Comments on: "Optical solitons in a parabolic law media with fourth order dispersion"[Appl. Math. Comput. 208(2009)209-302]

Commen ts on: “Optical solitons in a parab olic la w media with fourth order d isp ersion” [ Appl. Math. Comput. 208 (20 09) 209-302 ] Gui-Qiong Xu 1 Department of Information Management, Shangha i Universit y , Sha nghai 200 4 44, PR China Abstract. Recently , Biswas and Milovic [Appl. Math. Comput. 208 (2009 ) 209-30 2] hav e found optical o ne-soliton solutions o f a fourth or der dis per sive cubic-quintic non- linear Sc hr¨ odinger eq ua tion. In this comment, we firs t show there ar e mistakes in the pap er and demo ns trate that the obtained solutions do no t sa tis fy the considered equation. And then we r econstruct a s e ries o f analytical exact solutions by mea ns of a direct ans atz metho d and F-e x pansion metho d. These solutions include so lita ry wa ve solutions o f the b ell s hap e, solitary wa ve solutions of the kink s ha p e , and p erio dic wa ve solutions of Ja cobian elliptic function. 1. Analysis of the solutions giv en in Ref.[1] As is w ell kno wn, the in v estigation for solition solutions of n onlinear Sc hr ¨ odinger equation is alwa ys an imp ortan t and attractiv e topic. V ery recen tly , Bisw as and Milo vic[1 ] considered the higher order disp ersive cubic-quintic nonlinear Sc hr¨ odinger equation, i q t + a q xx − b q xxxx + c ( | q | 2 + d | q | 4 ) q = 0 , (1) and obtained the optical soliton solution of Eq.(1). Ho we ver, w e find th er e are mistak es in the pap er[1] and the obtained solution do es not satisfy Eq.(1). Bisw as et al.[1] first introdu ced the transformation, q ( x, t ) = P ( x, t ) e i ( − κ x + ω t + θ ) , (2) where P ( x, t ) is a real function to b e d etermined later, and κ, ω are r eal constan ts. By u sing the tran s formation (2), Eq .(1) is co n verted in to a complex differen tial equation of P ( x, t ), in 1 E-mail address: xugq@staff.shu.edu.cn (G.-Q. Xu) 1 whic h the r eal and im aginary p arts read, ∂ P ∂ t − 2 κ ( a + 2 bκ 2 ) ∂ P ∂ x + 4 b κ ∂ 3 P ∂ x 3 = 0 , (3) ( ω + a κ 2 + bκ 4 ) P − cP 3 − c d P 5 − ( a + 6 bκ 2 ) ∂ 2 P ∂ x 2 + b ∂ 4 P ∂ x 4 = 0 . (4) Then the solution of Eqs.(3)-(4) w as sup p osed as P = A ( λ + cosh τ ) p , τ = B ( x − ν t ) . (5) Substituting Eq.(5) in to Eqs.(3)-(4), the authors obtained tw o expressions with resp ect to cosh τ and sinh τ . W e ha ve not iced that t here are man y mistak es ab out the expressions (12)-(1 3) giv en in Ref.[1]. Equating the co efficien ts of 1 / ( λ + cosh τ ) p + j ( j = 0 , · · · , 4) of the obtained expressions, the v alues of A, B , ω , λ and ν we r e determined. A t last, the authors obtained the optical soliton solution of Eq.(1) as follo ws, q ( x, t ) = A λ + cosh( B ( x − ν t )) e i ( − κ x + ω t + θ ) , (6) where A, B , ω , λ and ν w ere give n by Eqs.(16)-(21) of R ef.[1 ]. Ho we ver, the “solution” (6) do es not satisfy E q .(1 ). W e ca n note this fact without su bsti- tuting (6) in to Eq.(1). The solitons are the results of a delicate balance b et wee n disp ersion and nonlinearit y , th u s it is imp ossible that the linear partial differentia l equation (3) admits the b ell t yp e solit ary wa ve (5). T o b e on the sa ve side we h a ve substituted Eq.(5) with p = 1 in to Eq.(3) and ha v e obtained the follo wing expression, E 1 = " ν + 2 aκ + 4 bκ 3 − 4 bκB 2 ( λ + cosh τ ) 2 + 24 bκB 2 cosh τ ( λ + cosh τ ) 3 − 24 bκB 2 sinh 2 τ ) ( λ + cosh τ ) 4 # A B sinh τ . W e can see that this expr ession is equal to zero only in tw o case s. One is A = 0 or B = 0, and the other is κ = ν = 0. This means that the “solution” (6) obtained b y Biswa s et al. in [1] is not correct. 2. New optical solitary w a ve solution of Eq.(1) In the follo w in g, we adopt the ansatz solution of Li et al.[2] in the form q ( x, t ) = E ( x, t ) e i ( k x − ω t + θ ) , (7) 2 where E ( x, t ) is the complex e n velo p e fu nction, a n d k , ω are real constan ts. Substituting Eq.(7) in to Eq.(1) and remo ving the exp onential term , w e can r ewrite Eq.(1) as i E t + 2 i k ( a + 2 b k 2 ) E x + ( a + 6 b k 2 ) E xx − 4 i b k E xxx − bE xxxx + ( ω − a k 2 − b k 4 ) E + c | E | 2 E + c d | E | 4 E = 0 . (8) W e no w tak e th e complex en v elop e ansatz function E ( x, t ) as E ( x, t ) = i β + λ tanh( ξ ) , ξ = p x + s t, (9) where β , λ, p, s are real constan ts. Substituting Eq.(9) int o Eq.(8) and setting the co efficien ts of tanh( ξ ) j ( j = 0 , · · · , 5) to zero, one obtains the follo w ing algebraic equations: λ ( dcλ 4 − 24 p 4 b ) = 0 , λ ( λ 3 β cd + 24 bp 3 k ) = 0 , λ ( s − 2 λ β 3 cd − λ β c + 2 k pa + 32 bp 3 k + 4 bpk 3 ) = 0 , λ ( λ 2 c + 2 λ 2 β 2 cd + 40 p 4 b + 2 p 2 a + 12 k 2 p 2 b ) = 0 , λ (2 p 2 a + k 2 a − β 4 cd − β 2 c + k 4 b + 16 p 4 b − ω + 12 k 2 p 2 b ) = 0 , β bk 4 − β ω − β 3 c − 2 apλ k − β 5 cd − sλ − 4 bp λ k 3 − 8 bp 3 k λ + β ak 2 = 0 . Solving it w e obtain one set of non trivial solution, s = 8 bp k ( k 2 + p 2 ) , β = − k λ p , ω = 2 p 2 a + 3 k 2 a + 37 k 4 b + 52 k 2 p 2 b + 16 p 4 b, c = − 2 p 2 (30 bk 2 + 20 bp 2 + a ) λ 2 , d = − 12 bp 2 λ 2 (30 bk 2 + 20 bp 2 + a ) . (10) F rom (7), (9) and (10), w e obtain the optical solita ry wa v e of Eq.(1), q 1 ( x, t ) =  − i k λ p + λ tanh( p x + 8 bpk ( k 2 + p 2 ) t )  e i ( k x − (2 p 2 a +3 k 2 a +37 k 4 b +52 k 2 p 2 b +16 p 4 b ) t + θ ) , where p, k are determined b y the last t wo iden tities of Eq.(10). F rom (9) and (10), the amplitude of the complex en v elop e fun ction E ( x, t ) reads, | E ( x, t ) | =  k 2 λ 2 p 2 + λ 2 tanh 2 ( p x + 8 bpk ( k 2 + p 2 ) t )  1 / 2 , whic h may approac h nonzero when the time v ariable ap p roac hes infinity . 3. A series of exact solutions for Eq.(1) b y using F-expansion metho d W e supp ose that the solution of (1) is of the form q ( x, t ) = P ( τ ) e i η , τ = B ( x − ν t ) , η = ( − κ x + ω t + θ ) , (11) 3 where P ( τ ) is a real f unction, and B , ν , κ, ω are r eal constan ts to b e d etermined. S u bstituting Eq.(11) to Eq.(1) and separating the r eal and imaginary parts, one m a y obtain the follo win g equations, − B ( ν + 2 aκ + 4 bκ 3 ) P ′ + 4 b κ B 3 P ′′′ = 0 , (12) ( ω + a κ 2 + bκ 4 ) P − cP 3 − c d P 5 − B 2 ( a + 6 bκ 2 ) P ′′ + bB 4 P ′′′′ = 0 . (13) The linear ordinary differenti al equatio n (12) has no solitary w a ve solutions, th u s w e h a ve to tak e κ = ν = 0. In this case Eq.(12) is satisfied identic ally , and Eq .(13) b ecomes, ω P − cP 3 − c d P 5 − a B 2 P ′′ + bB 4 P ′′′′ = 0 . (14) Eq.(14) can b e solv ed b y usin g the F-expansion met h o d[3]-[6]. According to t he F- expansion metho d, w e su pp ose, P ( τ ) = n X i =0 A i F i ( τ ) , A n 6 = 0 , (15) where A i ( i = 0 , · · · , n ) are real co n stan ts to b e determined , the in teger n is determined by balancing the linear highest order term and nonlinear term. And F ( τ ) in (15) sati sfies,  d F ( τ ) d τ  2 = h 0 + h 2 F ( τ ) 2 + h 4 F ( τ ) 4 , (16) where h 0 , h 2 , h 4 are r eal constants. By balancing the linear h ighest order deriv ativ e te rm P ′′′′ with nonlinear term P 5 in Eq.(14), w e find n = 1. Th us Eq.(15) b ecomes, P ( τ ) = A 0 + A 1 F ( τ ) . (17) Substituting Eq.(17) into Eq s .(14) along with Eq.(16), collecting all terms with the same p o w er of F j ( τ )( j = 0 , · · · , 5), and equati ng the co efficients o f these terms yields a s et of algebraic equations with resp ect to A 0 , A 1 , B , ω , a , b , c , d , h 0 , h 2 , h 4 : dcA 1 4 A 0 = 0 , ω A 0 − cA 0 3 − dcA 0 5 = 0 , 10 dcA 1 2 A 0 3 + 3 cA 1 2 A 0 = 0 , 24 bA 1 B 4 h 4 2 − d cA 1 5 = 0 , 20 bA 1 B 4 h 2 h 4 − 10 dcA 1 3 A 0 2 − cA 1 3 − 2 A 1 B 2 ah 4 = 0 , ω A 1 − 3 cA 1 A 0 2 − A 1 B 2 ah 2 + 12 bA 1 B 4 h 4 h 0 + bA 1 B 4 h 2 2 − 5 dcA 1 A 0 4 = 0 . Solving the ab o ve alge braic equations, we ha ve a set of non trivial solution, A 0 = 0 , A 1 = ± s 12 bB 2 h 4 d (10 bB 2 h 2 − a ) , ω = B 2 ( ah 2 − 12 bB 2 h 4 h 0 − bB 2 h 2 2 ) , c = d (10 bB 2 h 2 − a ) 2 6 b . (18) 4 Sp ecial analytical solutions to Eq.(16) exists for certain c hoices of the constants h 0 , h 2 and h 4 . When h 0 = 1, h 2 = − (1 + m 2 ), h 4 = m 2 , Eq.(16) has the solution F ( τ ) = sn( τ , m ). F rom Eq.(11) and Eq. (17), Eq.(1) h as the Jacobian elliptic sine function solution, q 2 ( x, t ) = ± v u u t − 12 B 2 bm 2 d (10 B 2 b + 10 B 2 bm 2 + a ) sn( B x, m ) e i ( − B 2 ( B 2 b + B 2 bm 4 +14 B 2 bm 2 + am 2 + a ) t + θ ) , where B is determined by d (10 bB 2 + 10 m 2 bB 2 + a ) 2 − 6 bc = 0. When h 0 = 1 − m 2 , h 2 = 2 m 2 − 1, h 4 = − m 2 , Eq.(16) has the solution F ( τ ) = cn( τ , m ). Inserting it i n to (17) and u sing the transform ation (11 ), Eq.(1) has the Jacobian elliptic cosine function solution, q 3 ( x, t ) = ± v u u t 12 B 2 bm 2 d (10 B 2 b − 20 B 2 bm 2 + a ) cn( B x, m ) e i ( B 2 (16 B 2 bm 2 − 16 B 2 bm 4 − B 2 b − a +2 am 2 ) t + θ ) , where B is determined by d ( a + 10 bB 2 − 20 bB 2 m 2 ) 2 − 6 bc = 0. Some solitary wa ve solutions can b e obtained if the mo du lus m approac h es to 1. F or example, w h en m → 1, the s olution q 2 ( x, t ) degenerates to the kink t yp e env elop e wa ve solution, q 4 ( x, t ) = ± v u u t − 12 b B 2 d (20 B 2 b + a ) tanh( B x ) e i ( − 2 B 2 (8 B 2 b + a ) t + θ ) , where B is determined by d ( a + 20 bB 2 ) 2 − 6 bc = 0. When m → 1, the solution q 3 ( x, t ) degenerates to the b ell type env elop e wa ve solution, q 5 ( x, t ) = ± v u u t 12 B 2 b d ( a − 10 B 2 b ) sec h( B x ) e i ( B 2 ( a − B 2 b ) t + θ ) , where B is determined by d ( a − 10 bB 2 ) 2 − 6 bc = 0. As p oin ted ou t in Ref.[3], Eq.(16) has many other Jacobi elliptic fu nction solutions in terms of dn( ξ ), n s( ξ ), n d( ξ ), nc( ξ ), sc( ξ ), cs( ξ ), sd ( ξ ), ds( ξ ), cd( ξ ), dc( ξ ) as well as the corresp ondin g solitary w a ve and trigonometric function solutions. F or simplicit y , su c h t yp es of solutions to Eq.(1) are not liste d here. With the aid of Maple, we ha v e chec ke d the solutions q j ( x, t )( j = 1 , · · · , 5) by putting them bac k into Eq.(1). 5 Ac kno wledgmen t This wo r k wa s supp orted by the National Natural Science F oundation of Ch ina ( No. 10801 037). References [1] A. Bisw as, D . Milo v ic, Optical solitons in a p arab olic la w media with fourth order disp ersion, Appl. Math. Comput. 208 (2009) 209. [2] Z.H. Li, L. Li, H.P . Tian and G.S. Zhou, N ew t yp es of solitary w a ve solutions for the higher order nonlinear S chr¨ odinger equation, Phys. Rev. Lett. 84 (2000) 4096. [3] Y.B. Zhou, M.L. W ang, Y.M. W ang, Periodic wa ve solutions to a coupled KdV equations with v ariable coefficients, Phys. Lett. A 308 (2003) 31. [4] M.L. W ang, Y.B. Zhou, The perio dic wa ve solutions for the Klein-Gordon-S c hr¨ odinger equations, Phys. Lett. A 318 (2003) 84. [5] E. Y om ba, The extended F-ex pansion metho d and its application for solving the nonlinear wa ve, CKGZ, GDS, DS and GZ equations, Phys. Lett. A 340 (2005) 149. [6] E. Y omba, A generalized auxiliary equation metho d and its application to n onlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations, Ph ys. Lett. A 372 (2008) 1048. 6