The Crocco transformation: order reduction and construction of Backlund transformations and new integrable equations

The Cr occo transformation: order r eduction and construction of B ¨ acklund transf ormations and new integrable equations A D P olyanin 1 and A I Zhurov 1 , 2 1 Institut e for Problems in Mech anics, Russian Academy of Sciences, 101 V ernadsk y A v enue, bldg 1, 119526 Mosco w , Russia. 2 School of Dentistry , Cardi ff Univ ersity , Heath Park, Cardi ff CF14 4XY , UK. E-mail: polyanin@ ipmnet.ru Abstract. W ide classes of nonl inear mathematic al physics equati ons are de scribed that admit order redu ction through the use of the Crocco transformat ion, with a first-order part ial deri v ati v e tak en as a ne w inde pendent vari able and a second -order partial deri va ti ve taken as the ne w depende nt varia ble. Associate d B ¨ acklund transformations are constructed for e volut ion equations of general form (special cases of which are Bur gers, Kort e weg–de V ries, and many other nonlinear equations of mathematical physics). The results obtaine d are used for order reducti on and construct ing exac t s olution s of hydrodynamics equations (Navier – Stoke s, Euler , and boundary layer). A numbe r of ne w int egra ble nonlinear equa tions, inclusi v e of the general ized Cal ogero equa tion, are considered. AMS classificat ion scheme numbers: 35Q58, 35K55, 35K40, 35Q53, 35Q30 The Cr occo tr ansformation : o r der r eduction and new inte gr able equations 2 1. Preliminary r emarks The Crocco transfo rmation is used in h ydrod ynamics fo r redu cing th e or der of the plan e bound ary-lay er equations [1– 3]. It is a transformation in which a first-order partial deriv ativ e taken as a new indepen dent v ariable and a second- order partial deriv ativ e taken as the ne w depend ent variable (this applies to the equ ation for the stream f unction) . So far, u sing the Crocco transfor mation h as been limited solely to the theory of boun dary layer . The present paper reveals that th e domain of application of the Crocco transformation is much bro ader . I t can be successfully used fo r reducing the ord er of wide classes of non linear equations with mixed der i vati ves and construc ting B ¨ acklund transfor mations for ev olution equations o f arbitrary or der and quite gener al for m, sp ecial cases of which include Burger s and K ortewe g–de Vries type eq uations as well as many other nonlinear equatio ns of mathematical physics. The B ¨ acklund tr ansforma tions o btained w ith th e Cro cco tran sformation may , in turn, be used fo r construc ting new in tegrable n onlinear equation s. Examples of the g eneralized Calogero eq uation and a num ber of other integrable n onlinear second -, third-, and fourth- order e quations ar e co nsidered. A gen eralization of th e Crocco transform ation to th e case of three indepen dent v ariables is gi ven. It is notew orthy that various B ¨ acklund tran sformation s and their ap plications to sp ecific equations of mathematical physics can be found , f or example, in [3–14]. In the present paper , the term inte grable equation applies to nonlinear par tial dif ferential equations that ad mit solution in terms of qu adratur es or solutions to linear d ifferential or linear integral equations. 2. Nonlinear equatio ns that admit order redu ction with the Crocco transformation Consider the n th-ord er nonlinear equation with a mixed d eriv ati ve ∂ 2 u ∂ t ∂ x + [ a ( t ) u + b ( t ) x ] ∂ 2 u ∂ x 2 = F  t, ∂ u ∂ x , ∂ 2 u ∂ x 2 , ∂ 3 u ∂ x 3 , . . . , ∂ n u ∂ x n  . (1) 1 ◦ . General prop erty: if e u ( t, x ) is a solutio n to equation (1), then the function u = e u ( t, x + ϕ ( t )) + 1 a ( t ) [ b ( t ) ϕ ( t ) − ϕ ′ t ( t )] , a ( t ) 6≡ 0 , (2) where ϕ ( t ) is an arbitrary functio n, is also a solution to equ ation (1). If a ( t ) ≡ 0 , then u = e u ( t, x ) + ϕ ( t ) is another solution to (1). 2 ◦ . Denote η = ∂ u ∂ x , Φ = ∂ 2 u ∂ x 2 . (3) Dividing ( 1) by u xx = Φ , differentiating with respect to x , and tak ing into ac count (3), we obtain Φ t Φ − u tx Φ x Φ 2 + a ( t ) η + b ( t ) = ∂ ∂ x F ( t, η , Φ , Φ x , . . . , Φ ( n − 2) x ) Φ . (4) Let us pass in (4) from the old variables to the Crocco variables: t, x, u = u ( t, x ) = ⇒ t, η , Φ = Φ( t, η ) , (5) where η and Φ are defined by (3). The deriv ati ves are tran sformed as follows: ∂ ∂ x = ∂ η ∂ x ∂ ∂ η = u xx ∂ ∂ η = Φ ∂ ∂ η , ∂ ∂ t = ∂ ∂ t + ∂ η ∂ t ∂ ∂ η = ∂ ∂ t + u tx ∂ ∂ η . The Cr occo tr ansformation : o r der r eduction and new in te grable eq uations 3 As a result, equation (4), and hence the origina l equatio n (1) , is redu ced to the ( n − 1) st-order equation a ( t ) η + b ( t ) Φ − ∂ ∂ t 1 Φ = ∂ ∂ η  1 Φ F  t, η , Φ , Φ ∂ Φ ∂ η , . . . , ∂ n − 2 Φ ∂ x n − 2  . (6) The higher deriv ativ es are calculated by the formulas ∂ k u ∂ x k = ∂ k − 2 Φ ∂ x k − 2 = Φ ∂ ∂ η ∂ k − 3 Φ ∂ x k − 3 , ∂ ∂ x = Φ ∂ ∂ η , k = 3 , . . . , n. Giv en a solu tion to th e or iginal equation ( 1), form ulas (3) define a solution to equation (6) in param etric f orm. Let Φ = Φ( t, η ) be a solution to eq uation (6). Then, in v iew of (3), the functio n u ( t, x ) satisfies the equation u xx = Φ( t, u x ) , (7) which c an b e treated as an or dinary differential equ ation in x with par ameter t . The general solution to equation (7) may be written in parametric form as x = Z η η 0 d s Φ( t, s ) + ϕ ( t ) , u = Z η η 0 s d s Φ( t, s ) + ψ ( t ) , (8) where ϕ ( t ) ψ ( t ) are ar bitrary f unctions a nd η 0 is an arbitrary constan t. Since the de riv ation of (6) i s b ased on dif ferentiating (1), o ne of th e arbitrar y function s in solution (8) is redundant. In order to rem ove this redu ndancy , it suffices to sub stitute (8) into (1). Ho wev er , it mo re conv enient to take advantage of the solutio n proper ty (2) and no te that solution (8) m ust also possess this pro perty . In view of th is, the g eneral solu tion to the o riginal equ ation (1) ca n b e rewritten in the parametric form x = Z η η 0 d s Φ( t, s ) + ϕ ( t ) , u = Z η η 0 s d s Φ( t, s ) + 1 a ( t ) [ b ( t ) ϕ ( t ) − ϕ ′ t ( t )] , (9) where ϕ ( t ) is an ar bitrary function. Example 1 (gene ralized Caloger o e quation) . W ith F = f ( t, u x ) u xx + g ( t, u x ) , which correspo nds to the nonlinear second-orde r equa tion u tx = [ f ( t, u x ) − a ( t ) u − b ( t ) x ] u xx + g ( t, u x ) , (10) passing to the Crocco variables (5), (3) leads to the first-order equation a ( t ) η + b ( t ) Φ − ∂ ∂ t 1 Φ = ∂ ∂ η  f ( t, η ) + g ( t, η ) Φ  , which become s linear with the substitution Φ = 1 / Ψ . In the special case o f a ( t ) = − 1 , b ( t ) = 0 , f ( t, u x ) = 0 , and g ( t, u x ) = g ( u x ) , equation (10) redu ces to the Calogero equation, which was considered in [15, 16] (see also [3, p. 4 33– 434]) . Example 2 (equ ation arising in gravitation theo ry). The nonlinea r th ird-or der eq uation u txx = k u u xxx , which is cr oss-disciplinary betwe en p rojective geometry and gr avitation theory [ 16, 17 ], can be reduced , by in tegrating with respect to x , to the form u tx = k u u xx − 1 2 k w 2 x + ψ ( t ) , (11) where ψ ( t ) is an ar bitrary functio n. E quation (11) is a sp ecial case o f equation (1 0), and h ence can be reduced to a linear first-ord er e quation. The Cr occo tr ansformation : o r der r eduction and new in te grable eq uations 4 Example 3 (Navier–Stokes and Euler equatio ns). An unstead y three -dimension al flow of a viscous incompr essible fluid may be described by the Na vier–Stokes and continuity equations ∂ V n ∂ t + V 1 ∂ V n ∂ x + V 2 ∂ V n ∂ y + V 3 ∂ V n ∂ z = − 1 ρ ∇ n P + ν  ∂ 2 V n ∂ x 2 + ∂ 2 V n ∂ y 2 + ∂ 2 V n ∂ z 2  , n = 1 , 2 , 3 , ∂ V 1 ∂ x + ∂ V 2 ∂ y + ∂ V 3 ∂ z = 0 , (12) where x , y , and z are Cartesian coordinates, t time, V 1 , V 2 , a nd V 3 the fluid velocity compon ents, P pressure, and ρ the fluid de nsity; also ∇ 1 P = ∂ P /∂ x , ∇ 2 P = ∂ P /∂ y , and ∇ 3 P = ∂ P /∂ z . Equatio ns (12) are ob tained un der th e assumption that the bulk fo rces are poten tial and includ ed into p ressure. In the degen erate case o f ν = 0 , equations (12) become the Euler equation s for an ideal (in viscid) fluid. The equation s of mo tion of a viscous incompressible fluid, (12), admit exact three- dimensiona l s olution s of t he fo rm V 1 = u, V 2 = − 1 2 y ∂ u ∂ x , V 3 = − 1 2 z ∂ u ∂ x , P ρ = 1 4 p ( t )( y 2 + z 2 ) + s ( t ) − 1 2 u 2 + ν ∂ u ∂ x − Z ∂ u ∂ t d x, where p ( t ) an d s ( t ) are arbitrar y f unctions o f time t , an d u = u ( t, x ) satisfies th e no nlinear third-or der equation ∂ 2 u ∂ t ∂ x + u ∂ 2 u ∂ x 2 − 1 2  ∂ u ∂ x  2 = ν ∂ 3 u ∂ x 3 + p ( t ) , (13) which is a special case of eq uation (1 ) with a ( t ) = 1 , b ( t ) = 0 , and F = ν u xxx + 1 2 u 2 x + p ( t ) . The Crocco transfor mation (5) brings (13) to the nonlinear second-orde r equa tion ∂ Φ ∂ t + [ 1 2 η 2 + p ( t )] ∂ Φ ∂ η = ν Φ 2 ∂ 2 Φ ∂ η 2 , (14) which can b e rewritten in the form of a n onlinear equ ation of conv ectiv e thermal con duction with a parabo lic, Poiseuille-type velocity p rofile: ∂ Ψ ∂ t + [ 1 2 η 2 + p ( t )] ∂ Ψ ∂ η = ν ∂ ∂ η  1 Φ 2 ∂ Ψ ∂ η  , Ψ = 1 Φ . It should be n oted that in the special case of in viscid fluid ( ν = 0 ) , the o riginal nonlinear equation (13) is reducib le to the linear first-order partial dif ferential equation (14), which can be solved by the method of characteristics. Example 4 (system of h ydrod ynamic-ty pe equ ations). Consider the system of equations ∂ 2 u ∂ t ∂ x + u ∂ 2 u ∂ x 2 −  ∂ u ∂ x  2 = ν ∂ 3 u ∂ x 3 + q ( t ) ∂ u ∂ x + p ( t ) , (15) ∂ v ∂ t + u ∂ v ∂ x − v ∂ u ∂ x = ν ∂ 2 v ∂ x 2 , (16) which describ es sev eral classes of exact so lutions to the Navier–Stokes equations in two a nd three dimen sions [ 3, 18–20 ]. The n onlinear equation (15) is independ ent of v and can b e treated separately . Althou gh linea r in v , equ ation (16) in volv es the function u , wh ich is governed b y equation (15). The Cr occo tr ansformation : o r der r eduction and new in te grable eq uations 5 The Crocco transfor mation (5) brings system (15)–(16) to the form ∂ Φ ∂ t + ( η 2 + qη + p ) ∂ Φ ∂ η = ( η + q )Φ + ν Φ 2 ∂ 2 Φ ∂ η 2 , (17) ∂ v ∂ t + ( η 2 + q η + p ) ∂ v ∂ η = η v + ν Φ 2 ∂ 2 v ∂ η 2 . (18) Here and he nceforth , the argumen ts o f p ( t ) a nd q ( t ) are omitted for brevity . Equ ation (18) was obtained using the representation of the mixed deri vati ve u tx obtained from (15). Equation (18) has exact solutions of the form v = Aη + B Φ + C, (19) where A = A ( t ) , B = B ( t ) , and C = C ( t ) are u nknown f unctions d etermined from an approp riate system of ordinary differential equations. This fact can be p roved b y substituting (19) into (18) and taking into account (17). Formula (19) allows one to arr iv e at the following imp ortant re sult with regard to solutions of the o riginal eq uation (1 6). Let u = u ( t, x ) b e a solution to equation (15). Then equation (16) admits the solution v = A ′ t + Aq + A ∂ u ∂ x + B ∂ 2 u ∂ x 2 , (20) where A = A ( t ) and B = B ( t ) satisfy the ordinar y dif ferential equations A ′′ tt + qA ′ t + ( p + q ′ t ) A = 0 , (21) B ′ t + qB = 0 . (22) The general solution to (22) is B = C 1 exp  − Z q d t  , where C 1 is an arbitrary constant. Listed below are some exact solutions to equatio n (15) r epresentable in terms of elementary function s an d suitable f or finding exact solution s to e quation (16) using formu las (20). 1 ◦ . Generalized separable solution rational in x : u = − α ′ t ( t ) + β ( t )[ x + α ( t )] − 6 ν x + α ( t ) , q = − 4 β , p = β ′ t + 3 β 2 , where α = α ( t ) a nd β = β ( t ) are arb itrary functions. 2 ◦ . Generalized separable solution exponen tial in x : u = α ( t )e − σx + β ( t ) , p = 0 , q = α ′ t α − σβ − σ 2 ν, where α = α ( t ) and β = β ( t ) are arb itrary fu nctions an d σ is an arbitrary co nstant. By choosing period ic functions a s α ( t ) an d β ( t ) , o ne obtains time-periodic solutions. 3 ◦ . Multiplicative separa ble solution periodic in x : u = α ( t ) sin ( σ x + C 1 ) , α ( t ) = C 2 exp  − ν σ 2 t + Z q ( t ) d t  , p = − σ 2 α 2 ( t ) , q = q ( t ) is an arbitra ry function , where C 1 , C 2 , an d σ are arb itrary co nstants. By setting q ( t ) = ν σ 2 + ϕ ′ t ( t ) with period ic ϕ ( t ) , one obtains a periodic solution in both x an d t . More complicate d so lutions to equation (15) can be found in [20]. The Cr occo tr ansformation : o r der r eduction and new in te grable eq uations 6 3. Some g eneralizations Consider the nonline ar n th-ord er e quation c ( t ) u tx + [ a ( t ) u + b ( t ) x ] u xx + d ( t )( u x u tx − u t u xx ) = F ( t, u x , u xx , . . . , u ( n ) x ) , (23) which become s (1) for c ( t ) = 1 and d ( t ) = 0 . 1 ◦ . General prop erty: if e u ( t, x ) is a solutio n to equation (23), then the function u = e u ( t, x + ϕ ( t )) + ψ ( t ) , where ϕ = ϕ ( t ) and ψ = ψ ( t ) are related by d ( t ) ψ ′ t − a ( t ) ψ = c ( t ) ϕ ′ t − b ( t ) ϕ (either fu nction can be chosen arbitrar ily), is als o a solution to (23). 2 ◦ . Let us d ivide (23) by u xx , differentiate the resulting equ ation with respect to x , and then pass to the Crocco variables (5 ), (3) to obtain the ( n − 1) st-or der equation a ( t ) η + b ( t ) Φ − [ d ( t ) η + c ( t )] ∂ ∂ t 1 Φ = ∂ ∂ η  1 Φ F  t, η , Φ , Φ ∂ Φ ∂ η , . . . , ∂ n − 2 Φ ∂ x n − 2  . Example. In the sp ecial case of n = 3 , a ( t ) = b ( t ) = c ( t ) = 0 , d ( t ) = 1 , an d F = [ f ( u xx )] x , (23) is a general bound ary layer equation for a no n-Newtonian fluid [3], with u be ing the stream function. By the Crocco transf ormation (5), this equ ation can be red uced to the secon d-ord er equation η Φ t = Φ 2 [ f (Φ)] ηη , whic h can be linearize d by th e substitution Ψ = 1 / Φ if f (Φ) = 1 / Φ . Remark. Equation (23) ca n be gen eralized b y addin g the argume nts J x , . . . , J ( m ) x , with J = u xx u txx − u tx u xxx , to the fun ction F . 4. Using the Crocco transformation for constructing RF-pairs and B ¨ acklund transformations Consider a fairly general n th-order e volution equation u t + [ a ( t ) u + b ( t ) x ] u x = F ( t, u x , u xx , u xxx , . . . , u ( n ) x ) . (24) General prop erty: if e u ( t, x ) is a solutio n to equation (24), then the function u = e u ( t, x + ψ ( t )) + C, where C is an ar bitrary constant an d ψ = ψ ( t ) satisfies the linear ordin ary differential equation ψ ′ t − b ( t ) ψ + C a ( t ) = 0 , is also a solution to (24). Differentiating (24) with respect to x yields an ( n + 1 ) st-order equation with a mixed deriv ativ e of th e form (1): u tx + [ a ( t ) u + b ( t ) x ] u xx = − a ( t ) u 2 x − b ( t ) u x + ∂ ∂ x F ( t, u x , u xx , u xxx , . . . , u ( n ) x ) . (25) By p assing in (25) fr om t , x , u to the Crocco variables (5), we one arrives at the n th-orde r equation 3 a ( t ) η + 2 b ( t ) Φ − ∂ ∂ t 1 Φ + [ a ( t ) η 2 + b ( t ) η ] ∂ ∂ η 1 Φ = ∂ 2 ∂ η 2 F  t, η , Φ , Φ ∂ Φ ∂ η , . . . , ∂ n − 2 Φ ∂ x n − 2  . (26) Equation s (24) and (26) are linked by the B ¨ ac klund transformation u t + [ a ( t ) u + b ( t ) x ] η = F ( t, η , Φ , Φ x , . . . , Φ ( n − 2) x ) , u x = η , u xx = Φ . (27) The Cr occo tr ansformation : o r der r eduction and new in te grable eq uations 7 Remark. Sometimes, it is con venient t o rewrite (26) in the form Ψ t − [ a ( t ) η 2 + b ( t ) η ]Ψ η − [3 a ( t ) η + 2 b ( t )]Ψ = − ∂ 2 ∂ η 2 F, Ψ = 1 Φ . Example 1. The unno rmalized Bu rgers equation u t + auu x = β u xx (28) is a special case of (24) with a ( t ) = a = const, b ( t ) = 0 , an d F = β u xx = β Φ . By the B ¨ acklund transformatio n (27), equation (28) can be reduced to Φ t − aη 2 Φ η + 3 aη Φ = β Φ 2 Φ ηη . Example 2. The nonlinea r seco nd-or der eq uation u t + [ a ( t ) u + b ( t ) x ] u x = f ( t, u x ) u xx + g ( t, u x ) (29) is a special case of (24). Th e B ¨ acklund transformation (27) reduces (29) to the equation 3 a ( t ) η + 2 b ( t ) Φ − ∂ ∂ t 1 Φ + [ a ( t ) η 2 + b ( t ) η ] ∂ ∂ η 1 Φ = ∂ 2 ∂ η 2  f ( t, η ) Φ + g ( t, η )  , which become s linear after substituting Φ = 1 / Ψ . Example 3. The unno rmalized Bu rgers K orteweg–de Vries eq uation u t + auu x = β u xxx (30) is a spec ial case o f (24) a ( t ) = con st, b ( t ) = 0 , and F = β u xxx = β ΦΦ η . The B ¨ acklun d transform ation (27) reduces (30) to the equation Φ t − aη 2 Φ η + 3 aη Φ = β Φ 2 (ΦΦ η ) ηη , which, after submitting Φ = θ 1 / 2 , becomes θ t − aη 2 θ η + 6 aη θ = β θ 3 / 2 θ ηη η . Example 4. The nonlinea r th ird-or der equation u t + auu x = f ( t, u x ) u 3 xx u xxx (31) can be redu ced, u sing the B ¨ acklund transfo rmation (27) with b ( t ) ≡ 0 and F = f ( t, u x ) u − 3 xx u xxx = f ( t, η )Φ − 2 Φ η followed by substituting Φ = 1 / Ψ , to the linear equation Ψ t − aη 2 Ψ η − 3 aη Ψ = [ f ( t, η )Ψ η ] ηη . Example 5. The linear third-o rder equation u t = αu xxx + β u xx (32) is a special case of (24) with F = αu xxx + β u xx = α P hi Φ η + β Φ , a ( t ) ≡ 0 , an d b ( t ) ≡ 0 . By applyin g to ( 32) th e B ¨ acklu nd tran sformation (27) and then substituting Φ = 1 / Ψ , on e arrives at the non linear equation Ψ t = α (Ψ − 3 Ψ η ) ηη + β (Ψ − 2 Ψ η ) η . (33) The special cases of (33) with α = 0 , β 6 = 0 an d β = 0 , α 6 = 0 were copn sidered in [2 1] and [3], respectively . Example 6. The linear fou rth-ord er equation u t = αu xxxx is reduced , using the same transform ation as in the p receding example an d substituting Φ = θ 1 / 2 , to th e nonlinear fourth- order equation θ t = αθ 3 / 2 ( θ 1 / 2 θ ηη ) ηη . The Cr occo transformation : order r ed uction and new inte grable equations 8 Remark. Equation (26) rema ins unch anged if the sum p ( t ) u + q ( t ) x + s ( t ) , with arbitrary function s p ( t ) , q ( t ) , and s ( t ) , is added to the righ t-hand side of (24) and that of the first equation in (27). Cor ollary . If equation (24) is integrable for some rig ht-hand side F , then the equation with the more complicated right-h and sid e F + p ( t ) u + q ( t ) x + s ( t ) is also integrable. Example 1. Since the Burgers equation u t + a uu x = bu xx is integrable, the more complicated equation u t + auu x = bu xx + p ( t ) u + q ( t ) x + s ( t ) is also integrable. Example 2. Like wise, since the Kortewe g –de Vr ies e quation u t + auu x = bu xxx is integrable, the more complicated equation u t + auu x = bu xxx + p ( t ) u + q ( t ) x + s ( t ) is also integrable. 5. Extension o f the Crocco transf o rmation to the case of three independent varia bles. Ap plication to unsteady boundary-la yer equations T r ansform ation (5 ) can be extended to the cases of more independ ent v ariab les. I n particular, it can be shown that the C rocco transfo rmation t, x, y , u = u ( t, x, y ) = ⇒ t, x, η , Φ = Φ( t, x, η ) , where η = u y , Φ = u y y , ( 34) reduces the order of the n th-or der equation [ a ( t, x ) u + b ( t, x ) y ] u y y + c 1 ( t, x ) u ty + c 2 ( t, x ) u xy + d 1 ( t, x )( u y u ty − u t u y y ) + d 2 ( t, x )( u y u xy − u x u y y ) = F ( t, x, u y , u y y , . . . , u ( n ) y ) . (35) Example. Consider the Prandtl system u t + uu x + vu y = ν u y y + f ( t, x ) , u x + v y = 0 , (36) which describes a flat unsteady bou ndary layer with pr essure g radient ( u and v the fluid velocity compo nents) [1– 3]. Equation s (36) can be reduc ed, by introdu cing a strea m function w such that u = w y and v = − w x , to a single third-o rder equation [1, 3]: w ty + w y w xy − w x w y y = ν w y y y + f ( t, x ) . (37) This equatio n is a special case of (35) (up to the obvious re naming u ⇄ w ). Dividing (37) by w y y followed by dif ferentiating with respect to y and passing f rom t , x , y , w to the Crocco variables t , x , η = w y , Φ = w y y , one arrives at the seco nd-or der eq uation ∂ Φ ∂ t + η ∂ Φ ∂ x + f ( t, x ) ∂ Φ ∂ η = ν Φ 2 ∂ 2 Φ ∂ η 2 , (38) which is reduced , with the substitution Φ = 1 / Ψ , to the nonlinear heat equation ∂ Ψ ∂ t + η ∂ Ψ ∂ x + f ( t, x ) ∂ Ψ ∂ η = ν ∂ ∂ η  1 Ψ 2 ∂ Ψ ∂ η  . (39) Remark. In the steady -state case with ∂ /∂ t = 0 and f ( t, x ) = 0 , eq uation (38) red uces to one considered in [1, 3]. The Cr occo transformation : order r ed uction and new inte grable equations 9 1 ◦ . In the special case f ( t, x ) = f ( t ) , eq uation (39) adm its a n exact solutio n o f the special form Ψ = Z ( ξ , τ ) , ξ = x − η t + Z tf ( t ) d t, τ = 1 3 t 3 . Hence we arrive at the in tegrable equation ∂ Z ∂ τ = ν ∂ ∂ ξ  1 Z 2 ∂ Z ∂ ξ  , (40) which can be reduced to the linear heat equation [3, 21]. 2 ◦ . In the more g eneral case f ( t, x ) = f ( t ) x + g ( t ) , we h av e solutions of the specia l form Ψ = Z ( ξ , τ ) , ξ = ϕ ( t ) x + ψ ( t ) η + θ ( t ) , τ = Z ψ 2 ( t ) d t, where ϕ = ϕ ( t ) , ψ = ψ ( t ) , and θ = θ ( t ) are determin ed by the linear system of ordin ary differential equations ϕ ′ t + f ψ = 0 , ψ ′ t + ϕ = 0 , θ ′ t + g ψ = 0 . As a result, we arrive at an integrable equation (40). Acknowledgments The work was carried out und er partial financial suppor t of the Russian Foundation for Basic Research (grants No. 08-0 1-00 553 , No. 08-0 8-005 30 and No. 09-0 1-003 43 ). References [1] Loitsyanskiy L G 1995 Mech anics of Liqui ds and Gases (New Y ork: Begel l House) p 971. [2] Schlicht ing H and Gersten M 2000 Boundary Layer Theory , 8th Edition (Heidel berg: Springer) p 801 [3] Polyanin A D and Zaitsev V F 2004 Handbook of Nonlinear P artial Differ ential Equations (Boca Raton: Chapman & Hall/CRC Press) p 840 [4] Ibragimo v N H 1994, 1995 CRC Handbook of Lie Grou p Anlysis of Diffe ren tial Equations, V ols 1, 2 (Boca Raton: CRC Press) p 448, 576 [5] Miura R M (ed) 1975 B ¨ a ckl und T ransformat ions (Berlin: Springer -V erlag) [6] Rogers C and Shadwick W F 1982 B ¨ ack lund T ransformat ions and Their Applications (New Y ork: Academic Press) p 174 [7] Nimmo J J C and Crighton D J 1982 Proc. Royal Soc. London A 384 381–401 [8] Kingston J G, Rogers C and W oodall D 1984 J . Physics A: Math. Gen. 17 L35–8 [9] W eiss J 1986 J . Math. Phys. 27 1293–305 [10] Kudrya shov N A 1991 Phys. Let. A 155 269–75 [11] Nucci M C 1993 J. Math. Anal. Appl. 178 294–300 [12] Karasu A and Sako vich S Y u 2001 J. Phys. A: Math. Gen. 34 7355–8 [13] W ang M, W ang Y and Zhou Y 2002 Phys. Let. A 303 45–51 [14] Y onga X, Zhangb Z and Chen Y 2008 Phys. Let. A 372 6273–9 [15] Caloger o F 1984 Stud. Appl. Math. 70 189–99 [16] Pa vlov M V 2001 Theor . & Math. Phys. 128 927–32 [17] Dryuma V S 2000 Pr oc. W orkshop on Nonlinearit y , Inte grabili ty , and All That: T wenty yea rs aft er NEEDS’79 (Lecce , Italy) ed M Boiti, L Martina et al (Singapore: W orld Scien tific) 109–16 [18] Polyanin A D 2001 Doklady Physics 46 (10) 726–31 [19] Pukhnache v V V 2006 Uspekhi Mekhanik i [in Russian] 4 (1) 6–76 [20] Aristov S N and Polyani n A D 2009 Dokl ady Physics 54 (7) 316–21 [21] Bluman G W and Kumei S 1980 J . Math. Phys. 21 1019–23