see also  ) , where it is noted that all corresponding solutions of the burgers equation are lie - invariant .
the system  ( 7 ) with  1=0 implies 1=x0 and 0=t0 + 20x0 + 3xx0 , and splitting the last eq .
( 5 ) with respect to  u we derive xx0=0,tt0 + 20tx0 + 4t0x0 + 40(x0)2=0 .
then we have 0=01(t)x+00(t ) , where the coefficients  01 and  00 satisfy the system tt01 + 601t01 + 4(01)3=0,tt00 + 401t00 + 2t0100 + 4(01)200=0 . the transformation 01=t/2
, 00=/ maps this system to the system of two simple uncoupled equations ttt=0 and  tt=0 for the functions =(t ) and  =(t ) . hence 01=c2t+c1c2t2 + 2c1t+c0,00=c4t+c3c2t2 + 2c1t+c0 , where c0, 
the substitution of the expressions obtained for 1 , 0 and 0=01x+00 into  ( 6 ) gives q=t+(c2t+c1)x+c4t+c3c2t2 + 2c1t+c0x+(c2t+c1)u+c2x+c4c2t2 + 2c1t+c0u .
it is easy to see that the operator  q differs from a lie symmetry operator of the burgers equation by the multiplier ( c2t2 + 2c1t+c0)1 .
proposition  1there is a bijection between reduction operators of the burgers equation of the general formq=t+(t , x)x+(1(t , x)u+0(t , x))uand one - dimensional algebras spanned by lie symmetry operators of this equation with nonzero coefficients of 
 t 
 .
there is a bijection between reduction operators of the burgers equation of the general formq=t+(t , x)x+(1(t , x)u+0(t , x))uand one - dimensional algebras spanned by lie symmetry operators of this equation with nonzero coefficients of 
 t 
 .
the bijection is established by the equivalence relation of reduction operators . in other words ,
any nonclassical reduction of the burgers equation with respect to a vector field with =1 and u=0 is in fact a lie reduction . at the same time , any lie solution of the burgers equation is obtained from a lie solution of the linear heat equation via the hopf  cole transformation . before presenting the corresponding rigorous assertion
, we recall that the maximal lie invariance algebra gh of the linear heat equation vt+vxx=0 is spanned by the vector fields pt=t , d=2tt+xx , k=t2t+txx+(14x212t)vv , px=x , g=tx+12xvv , i=vv , h(t , x)v , where h(t , x ) runs through the set of solutions of this equation .
we associate any vector field q = c0pt+c1d+c2k+c3px+c4 g from the maximal lie invariance algebra  gb of the eq .
proposition  2a solution 
 u 
 of the burgers equation  ( 1 )  is invariant with respect to a vector field 
 q 
 from 
 gb 
 if and only if 
 u=2vx / v 
 for some 
 q-invariant solution 
 v 
 of the linear heat equation 
 vt+vxx=0 , where 
  
 is a constant and 
 q=qigh . a solution 
 u 
 of the burgers equation  ( 1 )  is invariant with respect to a vector field 
 q 
 from 
 gb 
 if and only if 
 u=2vx / v 
 for some 
 q-invariant solution 
 v 
 of the linear heat equation 
 vt+vxx=0 , where 
  
 is a constant and 
 q=qigh .
proofif smooth functions u and v depending on  t and  x are related via the hopf  cole transformation , u=2vx / v , then q[u]=q[2vx
/ v]=2(q[v]/v)x.suppose that u is a q - invariant solution of  ( 1 ) .
then the function  v can be assumed to be a solution of the linear heat equation vt+vxx=0 , and ( q[v]/v)x = q[u]/2=0 , i.e.  q[v]=v for some smooth function  of  t. acting by the operator t = dt+dxx on both sides of the last equation , we derive tq[v]=q[tv]2(c2t+c1)tv=tv+tv . in view of tv=0 and tq[v]=0
as then q[v]v = q[v]=0 , the function  v is a q-invariant.conversely , if for some constant   the function  v is a q-invariant solution of the linear heat equation vt+vxx=0 then the function u=2vx / v is a solution of the burgers equation  ( 1 ) and q[u]=q[2vx / v]=2(q[v]/v)x=2(q[v]/v)x=0 , i.e. ,  the function  u is q - invariant . 
 if smooth functions u and v depending on  t and  x are related via the hopf  cole transformation , u=2vx / v , then q[u]=q[2vx / v]=2(q[v]/v)x .
then the function  v can be assumed to be a solution of the linear heat equation vt+vxx=0 , and ( q[v]/v)x = q[u]/2=0 , i.e.  q[v]=v for some smooth function  of  t. acting by the operator t = dt+dxx on both sides of the last equation , we derive tq[v]=q[tv]2(c2t+c1)tv=tv+tv . in view of tv=0 and tq[v]=0
conversely , if for some constant   the function  v is a q-invariant solution of the linear heat equation vt+vxx=0 then the function u=2vx / v is a solution of the burgers equation  ( 1 ) and q[u]=q[2vx / v]=2(q[v]/v)x=2(q[v]/v)x=0 , i.e. ,  the function  u is q - invariant . 
 the case 1=12 leads to reduction operators of the general form ( 8)q=t+(12u+0)x+(14u302u2+1u+0)u , where the coefficients 0 , 1 and  0 are smooth functions of  t and  x satisfying the system of differential equations ( 9)t0 + 20x0+xx02x1=0,t1 + 2x01+xx1+x0=0,t0 + 2x00+xx0=0 derived from  ( 7 ) .
as a differential substitution reduces the system  ( 9 ) to an uncoupled system of three copies of the linear heat equation  , the general solution of  ( 9 ) can not be represented in a closed form .
this result appears to directly follow from the fact that q is a reduction operator of eq .
moreover , we show that solutions of the system  ( 9 ) can be represented via solutions of the uncoupled system of three copies of the burgers equation .
theorem  1any solution of the determining system  ( 9 )  on the coefficients of reduction operators of the form  ( 8)  is represented as(10)0=(w(v))xw(v),1=|v,vxx , vxxx|w(v),0=2w(vx)w(v),where 
 v=(v1,v2,v3 ) 
 is a triple of linear independent solutions of the heat equation 
 vt+vxx=0 , 
 w(v)=|v,vx , vxx| 
 and 
 w(vx)=|vx , vxx , vxxx| 
 are the wronskians of this triple and the triple of the corresponding derivatives with respect to 
 x , respectively , and 
 |p,q,r| 
 denotes the determinant of the matrix constructed from ternary columns 
 p , 
 q 
 and 
 r 
 .
conversely , any triple 
 ( 0,1,0 ) 
 admitting the representation  ( 10 )  satisfies the system  ( 9 ) .
any solution of the determining system  ( 9 )  on the coefficients of reduction operators of the form  ( 8)  is represented as(10)0=(w(v))xw(v),1=|v,vxx , vxxx|w(v),0=2w(vx)w(v),where 
 v=(v1,v2,v3 ) 
 is a triple of linear independent solutions of the heat equation 
 vt+vxx=0 , 
 w(v)=|v,vx , vxx| 
 and 
 w(vx)=|vx , vxx , vxxx| 
 are the wronskians of this triple and the triple of the corresponding derivatives with respect to 
 x , respectively , and 
 |p,q,r| 
 denotes the determinant of the matrix constructed from ternary columns 
 p , 
 q 
 and 
 r 
 .
conversely , any triple 
 ( 0,1,0 ) 
 admitting the representation  ( 10 )  satisfies the system  ( 9 ) .
the set of q - invariant solutions of the burgers equation l[u]=0 coincides with the set of solutions of the system l[u]=0 and q[u]=0 , and it is parameterized by two arbitrary constants as q is a regular reduction operator of the burgers equation  . for convenience
we recombine the equations of the above system in the following way : l[u]=0 , l[u]+q[u]=0 .
the hopf  cole transformation u=2vx / v maps this system to the linear system ( 11)vt+vxx=0,vxxx0vxx+1vx+120v=0 .
 ,vn of t and x be linear independent solutions of the system  ( 11 ) . then the equation ( 12)u=2c1vx1++cnvxnc1v1++cnvn where c1, 
,cn are arbitrary constants which are not simultaneously zero , defines a family of q - invariant solutions of the burgers equation parameterized by n1 essential constants , and hence n3 as the number of such parameters can not be greater than two .
in other words , the dimension of the space  v of solutions of the system  ( 11 ) does not exceed three .
 ,vn of  t and  x form a basis of the space  v , where n = dimv .
then the expression  ( 12 ) represents the general solution of the system l[u]=0 and q[u]=0 , which contains n1 essential constant parameters .
therefore , n1=2 , i.e.  n=3.consider a basis { v1,v2,v3 } of the space  v. by definition , the elements of  v are solutions of the heat equation vt+vxx=0 , which is linear and evolutionary . hence the usual linear independence of them implies the linear independence of them over the ring of smooth functions of  t , i.e.  the wronskian  w(v ) of the functions  v1 , v2 and v3 with respect to  x does not vanish .
see , e.g. ,  note  5 in  . substituting the elements of the basis into the second equation of  ( 11 ) , we obtain a well - defined system of linear algebraic equations , vxxxi0vxxi+1vxi+120vi=0,i=1,2,3 , for the coefficients  0 , 1 , 0 , or , in the matrix form , mq=vxxx , where v=(v1v2v3),m=(v1vx1vxx1v2vx2vxx2v3vx3vxx3),q=(12010 ) . solving this system with respect to  0 , 1 and  0
, we derive the representation  ( 10).as the proof can be turned back , the inverse statement is also true . 
the set of q - invariant solutions of the burgers equation l[u]=0 coincides with the set of solutions of the system l[u]=0 and q[u]=0 , and it is parameterized by two arbitrary constants as q is a regular reduction operator of the burgers equation  . for convenience
we recombine the equations of the above system in the following way : l[u]=0 , l[u]+q[u]=0 . the hopf  cole transformation u=2vx
 ,vn of t and x be linear independent solutions of the system  ( 11 ) . then the equation ( 12)u=2c1vx1++cnvxnc1v1++cnvn where c1, 
,cn are arbitrary constants which are not simultaneously zero , defines a family of q - invariant solutions of the burgers equation parameterized by n1 essential constants , and hence n3 as the number of such parameters can not be greater than two . in other words ,
the dimension of the space  v of solutions of the system  ( 11 ) does not exceed three .
 ,vn of  t and  x form a basis of the space  v , where n = dimv .
then the expression  ( 12 ) represents the general solution of the system l[u]=0 and q[u]=0 , which contains n1 essential constant parameters .
consider a basis { v1,v2,v3 } of the space  v. by definition , the elements of  v are solutions of the heat equation vt+vxx=0 , which is linear and evolutionary . hence the usual linear independence of them implies the linear independence of them over the ring of smooth functions of  t , i.e.  the wronskian  w(v ) of the functions  v1 , v2 and v3 with respect to  x does not vanish .
see , e.g. ,  note  5 in  . substituting the elements of the basis into the second equation of  ( 11 ) , we obtain a well - defined system of linear algebraic equations , vxxxi0vxxi+1vxi+120vi=0,i=1,2,3 , for the coefficients  0 , 1 , 0 , or , in the matrix form , mq=vxxx , where v=(v1v2v3),m=(v1vx1vxx1v2vx2vxx2v3vx3vxx3),q=(12010 ) . solving this system with respect to  0 , 1 and  0 , we derive the representation  ( 10 ) .
as the proof can be turned back , the inverse statement is also true . 
 corollary  1the coefficients of the reduction operator  ( 8)  of the burgers equation can be represented in the form(13)0=12|e,u,z||e,u,y|,1=14|e,y,z||e,u,y|,0=14|u,y,z||e,u,y|,where the columns 
 e , 
 y 
 and 
 z 
 consist of three units , the expressions 
 yi=2uxi+(ui)2 
 and 
 zi=4uxxi+6uiuxi+(ui)3 , respectively , 
 i=1,2,3 , and 
 u 
 is a column of three solutions of the burgers equation with 
 |e,u,y|0 .
the coefficients of the reduction operator  ( 8)  of the burgers equation can be represented in the form(13)0=12|e,u,z||e,u,y|,1=14|e,y,z||e,u,y|,0=14|u,y,z||e,u,y|,where the columns 
 e , 
 y 
 and 
 z 
 consist of three units , the expressions 
 yi=2uxi+(ui)2 
 and 
 zi=4uxxi+6uiuxi+(ui)3 , respectively , 
 i=1,2,3 , and 
 u 
 is a column of three solutions of the burgers equation with 
 |e,u,y|0 .
proofthe connection between solutions of the heat equation and the burgers equation via the hopf  cole transformation 2vxi / vi = ui gives the expressions vxxivi=12uxi+14(ui)2,vxxxivi=34uiuxi+18(ui)3
note that the determinant |e,u,y| is nonvanishing as the wronskian  w(v ) is the same . 
 the connection between solutions of the heat equation and the burgers equation via the hopf  cole transformation 2vxi / vi = ui gives the expressions vxxivi=12uxi+14(ui)2,vxxxivi=34uiuxi+18(ui)3
note that the determinant |e,u,y| is nonvanishing as the wronskian  w(v ) is the same . 
 corollary  2the representations  ( 10 )  and  ( 12 ) , where 
 n=3 , explicitly define the one - to - one correspondence between reduction operators of the form  ( 8)  and families of solutions of the burgers equation that are invariant with respect to these operators . the representations  ( 10 )  and  ( 12 ) , where 
 n=3 , explicitly define the one - to - one correspondence between reduction operators of the form  ( 8)  and families of solutions of the burgers equation that are invariant with respect to these operators .
it looks very difficult to explicitly construct an ansatz for  u by the direct integration of the invariant surface condition that corresponds to an operator of the form  ( 8) in the case of an arbitrary solution of system  ( 9 ) . even for a simple solution of system  ( 9 ) , carrying out the corresponding reduction of the burgers equation  ( 1 )
see , e.g. ,  , where a few such reductions were implemented . a better way for using an operator of the form  ( 8) for reducing the burgers equation is to consider , instead of  q , the generalized vector field q=(l[u]+q[u])u which is equivalent to the evolutionary representative q[u]u of  q and is a generalized conditional symmetry of  ( 1 ) , cf . 
( 1 ) with respect to the generalized vector field q is somehow presented in the proof of theorem  1 , and corollary  2 exhaustively describes the family of q - invariant solutions of  ( 1 ) . at the same time , knowing the representation  ( 12 ) , where n=3 , for q - invariant solutions of  ( 1 ) allows us to easily construct an ansatz for  u associated with the operator  q and then to reduce the burgers equation  ( 1 ) using this ansatz . setting c1=1 and c2=0 ( resp . 
c1=0 and c2=1 ) and assuming c3 to be an arbitrary constant , we derive two integrals of the equation q[u]=0 , =v1u2vx1v3u2vx3,=v2u2vx2v3u2vx3 , which are correctly defined for u2vx3/v3 ( we can always assume this condition to be satisfied for a specific solution  u up to renumbering the functions  v1 , v2 and v3 ) .
therefore , the general solution of the equation q[u]=0 is implicitly represented in the form f(,)=0 , where f is an arbitrary nonconstant function of its arguments . up to permutation of   and  
then the equality f(,)=0 implies the implicit ansatz =( ) for the unknown function u = u(t , x ) , where  and   are the new dependent and independent variables , respectively . the standard way for deriving the corresponding reduced equation via the computation of the expressions for derivatives
ut , ux and uxx implied by the ansatz and the subsequent substitution of the expressions into  ( 1 ) is too cumbersome . instead of this
, we act on the ansatz =( ) by the operator dt+dxx and then make the substitutions ut+uxx=uux and vti+vxxi=0 , i=1,2,3 . as a result , we derive the equation dx=0 .
the differential function  dx does not vanish for solutions of  ( 1 ) which are implicitly represented in the form =( ) for some smooth function   of  . indeed , suppose that dx=0
( i.e. ,  =0(t ) ) for such a solution u = u0(t , x )
. then we have u0=2(vx20vx3)/(v20v3 ) , which implies that for some nonvanishing smooth function   of  t the expression ( v2+0v3) gives a solution of the linear heat equation vt+vxx=0 . as the functions  v2 and 
v3 are linearly independent solutions of the same equation , this is possible only if 0 is a constant .
therefore , the substitution u = u0(t , x ) into   results in the constant value 0=(0 ) and hence u0=2(vx10vx3)/(v10v3 ) . comparing the two expressions for the solution
u = u0(t , x ) , we conclude that the functions  v1 , v2 and  v3 are linearly dependent and thus arrive at a contradiction . as dx0 , the reduced equation takes the form =0 .
its general solution is =c1+c2 , which completely agrees with the representation  ( 12 ) and the ansatz =( ) .
the above reduced equation is obtained for an arbitrary operator  q of the form  ( 8) and it is much simpler than particular reduced equations derived in  . this fact is explained by that the ansatz =( ) , whose construction is based on the representation  ( 12 ) for q - invariant solutions of the burgers equation  ( 1 ) , is in better agreement with the structure of this equation than particular ansatzes given in  .
the corresponding vector field q1=t+ux is a unique reduction operator ( up to the equivalence relation ) for the burgers equation in this case .
the set of q1-invariant solutions consists of two families , one of which is two - parameter and the other is one - parameter .
the two - parameter family is formed by the functions u=(x+c1)/(t+c0 ) , where  c1 and  c0 are arbitrary constants , and all these solutions are lie - invariant and equivalent to the scale - invariant solution u = x / t .
they are invariant with respect to shifts of both  t and  x. the optimal way for the construction of q1-invariant solutions of  ( 1 ) is to integrate at first the equation l[u]+q1[u]=uxx=0 , which gives the representation u=(t)x+(t ) with some smooth functions   and   of  t. the generalized vector field q=(l[u]+q1[u])u = uxxu , is equivalent to the evolutionary representative q1[u]u of  q1 on the set of solutions of  ( 1 ) and hence it is a generalized conditional symmetry of  ( 1 ) , cf . 
this is why the associated ansatz u=(t)x+(t ) reduces the burgers equation  ( 1 ) to a system of two ordinary differential equations with respect to the functions   and   , t+2=0,t+=0 . a complete set of functionally independent integrals of the equation q1[u]=0 consists of xut and  u. therefore ,
directly with  q1 we construct the implicit ansatz u=( ) , where =xut ( resp . 
the case 1=0 is discussed in   in detail ( see also  ) , where it is noted that all corresponding solutions of the burgers equation are lie - invariant .
the system  ( 7 ) with  1=0 implies 1=x0 and 0=t0 + 20x0 + 3xx0 , and splitting the last eq .
( 5 ) with respect to  u we derive xx0=0,tt0 + 20tx0 + 4t0x0 + 40(x0)2=0 .
then we have 0=01(t)x+00(t ) , where the coefficients  01 and  00 satisfy the system tt01 + 601t01 + 4(01)3=0,tt00 + 401t00 + 2t0100 + 4(01)200=0 . the transformation 01=t/2 , 00=/ maps this system to the system of two simple uncoupled equations ttt=0 and  tt=0 for the functions =(t ) and  =(t ) . hence
the substitution of the expressions obtained for 1 , 0 and 0=01x+00 into  ( 6 ) gives q=t+(c2t+c1)x+c4t+c3c2t2 + 2c1t+c0x+(c2t+c1)u+c2x+c4c2t2 + 2c1t+c0u .
it is easy to see that the operator  q differs from a lie symmetry operator of the burgers equation by the multiplier ( c2t2 + 2c1t+c0)1 .
proposition  1there is a bijection between reduction operators of the burgers equation of the general formq=t+(t , x)x+(1(t , x)u+0(t , x))uand one - dimensional algebras spanned by lie symmetry operators of this equation with nonzero coefficients of 
 t 
 .
there is a bijection between reduction operators of the burgers equation of the general formq=t+(t , x)x+(1(t , x)u+0(t , x))uand one - dimensional algebras spanned by lie symmetry operators of this equation with nonzero coefficients of 
 t 
 .
the bijection is established by the equivalence relation of reduction operators . in other words ,
any nonclassical reduction of the burgers equation with respect to a vector field with =1 and u=0 is in fact a lie reduction . at the same time
, any lie solution of the burgers equation is obtained from a lie solution of the linear heat equation via the hopf  cole transformation . before presenting the corresponding rigorous assertion
, we recall that the maximal lie invariance algebra gh of the linear heat equation vt+vxx=0 is spanned by the vector fields pt=t , d=2tt+xx , k=t2t+txx+(14x212t)vv , px=x , g=tx+12xvv , i=vv , h(t , x)v , where h(t , x ) runs through the set of solutions of this equation .
we associate any vector field q = c0pt+c1d+c2k+c3px+c4 g from the maximal lie invariance algebra  gb of the eq .
proposition  2a solution 
 u 
 of the burgers equation  ( 1 )  is invariant with respect to a vector field 
 q 
 from 
 gb 
 if and only if 
 u=2vx / v 
 for some 
 q-invariant solution 
 v 
 of the linear heat equation 
 vt+vxx=0 , where 
  
 is a constant and 
 q=qigh .
a solution 
 u 
 of the burgers equation  ( 1 )  is invariant with respect to a vector field 
 q 
 from 
 gb 
 if and only if 
 u=2vx / v 
 for some 
 q-invariant solution 
 v 
 of the linear heat equation 
 vt+vxx=0 , where 
  
 is a constant and 
 q=qigh .
proofif smooth functions u and v depending on  t and  x are related via the hopf  cole transformation , u=2vx / v , then q[u]=q[2vx / v]=2(q[v]/v)x.suppose that u is a q - invariant solution of  ( 1 ) . then the function  v can be assumed to be a solution of the linear heat equation vt+vxx=0 , and ( q[v]/v)x = q[u]/2=0 , i.e.  q[v]=v for some smooth function  of  t. acting by the operator t = dt+dxx on both sides of the last equation , we derive tq[v]=q[tv]2(c2t+c1)tv=tv+tv . in view of tv=0 and tq[v]=0 , this implies that t=0 , i.e. ,   is a constant .
as then q[v]v = q[v]=0 , the function  v is a q-invariant.conversely , if for some constant   the function  v is a q-invariant solution of the linear heat equation vt+vxx=0 then the function u=2vx / v is a solution of the burgers equation  ( 1 ) and q[u]=q[2vx / v]=2(q[v]/v)x=2(q[v]/v)x=0 , i.e. ,  the function  u is q - invariant . 
 if smooth functions u and v depending on  t and  x are related via the hopf  cole transformation , u=2vx / v , then q[u]=q[2vx
then the function  v can be assumed to be a solution of the linear heat equation vt+vxx=0 , and ( q[v]/v)x = q[u]/2=0 , i.e.  q[v]=v for some smooth function  of  t. acting by the operator t = dt+dxx on both sides of the last equation , we derive tq[v]=q[tv]2(c2t+c1)tv=tv+tv . in view of tv=0 and tq[v]=0 , this implies that t=0 , i.e. ,   is a constant .
conversely , if for some constant   the function  v is a q-invariant solution of the linear heat equation vt+vxx=0 then the function u=2vx / v is a solution of the burgers equation  ( 1 ) and q[u]=q[2vx / v]=2(q[v]/v)x=2(q[v]/v)x=0 , i.e. ,  the function  u is q - invariant . 
the case 1=12 leads to reduction operators of the general form ( 8)q=t+(12u+0)x+(14u302u2+1u+0)u , where the coefficients 0 , 1 and  0 are smooth functions of  t and  x satisfying the system of differential equations ( 9)t0 + 20x0+xx02x1=0,t1 + 2x01+xx1+x0=0,t0 + 2x00+xx0=0 derived from  ( 7 ) .
as a differential substitution reduces the system  ( 9 ) to an uncoupled system of three copies of the linear heat equation  , the general solution of  ( 9 ) can not be represented in a closed form .
this result appears to directly follow from the fact that q is a reduction operator of eq .
moreover , we show that solutions of the system  ( 9 ) can be represented via solutions of the uncoupled system of three copies of the burgers equation .
theorem  1any solution of the determining system  ( 9 )  on the coefficients of reduction operators of the form  ( 8)  is represented as(10)0=(w(v))xw(v),1=|v,vxx , vxxx|w(v),0=2w(vx)w(v),where 
 v=(v1,v2,v3 ) 
 is a triple of linear independent solutions of the heat equation 
 vt+vxx=0 , 
 w(v)=|v,vx , vxx| 
 and 
 w(vx)=|vx , vxx , vxxx| 
 are the wronskians of this triple and the triple of the corresponding derivatives with respect to 
 x , respectively , and 
 |p,q,r| 
 denotes the determinant of the matrix constructed from ternary columns 
 p , 
 q 
 and 
 r 
 .
conversely , any triple 
 ( 0,1,0 ) 
 admitting the representation  ( 10 )  satisfies the system  ( 9 ) .
any solution of the determining system  ( 9 )  on the coefficients of reduction operators of the form  ( 8)  is represented as(10)0=(w(v))xw(v),1=|v,vxx , vxxx|w(v),0=2w(vx)w(v),where 
 v=(v1,v2,v3 ) 
 is a triple of linear independent solutions of the heat equation 
 vt+vxx=0 , 
 w(v)=|v,vx , vxx| 
 and 
 w(vx)=|vx , vxx , vxxx| 
 are the wronskians of this triple and the triple of the corresponding derivatives with respect to 
 x , respectively , and 
 |p,q,r| 
 denotes the determinant of the matrix constructed from ternary columns 
 p , 
 q 
 and 
 r 
 .
conversely , any triple 
 ( 0,1,0 ) 
 admitting the representation  ( 10 )  satisfies the system  ( 9 ) .
the set of q - invariant solutions of the burgers equation l[u]=0 coincides with the set of solutions of the system l[u]=0 and q[u]=0 , and it is parameterized by two arbitrary constants as q is a regular reduction operator of the burgers equation  . for convenience
we recombine the equations of the above system in the following way : l[u]=0 , l[u]+q[u]=0 . the hopf  cole transformation u=2vx
 ,vn of t and x be linear independent solutions of the system  ( 11 ) . then the equation ( 12)u=2c1vx1++cnvxnc1v1++cnvn where c1, 
,cn are arbitrary constants which are not simultaneously zero , defines a family of q - invariant solutions of the burgers equation parameterized by n1 essential constants , and hence n3 as the number of such parameters can not be greater than two .
in other words , the dimension of the space  v of solutions of the system  ( 11 ) does not exceed three .
 ,vn of  t and  x form a basis of the space  v , where n = dimv .
then the expression  ( 12 ) represents the general solution of the system l[u]=0 and q[u]=0 , which contains n1 essential constant parameters .
therefore , n1=2 , i.e.  n=3.consider a basis { v1,v2,v3 } of the space  v. by definition , the elements of  v are solutions of the heat equation vt+vxx=0 , which is linear and evolutionary . hence the usual linear independence of them implies the linear independence of them over the ring of smooth functions of  t , i.e.  the wronskian  w(v ) of the functions  v1 , v2 and v3 with respect to  x does not vanish .
see , e.g. ,  note  5 in  . substituting the elements of the basis into the second equation of  ( 11 )
, we obtain a well - defined system of linear algebraic equations , vxxxi0vxxi+1vxi+120vi=0,i=1,2,3 , for the coefficients  0 , 1 , 0 , or , in the matrix form , mq=vxxx , where v=(v1v2v3),m=(v1vx1vxx1v2vx2vxx2v3vx3vxx3),q=(12010 ) . solving this system with respect to  0 , 1 and  0 ,
we derive the representation  ( 10).as the proof can be turned back , the inverse statement is also true . 
the set of q - invariant solutions of the burgers equation l[u]=0 coincides with the set of solutions of the system l[u]=0 and q[u]=0 , and it is parameterized by two arbitrary constants as q is a regular reduction operator of the burgers equation  . for convenience we recombine the equations of the above system in the following way : l[u]=0 , l[u]+q[u]=0 . the hopf  cole transformation u=2vx
let for some integer  n functions v1,  ,vn of t and x be linear independent solutions of the system  ( 11 ) .
,cn are arbitrary constants which are not simultaneously zero , defines a family of q - invariant solutions of the burgers equation parameterized by n1 essential constants , and hence n3 as the number of such parameters can not be greater than two . in other words ,
the dimension of the space  v of solutions of the system  ( 11 ) does not exceed three .
 ,vn of  t and  x form a basis of the space  v , where n = dimv .
then the expression  ( 12 ) represents the general solution of the system l[u]=0 and q[u]=0 , which contains n1 essential constant parameters .
consider a basis { v1,v2,v3 } of the space  v. by definition , the elements of  v are solutions of the heat equation vt+vxx=0 , which is linear and evolutionary .
hence the usual linear independence of them implies the linear independence of them over the ring of smooth functions of  t , i.e.  the wronskian  w(v ) of the functions  v1 , v2 and v3 with respect to  x does not vanish .
see , e.g. ,  note  5 in  . substituting the elements of the basis into the second equation of  ( 11 ) , we obtain a well - defined system of linear algebraic equations , vxxxi0vxxi+1vxi+120vi=0,i=1,2,3 , for the coefficients  0 , 1 , 0 , or , in the matrix form , mq=vxxx , where v=(v1v2v3),m=(v1vx1vxx1v2vx2vxx2v3vx3vxx3),q=(12010 ) . solving this system with respect to  0 , 1 and  0
corollary  1the coefficients of the reduction operator  ( 8)  of the burgers equation can be represented in the form(13)0=12|e,u,z||e,u,y|,1=14|e,y,z||e,u,y|,0=14|u,y,z||e,u,y|,where the columns 
 e , 
 y 
 and 
 z 
 consist of three units , the expressions 
 yi=2uxi+(ui)2 
 and 
 zi=4uxxi+6uiuxi+(ui)3 , respectively , 
 i=1,2,3 , and 
 u 
 is a column of three solutions of the burgers equation with 
 |e,u,y|0 .
the coefficients of the reduction operator  ( 8)  of the burgers equation can be represented in the form(13)0=12|e,u,z||e,u,y|,1=14|e,y,z||e,u,y|,0=14|u,y,z||e,u,y|,where the columns 
 e , 
 y 
 and 
 z 
 consist of three units , the expressions 
 yi=2uxi+(ui)2 
 and 
 zi=4uxxi+6uiuxi+(ui)3 , respectively , 
 i=1,2,3 , and 
 u 
 is a column of three solutions of the burgers equation with 
 |e,u,y|0 .
proofthe connection between solutions of the heat equation and the burgers equation via the hopf  cole transformation 2vxi / vi = ui gives the expressions vxxivi=12uxi+14(ui)2,vxxxivi=34uiuxi+18(ui)3
note that the determinant |e,u,y| is nonvanishing as the wronskian  w(v ) is the same . 
 the connection between solutions of the heat equation and the burgers equation via the hopf  cole transformation 2vxi / vi = ui gives the expressions vxxivi=12uxi+14(ui)2,vxxxivi=34uiuxi+18(ui)3
note that the determinant |e,u,y| is nonvanishing as the wronskian  w(v ) is the same . 
 corollary  2the representations  ( 10 )  and  ( 12 ) , where 
 n=3 , explicitly define the one - to - one correspondence between reduction operators of the form  ( 8)  and families of solutions of the burgers equation that are invariant with respect to these operators . the representations  ( 10 )  and  ( 12 ) , where 
 n=3 , explicitly define the one - to - one correspondence between reduction operators of the form  ( 8)  and families of solutions of the burgers equation that are invariant with respect to these operators .
it looks very difficult to explicitly construct an ansatz for  u by the direct integration of the invariant surface condition that corresponds to an operator of the form  ( 8) in the case of an arbitrary solution of system  ( 9 ) . even for a simple solution of system  ( 9 ) , carrying out the corresponding reduction of the burgers equation  ( 1 )
see , e.g. ,  , where a few such reductions were implemented . a better way for using an operator of the form  ( 8) for reducing the burgers equation is to consider , instead of  q , the generalized vector field q=(l[u]+q[u])u which is equivalent to the evolutionary representative q[u]u of  q and is a generalized conditional symmetry of  ( 1 ) , cf . 
( 1 ) with respect to the generalized vector field q is somehow presented in the proof of theorem  1 , and corollary  2 exhaustively describes the family of q - invariant solutions of  ( 1 ) . at the same time , knowing the representation  ( 12 ) , where n=3 , for q - invariant solutions of  ( 1 ) allows us to easily construct an ansatz for  u associated with the operator  q and then to reduce the burgers equation  ( 1 ) using this ansatz . setting c1=1 and c2=0 ( resp . 
c1=0 and c2=1 ) and assuming c3 to be an arbitrary constant , we derive two integrals of the equation q[u]=0 , =v1u2vx1v3u2vx3,=v2u2vx2v3u2vx3 , which are correctly defined for u2vx3/v3 ( we can always assume this condition to be satisfied for a specific solution  u up to renumbering the functions  v1 , v2 and v3 ) .
therefore , the general solution of the equation q[u]=0 is implicitly represented in the form f(,)=0 , where f is an arbitrary nonconstant function of its arguments . up to permutation of   and  
then the equality f(,)=0 implies the implicit ansatz =( ) for the unknown function u = u(t , x ) , where  and   are the new dependent and independent variables , respectively . the standard way for deriving the corresponding reduced equation via the computation of the expressions for derivatives
ut , ux and uxx implied by the ansatz and the subsequent substitution of the expressions into  ( 1 ) is too cumbersome . instead of this , we act on the ansatz =( ) by the operator dt+dxx and then make the substitutions ut+uxx=uux and vti+vxxi=0 , i=1,2,3 . as a result
the differential function  dx does not vanish for solutions of  ( 1 ) which are implicitly represented in the form =( ) for some smooth function   of  . indeed , suppose that dx=0
( i.e. ,  =0(t ) ) for such a solution u = u0(t , x ) .
then we have u0=2(vx20vx3)/(v20v3 ) , which implies that for some nonvanishing smooth function   of  t the expression ( v2+0v3) gives a solution of the linear heat equation vt+vxx=0 . as the functions  v2 and 
v3 are linearly independent solutions of the same equation , this is possible only if 0 is a constant .
therefore , the substitution u = u0(t , x ) into   results in the constant value 0=(0 ) and hence u0=2(vx10vx3)/(v10v3 ) .
comparing the two expressions for the solution u = u0(t , x ) , we conclude that the functions  v1 , v2 and  v3 are linearly dependent and thus arrive at a contradiction .
its general solution is =c1+c2 , which completely agrees with the representation  ( 12 ) and the ansatz =( ) .
the above reduced equation is obtained for an arbitrary operator  q of the form  ( 8) and it is much simpler than particular reduced equations derived in  .
this fact is explained by that the ansatz =( ) , whose construction is based on the representation  ( 12 ) for q - invariant solutions of the burgers equation  ( 1 ) , is in better agreement with the structure of this equation than particular ansatzes given in  .
the aim of this paper is to arrange , enhance and complete the description of the nonclassical reductions of the burgers equation , including lie reductions .
although this problem had been considered in a number of papers , certain of its aspects needed additional investigation .
the set of reduction operators of the burgers equation is naturally partitioned into two subsets , which consist of singular and regular reduction operators , respectively .
basic properties of singular reduction operators , whose coefficients of  t vanish , are quite common for ( 1 + 1)-dimensional evolution equations and properly formulated in no - go terms , and the main property is the existence of a bijection between equivalence classes of singular reduction operators and one - parameter families of solutions which are different up to re - parameterization . at the same time , the subset of regular reduction operators , whose coefficients of  t do not vanish , is of specific structure . the representatives of equivalence classes of such operators , whose coefficients of  t equal one , are partitioned into three sets , namely , the unique operator  t+ux with  u=1 , the family of operators with u=0 , each of which is equivalent to a lie symmetry operator , and the family of operators with  u=12 , which have the form  ( 8) with the coefficients  0 , 1 and 0 satisfying the system  ( 9 ) .
we present a new optimized proof of the no - go theorem on the system  ( 9 ) of determining equations for reduction operators with =1 and u=12 , which is directly based on properties of such operators . as a consequence of the theorem
, it is detected that the coefficients of reduction operators in this case admit the representation in terms of solutions of the uncoupled system formed by three copies of the burgers equation .
any lie reduction of the burgers equation proves to be equivalent via the hopf  cole transformation to a parameterized family of lie reductions of the linear heat equation .
we also carry out all possible nonclassical reductions of the burgers equation to single ordinary differential equations .
regular reduction operators with the coefficient   gauged to one give purely nonclassical reductions of the burgers equation only if u{1,12}. it is obvious that all corresponding ansatzes are necessarily implicit and hence they can not be constructed using the direct method by clarkson and kruskal .
see related discussions on the connection between nonclassical symmetries and the direct method , e.g. ,  in  . finding reduction operators for the burgers equation , one faces two kinds of no - go cases .
the first kind is given by singular reduction operators and is in fact related to lowering the equation order to one in the course of reduction .
this is why similar no - go results are true for any ( 1 + 1)-dimensional partial differential equation possessing a family of reduction operators of singularity co - order one which is parameterized by an arbitrary smooth function of all independent and dependent variables  . for each such family ,
the system of determining equations consists of a single partial differential equation on the function parameterizing the family , and this equation is equivalent , in a certain sense , to the original equation , where the variable tuple is implicitly augmented with an additional parametric variable .
the second kind given by regular reduction operators with =1 and u=12 is more specific and definitely related to the fact that the burgers equation is linearized by the hopf  cole transformation to the linear heat equation .
the corresponding system of determining equations is a system of three ( 1 + 1)-dimensional evolution equations , which is reduced by differential substitutions to the uncoupled system of three copies of the linear heat equation as well as to the uncoupled system of three copies of the burgers equation .
similar no - go results are known only for linear ( 1 + 1)-dimensional evolution equations of the second order  .
it looks possible to extend these results to the entire class of generalized burgers equations which are linearized by the hopf  cole transformation , and the optimized proof of theorem  1 creates a significant prerequisite for this . the question whether there exist no - go cases of other kinds related to regular reduction operators of single evolution equations is still open .
 the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed . 
 a  new proof of the theorem on the special  no - go  case of regular reduction operators is presented , and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case . 
 all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described . 
 any lie reduction of the burgers equation proves to be equivalent via the hopf  cole transformation to a parameterized family of lie reductions of the linear heat equation . 
Introduction
Singular reduction operators
Regular reduction operators
Trivial case
Case related to Lie symmetries
No-go case
Conclusion