A note on "New abundant solutions for the Burgers equation"

A note on ”New ab undan t solutions for the Burgers equation” Nikolai A. Kudryasho v ∗ and Dmitry I. Sinelshc hiko v Department of Applied Mathemati cs, Natio nal Resear ch Nuclear Univ ersity ”MEPhI”, Kashirskoe sh. 31, Mosco w 1154 0 9, Russia Abstract Salas, Gomez and Herana ´ ndez [A.Y. Salas S., C.A. Gomez S., J.E.C Herna ´ ndez, New abund an t solutions for tha Burgers equation, Computers and Mathematics with Applications 58 (2009) 514 -520] present ed 70 ”new exact solutions” of a ”generali zed version” of th e Burgers equation. In th is commen t we sho w that all 70 solutions by these authors are not new an d cannot b e new. P A CS: 02.30.Jr - Ordinary differen tial equations Key w ords: Nonlinear ev olution equations; Exact solution; Burgers equation; Riccati equation. 1 In tr o d u ction Recen tly Sala s, Gomez and Herana ´ ndez in [1 ] considered the Burgers equa- tion in the form u t + α u u x + β u xx = 0 . (1) The authors [1] b elieve that they studied a ” generalized v ersion” o f the Burgers equation but they are wrong here. Eq.(1) can b e transformed to the usual form of the Burgers equation [2–4] u t + u u x = β u xx (2) ∗ E-mail: Kudryashov@mephi.ru 1 if we use the followin g transformations u = 1 α u ′ , x = − x ′ , t = − t ′ (3) (primes in (2) are o mitt ed). Eq. (2) w as firstly in tro duced in [5]. But this equation b ecame p opular after w ork [2] for describing turbulence pro cesses. It is we ll kno wn that the Burgers equation can b e linearized b y the Cole—Hopf transformation [6, 7] u = − 2 β ∂ ln z ∂ x (4) As a result of application of transformation (4), w e hav e u t + u u x − β u xx = − 2 β ∂ ∂ x  z t − β z xx z  (5) Th us, solution of the Burgers equation (2) can b e expressed via solutions of the linear heat equation z t − β z xx = 0 (6) Solving the Cauc h y problem for Eq. (6) w e can obtain the solution of the Cauc hy pro blem for the Burgers equation (2) [8 , 9]. 2 Analysis of 70 exact solutio ns of the Riccati equation b y Salas, Gomez and He r ana ´ ndez T aking a ”new mo dified Exp-function metho d” in to accoun t Salas, Gomez and Herana ´ ndez in [1] ha v e used the tra v eling w av e ξ = x + λ t fo r t he Burgers equation (1) and obtained 70 solutions o f the no nlinear ordina r y differen tial equation λ u ′ ( ξ ) + α u ( ξ ) u ′ ( ξ ) + β u ′′ ( ξ ) = 0 . (7) A t this stage the authors [1] essen tially reduced a class of p ossible solu- tions for Eq.(1 ) b ecause the authors studied the nonlinear ordinary differen- tial equation (7) but not the partial differen tia l equation (1). The authors [1] did not note t ha t Eq.(7) can b e in tegrated. Integrating Eq. (7) with r esp ect t o ξ we obtain the famous Riccati equation − C + λ u ( ξ ) + α 2 u ( ξ ) 2 + β u ′ ( ξ ) = 0 , (8) where C is a constant of in tegration. 2 Equation (8) was introduced b y Ita lia n mathematician Jacop o F rancesco Riccati in 1724 . After that Eq. (8) w as studied man y times (see [10–15]). The general solution of the Riccati equation is w ell kno wn a nd is describ ed b y the formulae ( see for example [9, 13] ) u ( ξ ) = − λ α + 2 β K α tanh { K ( ξ + C 1 ) } , K = √ 2 C α + λ 2 2 β , λ 2 + C 2 6 = 0 , (9) u ( ξ ) = 2 β α ξ + 2 β C 1 , C = λ = 0 , (10) where C 1 is an arbitrary constant. These solutions were found more than one cen tury ago and nob o dy can find new solutions of Eq. (8). An alternative f orm o f expression (9) is u ( ξ ) = 1 α  2 β K − λ − 4 β K 1 + C 2 e 2 K ξ  , (11) where C 2 = e 2 K C 1 . Solution (11) follow s from the set o f iden tities u ( ξ ) = − λ α + 2 β K α tanh { K ( ξ + C 1 ) } = = − λ α + 2 β K α  e K ( ξ + C 1 ) − e − K ( ξ + C 1 ) e K ( ξ + C 1 ) + e − K ( ξ + C 1 )  = = − λ α + 2 β K α  1 − 2 e − K ( ξ + C 1 ) e K ( ξ + C 1 ) + e − K ( ξ + C 1 )  = = 1 α  2 β K − λ − 4 β K 1 + C 2 e 2 K ξ  . (12) F ollowin g to the rep ort b y o ne of the r eferees let us sho w t ha t all solutions b y Salas, Gomez and Herana ´ ndez in [1] can b e reduced to the form ulae (9) or ( 1 1). In [1], the solutions u 2 m ( m = 1 , . . . , 35) are obtained from the solutions u 2 m − 1 b y replacing µ by i µ . (There is t yp ographical error in u 8 : ′ x + ′ should 3 b e ′ x − ′ .) Consequen tly , it is only necessary to show that solutions u 2 m − 1 ( m = 1 , . . . , 35) a re just sp ecial cases of (9) or (11). W e hav e u 1 is (11) with K = − µ/ 2 , λ = − β µ , C 2 = b 2 ; u 3 is (11) with K = µ / 2, λ = β µ , C 2 = b 2 ; u 5 is (11) with K = − µ/ 2 , λ = − ( β µ + p α ), C 2 = b 2 ; u 7 is (11) with K = − µ/ 2 , λ = − ( β µ + a 2 b 2 α ), C 2 = b 2 ; u 9 is (11) with K = µ / 2, λ = β µ − a 2 b 2 α , C 2 = b 2 ; u 11 is (11) with K = − µ/ 2, λ = − ( β µ + p α + a 2 b 2 α ), C 2 = b 2 ; u 13 is (11) with K = µ/ 2, λ = β µ − p α − a 2 b 2 α , C 2 = b 2 ; u 15 is (9) with K = µ , λ = − p α , K C 1 = i π / 2 ; u 17 is (9) with K = µ , λ = − p α , C 1 = 0; u 19 is (9) with K = µ/ 2, λ = − p α , K C 1 = i π / 2; u 21 is (9) with K = µ/ 2, λ = − p α , C 1 = 0; u 23 is (9) with K = µ , λ = − a 1 b 1 α , K C 1 = i π / 2 ; u 25 is (9) with K = µ , λ = − a 2 b 2 α , C 1 = 0; u 27 is (9) with K = µ , λ = −  p + a 1 b 1  α , K C 1 = i π / 2; u 29 is (9) with K = µ , λ = −  p + a 2 b 2  α , C 1 = 0; u 31 is (9) with K = µ/ 2, λ = − β µ a 1 a 2 , C 1 = 0; u 33 is (9) with K = a 2 µ/ 2, λ = − β µ a 1 a 2 , K C 1 = i π / 2 ; u 35 is (9) with K = µ/ 2, 2 K C 1 = θ 0 + i π , where tanh θ 0 = p α + λ β µ ; u 37 is (9) with K = µ/ 2, 2 K C 1 = θ 0 , where tanh θ 0 = p α + λ β µ ; u 39 is (9) with K = µ / 2, λ = − p α , 2 K C 1 = θ 0 + iπ , where tanh θ 0 = a 0 α β µ ; u 41 is (9) with K = µ/ 2, λ = − p α , 2 K C 1 = θ 0 , where tanh θ 0 = a 0 α β µ ; u 43 is (11) with K = µ/ 2, λ = − ( − β µ + p α ), C 2 = b 2 ; u 45 is (9) with K = µ/ 2, λ = − p α + iβ µ b 2 , K C 1 = θ 0 + iπ / 4, where tanh θ 0 = 1 i b 2 ; u 47 is ( 9) with K = µ/ 2 , λ = − p α − iβ µ b 2 , K C 1 = − θ 0 − iπ / 4, where tanh θ 0 = 1 i b 2 ; u 49 is (9) with K = µ/ 2, 2 K C 1 = θ 0 + iπ , where tanh θ 0 = λ β µ ; u 51 is (9) with K = µ/ 2, 2 K C 1 = θ 0 , where tanh θ 0 = λ β µ ; u 53 is (9) with K = µ/ 2, 2 K C 1 = θ 0 , where tanh θ 0 = a 0 α + λ β µ ; u 55 is (9) with K = µ/ 2 , λ = i β µ b 2 , K C 1 = θ 0 + i π / 4, where tanh θ 0 = 1 i b 2 ; u 57 is (9) with K = µ/ 2, λ = − i β µ b 2 , K C 1 = − θ 0 − i π / 4, where tanh θ 0 = 1 i b 2 ; u 59 is (9) with K = µ/ 2, λ = − p α , K C 1 = − i π / 4; u 61 is (9) with K = µ/ 2, λ = − p α , K C 1 = i π / 4; u 63 is (9) with K = µ/ 2, λ = − p α − β µ a 2 a 1 , K C 1 = − i π / 4; 4 u 65 is (9) with K = µ/ 2, λ = − p α − β µ a 2 a 1 , K C 1 = i π / 4; u 67 is (9) with K = µ/ 2, λ = − β µ a 2 a 1 , K C 1 = − i π / 4; u 69 is (9) with K = µ/ 2, λ = − β µ a 2 a 1 , K C 1 = i π / 4. In considering u 59 , u 61 , u 63 , u 65 , u 67 and u 69 it was used the iden tities tanh z − i sec h z = coth  z − iπ 2  − cosech  z − iπ 2  = tanh  z 2 − iπ 4  (13) Th us, the analysis of ’man y new solutions’ of the Riccati equation (7) sho ws that all 70 exact solutions b y Salas, Gomez and Herana ´ ndez [1] can b e found from the general solution (9) of Eq.(7). A t first glance w e ha v e the only negative momen t of w ork [1]. Ho w ev er the authors obtained 70 differen t forms of the solution of the Riccati equation. T aking the Riccati equation as the simplest equation in the metho d discuss ed in [16, 17] we can imagine ho w many metho ds can b e suggested to searc h for exact solutions of nonlinear differen tia l equations. Ev ery form of the solution for t he Riccati equation can b e used in finding exact solutions of nonlinear ordinary differential equations. Ho w ev er w e ho p e the researc hes will not use this dubious idea. Salas, Gomez and Herana ´ ndez wrote in [1] ”w e conclude that the v ari- an t of the Exp - metho d here used is a v ery p ow erful mathematical to ol for solving other nonlinear equations”. How ev er the analysis o f the solutions f o r the Riccati equation by the pap er [1] p oints clearly to the obv ious deficiency of the Exp - function metho d in finding exact solutions of nonlinear ordi- nary differen tia l equations: this metho d allows us to find man y redundan t solutions. W e affirm that Salas, Gomez and Herana ´ ndez in [1] made the errors that w ere discussed in w orks [18–26]. W e are gr a teful to one of referee for the useful remarks a nd his car eful consideration. References [1] Alv ar o H. Salas S., Cesar A. 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