Comment on: "New exact solutions of the (3+1)-dimensional Burgers System" [Phys. Lett. A 373 (2009) 181]

Commen t on: ”New exact solutions of the (3+1)-dimensiona l Burgers System” [Ph ys. Lett. A 373 (2009) 181] Nikolai A. Kudryasho v ∗ and Mikhail B. So ukharev † Department of Applied Mathemat ics, Moscow Engi neering a nd Ph ysics Institute, Kashirskoe sh. 31, Mosco w 11540 9, Russia Abstract W e demonstrate that all exa ct s olutions of the Ricc ati equation b y Dai and W ang [C.-Q. Dai, Y.-Y. W ang, Phys. Lett. A 373 (20 09) 181 – 187] are not new and cannot b e new b ecause the general solution of this equation wa s obtained m ore than one cen tury ago. Moreo v er we sho w that some ”new solutio ns” by D ai and W ang of the Ricc ati equation do n ot satisfy this equation. W e also illustr ate that the auth ors d id not obtain any new solutions f or solution of th e (3+1)-dimensional Burgers system. MSC2000 num bers: 34A05, 34A25, 34M05, 35C05 Key w ords: Nonlinear evolution equation; Riccati equation; Exact so- lution; Exp-function metho d; Burgers system 1 On ”new” so lutions of the Riccati equation b y Dai and W ang Dai and W ang [1 ] lo ok ed for exact solutions of the Riccati equation dϕ ( ξ ) dξ = l 0 + ϕ 2 ( ξ ) , (1) ∗ E-mail: Kudryashov@mephi.ru † E-mail: Soukhar ev@mephi.ru 1 where l 0 is constan t. In their letter these authors say : ”w e firstly use the the Exp—function metho d [2] to seek new exact solutions of the Riccati equation (1)”. Firstly Dai and W ang are wrong here b ecause they are not first who applied the Exp-function metho d to find ”new solutions” of the Ricatti equation. Z ha ng w as first in [3] who used the Exp - function method to find ”new generalized solitonary solutions of Riccati equation” . Criticism of t he pap er by Zhang [3] w as giv en in our recen t w ork [4]. Secondly Dai and W ang claim: ”w e obtain some new exact solutions of the Riccati equation”. The Riccati equation (1) has b een studied during sev - eral cen turies, t herefore statemen t b y D a i and W ang may cause a sensation. Unfortunately there is no sensation b ecause t his statemen t is wrong as well. Let us sho w that some ”solutions” by Dai and W ang are not new and some of them do no t satisfy the Riccati equation. ”In tro ducing a complex v ariable” η = k ξ + ξ 0 Dai and W ang r ewrite equation (1) in the for m k ϕ ′ − l 0 − ϕ 2 = 0 , (2) where k is a constan t, ξ 0 is an arbitrary constan t. It is we ll kno wn (see an y textb o ok on differen tial equations) that this equation has the general solution in the fo r m ϕ ( η ) = − p − l 0 tanh  √ − l 0 k η  , η = k ξ + ξ 0 . (3) This solution dep ends on one arbitrary constant ξ 0 . In the limit k → ∓ 0 this form ula degenerates to the constant solution ϕ ( η ) = ± p − l 0 . (4) There is no any other solution of equation (2) b esides (3) and ( 4). Using the Exp-function metho d Dai and W ang fo und five exact solutions of t he Riccati equation (2). These solutio ns a re as follows : ϕ 1 = − √ − l 0 b 1 e η + a − 1 e − η b 1 e η + a − 1 √ − l 0 e − η , η = p − l 0 ξ + ξ 0 , (5) ϕ 2 = − i √ l 0 b 1 e η + a − 1 e − η b 1 e η − i a − 1 √ l 0 e − η , η = i p l 0 ξ + ξ 0 , (6) ϕ 3 = − √ − l 0 b 1 e η + a 0 − √ − l 0 b − 1 e − η a 2 0 + l 0 b 2 0 4 l 0 b − 1 e η + b 0 + b − 1 e − η , η = 2 p − l 0 ξ + ξ 0 , (7) 2 ϕ 4 = √ − l 0 b 2 e 2 η + a 1 e η + a 0 − √ − l 0 l 0 b − 1 e − η b 2 e 2 η + b 1 e η + a 2 1 + l 0 b 2 1 +2 √ − l 0 a 0 b 2 2 l 0 b 2 + ( a 1 + √ − l 0 b 1 )( 4 a 0 b 2 − √ − l 0 a 2 1 − l 0 √ − l 0 b 2 1 ) 8 l 2 0 b 2 2 e − η , η = 2 p − l 0 ξ + ξ 0 , (8) ϕ 5 = √ − l 0 b 2 e 2 η − √ − l 0 b 1 e η − a − 1 b 2 b 1 + a − 1 e − η b 2 exp(2 η ) + b 1 e η + √ − l 0 a − 1 b 2 l 0 b 1 + √ − l 0 l 0 e − η , η = − 2 p − l 0 ξ + ξ 0 . (9) Solutions (5)–( 9) corresp ond t o form ulae (17), (20), (23), (27) and (28) in the w o rk [1]. Note that the Riccati equation (2) is the first-or der differen t ia l equation and all solutions of this equation can ha v e only one arbitr ary c ons tant [5 –8]. Therefore form ulae (5)–(9) can con tain only one arbitrary constant. But Dai and W ang believ e that there are more a rbitrary constan t s in eac h expre ssion (5)–(9). Namely , they claim that there are: tw o arbitrary constants a − 1 and b 1 in ϕ 1 and ϕ 2 , three arbitra r y constants a 0 , b 0 , and b − 1 in ϕ 3 , four arbitra r y constan ts a 0 , a 1 , b 1 , and b 2 in ϕ 4 , three arbitrary constan ts a − 1 , b 1 , and b 2 in ϕ 5 . Let us demonstrate that t his is no t the case b ecause it is not p ossible nev er. Rewriting expression (5) w e hav e ϕ 1 = − √ − l 0 b 1 e η + a − 1 e − η b 1 e η + a − 1 √ − l 0 e − η = − p l 0 1 − a − 1 √ − l 0 b 1 e 2 η 1 + a − 1 √ − l 0 b 1 e 2 η = = − p − l 0 tanh  η − 1 2 log a − 1 √ − l 0 b 1  . (10) Due to η = √ − l 0 ξ + ξ 0 w e o btain that there is only one arbit r a ry constant ξ 0 − 1 2 log a − 1 √ − l 0 b 1 in the argumen t of tanh. Therefore solution ϕ 1 b y Da i and W ang coincides with know n solution (3). W e can see that solution ϕ 2 b y D ai and W ang is equal to ϕ 1 b ecause √ − l 0 = i √ l 0 . Therefore ϕ 2 coincides with solution (3 ) . Substituting ϕ 3 in Eq.(2) we do not get zero. Therefore expression ϕ 3 do es not satisfy Eq.(2) and this expres sion is not a solution of the Riccati equation (2 ) if a 0 , b 0 , b − 1 are arbitrar y constan t s. F unction ϕ 3 can b e solution of t he Riccati equation (2) only if w e assume additional constrain ts a 2 0 + b 2 0 l 0 = 0 , b − 1  b 0 l 0 − a 0 p − l 0  = 0 (11) 3 Ho w ev er in this case expres sion ϕ 3 is the trivial solution (4). Substituting ϕ 4 in equation (2) w e do not obtain zero as we ll. So function ϕ 4 is not a solution of the Riccati equation (2) if a 0 , a 1 , b 1 , a nd b 2 are arbit r a ry constan ts. Express ion ϕ 4 is a solution of the Riccati equation (2) only if we tak e additional constrain ts in to accoun t a 1 = − p − l 0 b 1 , b 1 b 2 = 0 , a 0 b 2 = 0 , (12) or a 1 = p − l 0 b 1 , a 0 b 2 = 0 , a 0 b 1 = 0 . (13) In these cases expression ϕ 4 is the trivial solution (4) again. Expression (9) can b e presen t ed as the follo wing ϕ 5 = √ − l 0 b 2 e 2 η − √ − l 0 b 1 e η − a − 1 b 2 b 1 + a − 1 e − η b 2 e 2 η + b 1 e η + √ − l 0 a − 1 b 2 l 0 b 1 + √ − l 0 l 0 a − 1 e − η = = p − l 0 ( b 2 e η − b 1 )  e η − a − 1 b 1 √ − l 0 e − η  ( b 2 e η + b 1 )  e η − a − 1 b 1 √ − l 0 e − η  = = p − l 0 ( b 2 e η − b 1 ) ( b 2 e η + b 1 ) = p − l 0 tanh  η 2 + 1 2 log b 2 b 1  . (14) Due to η = − 2 √ − l 0 ξ + ξ 0 w e obtain tha t there is only one a r bit r a ry constan t ξ 0 2 + 1 2 log b 2 b 1 in arg ument of tanh. Therefore solution ϕ 5 b y Da i and W ang coincides with kno wn solution (3 ) . Th us w e ha v e pro v ed that Dai and W ang did not find new solutions of the Riccati equation (2 ). Moreov er, functions ϕ 3 and ϕ 4 in general case are not solutions of the Riccati equation (2). What is more w e state that nob o dy can not o bt a in new exact solutions of the Riccati equation (2). The statemen t b y Dai and W ang in [1] on t he Riccati equation is wrong. 2 On ”new” solutio ns of the (3+1)-di mensional Burgers system b y Dai and W ang Dai and W ang ha v e considered the (3 +1)-dimensional Burgers system in [1] as w ell u t − 2 uu y − 2 v u x − 2 w u z − u xx − u y y − u z z = 0 , u x − v y = 0 , u z − w y = 0 . (15) 4 Authors [1] claim: ”based on the R iccati equation and its new exact solutions, w e find new and more general exact solutions with t w o arbitrary functions of the (3+1)-dimensional Burgers system”. In this section we show that this statemen t is wrong. All solutions of the (3+1)-dimensional Burgers system b y Dai and W ang are not new. Authors [1] hav e o bt a ined t w o f o rmal solutions of the system of equations (15) u = − hϕ ( ξ ) , v = p t − p xx − p z z 2 p x − p x ϕ ( ξ ) , w = − p z ϕ ( ξ ) (16) and u = − 1 2 hϕ ( ξ ) + 1 2 h p l 0 + ϕ 2 ( ξ ) , v = p t − p xx − p z z 2 p x − 1 2 p x ϕ ( ξ ) + 1 2 p x p l 0 + ϕ 2 ( ξ ) , w = − 1 2 p z ϕ ( ξ ) + 1 2 p z p l 0 + ϕ 2 ( ξ ) . (17) Here ϕ ( ξ ) is solution of the Riccati equation (1). In formulae (1 6)–(17) Dai and W ang tak e ξ = p ( x, z , t ) + hy , where p ( x, z , t ) is an arbitrary function, h is an arbitrar y constan t. Substituting ex pressions (5)–(9) to the fo rm ulae (16)–(17) Dai and W ang obtained ten ”new solutions” of the Burgers system (15). W e pro v ed in previous section that expressions (5) –(9) b y Dai and W a ng are equiv alent to t w o solutions (3) and (4) o f the Riccati equation. Therefore Dai and W ang can get only four solutions of the Burgers sys tem (15). Let us show that only t w o distinct solutions of (15) can b e obtained in such a w ay . T aking solution ( 3) o f the Riccati equation (1) in the form ϕ ( ξ ) = − p − l 0 tanh  p − l 0 ( p + hy ) + ξ 0  , ξ = p + hy , (18) and substituting (18) in to the f ormal solution (1 6 ) we hav e u = h p − l 0 tanh  p − l 0 ( p + hy ) + ξ 0  , v = p t − p xx − p z z 2 p x + p x p − l 0 tanh  p − l 0 ( p + hy ) + ξ 0  , w = p z p − l 0 tanh  p − l 0 ( p + hy ) + ξ 0  . (19) Without loss of generalit y we can in tro duce new arbitrar y function q ( x, z , t ) = √ − l 0 p ( x, z , t ) + ξ 0 and new arbitrary constan t r = √ − l 0 h . Then ex pressions 5 (19) tak e the f orm u = r tanh( q + r y ) , v = q t − q xx − q z z 2 q x + q x tanh( q + r y ) , w = q z tanh( q + r y ) , (20) No w let us substitute the function (1 8 ) in the formal solution (17). Aft er some transformations w e hav e u = 1 2 h p − l 0 tanh  √ − l 0 2 ( p + hy ) + ξ 0 2 − iπ 4  , v = p t − p xx − p z z 2 p x + 1 2 p x p − l 0 tanh  √ − l 0 2 ( p + hy ) + ξ 0 2 − iπ 4  , w = 1 2 p z p − l 0 tanh  √ − l 0 2 ( p + hy ) + ξ 0 2 − iπ 4  . (21) In tro ducing new arbitrary f unction q ( x, z , t ) = √ − l 0 2 p ( x, z , t ) + ξ 0 2 − iπ 4 and new arbitr a ry constan t r = √ − l 0 h 2 w e can rew rite ex pressions (21) in the fo r m (20). Therefore solutions (19) and ( 2 1) are the same. Ho w ev er solution (20) of the Burgers system (15) is not new. This solution w as found b y Li et al in w ork [9]. T aking constant solution ϕ ( ξ ) = ± √ − l 0 of the Riccati equation (1 ) and substituting it in (1 6) w e hav e u = ± h p − l 0 , v = p t − p xx − p z z 2 p x ± p x p − l 0 , w = ± p z p − l 0 . (22) Substituting the same function in (17) w e obtain u = ± h 2 p − l 0 , v = p t − p xx − p z z 2 p x ± p x 2 p − l 0 , w = ± p z 2 p − l 0 . (23) W e can write solutions (22)–(23) in more g eneral form u = r , v = q , w = s, (24) where q = q ( x, z , t ) and s = s ( x, z , t ) are arbitrary functions, r is an arbitrary constan t. Therefore solutions (22 )–(23) a re the same. Solution (24 ) of the system (15) is obv ious and can b e obtained without an y calculations. Th us w e see that Dai and W ang get only tw o distinct solutions (20) and (24) of the Burgers s ystem (15). Solution (20) is not new, solution (24) is the 6 trivial solution. So the statemen t by Dai and W ang cited in the beginning of this section is wrong. Authors [1] did not obtain new results for the Riccati equation (2) and for the Burgers system (15). Some statemen ts and some results b y Dai a nd W ang are wrong. References [1] C .-Q. Dai, Y.-Y. W ang ,, Phy s. L ett. A 373 (2009) 1 81. [2] J.H . He, X.H. Wu , Chaos, Solitons and F ractals, 30 (2 006) 700 [3] S. Zhang , Phy sics Letters A 372 (2 008) 1873 700 - 708 [4] N.A. Kudryasho v, N.B. 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