A Note on "Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation"

A Note on ”Exp-function metho d for the exact solutions of fifth order KdV equation and mo dified Burgers equation” Nik ola y A. Kudry asho v ∗ , Dmitry I. Sinelshc hik o v Departmen t of Applied Mathematics, National Researc h Nuclear Univ ersit y MEPHI, 3 1 Kashirsk o e Shosse, 11 5409 Mosco w, Russian F ederation Abstract W e discuss the recent pap er by I nan and Ugurlu [Inan I.E., Ugurlu Y., Exp-function metho d for the exact sol utions of fifth order KdV equation and mod ifi ed Burgers equ ation, A ppl. Math. Comp. 217 (2010) 1294 – 1299]. W e demonstrate that al l exact solutions of fifth order KdV equation and mo dified Burgers equation by Inan and Ugurlu are trivial solutions that are reduced to constants. Moreo ver, w e show exact solutions of the fifth – order equation studied by In an and Ugurlu cannot b e found by th e Exp-function method. Keywor ds: Nonlinear evolution equatio ns, ex act s o lutions, fifth order KdV equation, mo dified Burg ers equation. P ACS 02.30.Jr - Ordinary differential equations In the recent pape r [1] Inan and Ugurlu considered the fifth – o r der KdV equation u t + u u x + u xxxxx = 0 (1) In [1]I na n a nd Ugur lu used the trav eling wav e solutions u ( x, t ) = u ( ξ ), ξ = k x − w t in Eq.(1) and obta ined nonlinear ordinar y differen tial equation in the form w u ′ + k u u ′ + k 5 u (5) = 0 , (2) where u ′ = du dξ , u (5) = d 5 u dξ 5 . After integration Eq. (2) with resp ect to ξ we can ha ve C + w u + k 2 u 2 + k 5 u (4) = 0 . (3) ∗ nakudry ashov@mep hi.r u 1 Inan and Ugurlu omitted the cons tant of integration C in [1] and es s ent ially reduced a class for exact solutions of Eq.(1). It is known that there are some exact solutions of E q.(1) express ed thro ugh r ational and elliptic functions [2–4]. How ever taking the Exp-function metho d in to co nsideration we cannot find these solutions. Using Exp-function method Ina n and Ugurlu found the following five solu- tions of Eq.(1) in [1] u 1 ( x, t ) = − 2 w k exp( k x + w t ) − 2 b 0 w k − 2 b − 1 w k exp( − k x − w t ) exp( k x + w t ) + b 0 + b − 1 exp( − k x − wt ) , (4) u 2 ( x, t ) = − 2 w k exp( k x + w t ) − 2 b 0 w k exp( k x + w t ) + b 0 , (5) u 3 ( x, t ) = − 2 k 4 exp( k x + k 5 t ) − 2 b 0 k 4 exp( k x + k 5 t ) + b 0 , (6) u 4 ( x, t ) = − 32 k 4 exp( kx + 1 6 k 5 t ) exp( k x + 1 6 k 5 t ) , (7) u 5 ( x, t ) = − 32 k 4 exp( k x + 1 6 k 5 t ) − 32 b 0 k 4 exp( k x + 1 6 k 5 t ) + b 0 . (8) How ever it is easy to note that all these ”solutions” can be written as co n- stants u 1 ( x, t ) = 2 w k , (9) u 2 ( x, t ) = − 2 w k , (10) u 3 ( x, t ) = − 2 k 4 , (11) u 4 ( x, t ) = − 32 k 4 , (12) u 5 ( x, t ) = − 32 k 4 . (13) It is wonderful but even trivia l ”solutions” (9 ), (11), (12) and (13) do not satisfy Eq.(3) at C = 0. Inan a nd Ugurlu in [1] also pr esented the exact solutions of the mo dified Burgers equation u t + u 2 u x + u xx = 0 (14) Authors [1] have used again the traveling wa ve solutio ns for E q. (14) . They also hav e omitted the constan t of in tegra tio n in r e duction of (14). Finally , Inan and Ugurlu hav e considered following equation w u + k 3 u 3 + k 2 u ′ = 0 . (15) 2 Equation (1 5) is well known and was introduced by Jakob Bernoulli in 16 95. Eq. (15) can b e reduced to the linear form using transforma tion u ( ξ ) = 1 p ( v ( ξ )) (16) In this cas e E q.(15) is reduced to the fo rm k 2 v ξ = 2 k 3 + 2 w v (17) The genera l solution of Eq.(15) can b e pre sented in the form u ( ξ ) = ± 3 q 9 C 2 e 2 wξ k 2 − 3 k w , (18) where C 2 is a arbitr ary constant . In [1] Inan a nd Ugurlu applied ag a in the Exp-function metho d and found ”three solutions” of E q. (14). The fir st ”solution” is u 1 ( x, t ) = q 3 k 2 exp( k x − k 2 2 t ) − q 3 k 2 b 0 exp( k x − k 2 2 t ) + b 0 (19) How ever, solution (19) do not s atisfy Eq. (14). Substituting function (19) into Eq. (14) we obtain the following expres sion E = − 2 √ 6 b 2 0 k 5 / 2 exp( k x − k 2 2 t )  2 exp( k x − k 2 2 t ) − b 0   exp( k x − k 2 2 t ) + b 0  4 (20) W e can see that this express ion is not equal to zero in the g eneral case. The second and the third solutions can b e wr itten as u 2 ( x, t ) = √ 3 k exp( k x − k 2 2 t ) exp( k x − k 2 2 t ) (21) u 3 ( x, t ) = q 3 k 2 exp( k x − k 2 2 t ) + q 3 k 2 b 0 + q 3 k 2 b − 1 exp( − k x + k 2 2 t ) exp( kx − k 2 2 t ) + b 0 + b − 1 exp( − k x + k 2 2 t ) (22) How ever these expr essions can b e pres ent ed as the trivial solutio ns o f Eq .(1 5). W e have u 2 ( x, t ) = √ 3 k , (23) u 3 ( x, t ) = r 3 k 2 . (24) How ever the trivial solutions (23) a nd (24) do not sa tis fy E q .(15) as w ell. W e can see that a ll exact s o lutions obtained by Inan a nd Ugurlu in [1 ] are trivial and we c a n obtain these solutions without the Exp-function method. What is more these tr ivial solution do not sa tisfy equations as well. In conclusion we stro ngly r ecommend to a uthors and referees to loo k at pap ers [5 – 23] carefully b efore finding exact solutio ns of nonlinea r differential equations. 3 References [1] Ina n I.E., Ugurlu Y. Exp-function metho d for the exa ct so lutions of fifth order KdV equa tion and mo dified B ur gers equation, Appl. Math. Co mp. 217 (201 0) 1294 – 1299 [2] K udryasho v N.A. Analitical theory of nonlinear differential eq ua tions, Mosko w - Igevsk, Institute of co mputer in vestigations, 2004, (in Russian) [3] K udryasho v N.A., Nonlinea r Differential Equatio ns with E xact Solutions Expresse d v ia the W eierstra s s F unction, Z. Naturforsch. 5 9a (200 4) 4 4 3 454 [4] Demina M.V., Kudryashov N.A. 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