Adomian Decomposition Method for Approximating the Solution of the High-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

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1 Adomian Decomposiion Mehod for Approximaing he Soluion of he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion Xian-Jing Lai a, Jie-Fang Zhang b, and Jian-Fei Luo c a Deparmen of Basic Science, Zhejiang Shuren Universiy, Hanghou, 35, Zhejiang, China b Insiue of Theoreical Physics, Zhejiang Normal Universiy, Jinhua, 34, Zhejiang, China c Presiden s Office, Zhejiang Normal Universiy, Jinhua, 34, Zhejiang, China Reprin requess o X.-J. L.; laixianjing@63.com Z. Naurforsch. 6a, 5 5 (6); received December, 5 In his paper, he decomposiion mehod is implemened for solving he high-order dispersive cubic-quinic nonlinear Schrödinger equaion. By means of Maple he Adomian polynomials of obained series soluion have been calculaed. The resuls repored in his aricle provide furher evidence of he usefulness of Adomain decomposiion for obaining soluions of nonlinear problems. PACS numbers:.3.jr;.6.cb; 4.65.Tg Key words: Adomian Decomposiion Mehod; High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion; Adomian Polynomials.. Inroducion The dynamics of solions in Kerr media are in general described by he nonlinear Schrödinger (NLS) family of equaions wih cubic nonlinear erms 5]. However, when he inensiy of he inciden ligh field ges sranger and sranger, one can no neglec he non-kerr nonlineariy effecs and NLS equaions wih higher-order dispersion erms are needed o describe he propagaion of opical pulses in fibers 6, 7]. The general higher-order nonlinear Schrödinger (HONLS) equaion models proposed in he lieraure are no compleely inegrable and canno be exacly solved by he inverse scaering ransform mehod 8]. The noninegrabiliy usually originaes no only from he higherorder nonlinear erms bu also he higher dispersion erms. The analyical and numerical soluions for he NLS equaions wih higher nonlineariy and dispersion have been acively invesigaed by many auhors in several differen models 6 ]. In his paper we consider a higher-order NLS equaion including fourh-order dispersion wih a parabolic nonlineariy law. This high dispersive cubic-quinic nonlinear Schrödinger (HDCNS) equaion can be wrien in he form iψ = β Ψ +i β 3 6 Ψ + 4 Ψ γ Ψ Ψ γ Ψ 4 Ψ, () wih he iniial condiion Ψ(,)= f (), where i =, Ψ(,) is he slowly varying envelope of he elecromagneic field, represens he ime (in he group-velociy frame), represens he disance along he direcion of propagaion (he longiudinal coordinae), β, β 3 and represen coefficiens of second-order dispersive (GVD), hird-order dispersive (TOD), fourh-order dispersive (FOD), respecively, and γ and γ are coefficiens of he cubic and quinic nonlineariies, respecively. In ] he modulaional insabiliy of opical waves o () is invesigaed. For picosecond ligh pulses, he higher-order erms of () can be omied, i.e., β 3 = = γ =, and () can reduce o he NLS equaion. The NLS equaion including only he GVD and he self-phase modulaion (SPM) is well known in he fiber, and i admis brigh and dark solion-ype pulse propagaion in anomalous and normal dispersion regimes, respecively. However, for femosecond ligh pulses, whose duraion is shorer han fs, he higher-order erms are nonnegligible and should be reained. When β 3 = = in (), i was shown ha he dark and brigh soliary wave soluions exis even in he normal dispersion regime. Recenly, dark and brigh solion soluions have been proposed for () wih hird-order dispersion being nil (β 3 = ) 3]. Besides, in he absence of he quinic / 6 / 5 5 $ 6. c 6 Verlag der Zeischrif für Naurforschung, Tübingen hp://naurforsch.com

2 6 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion nonlinear erm (γ = ) of (), Shagalov 4] has invesigaed he effec of he hird- and fourh-order dispersion erms on he modulaional insabiliy, and Karpman and Shagalov 5 6] have sudied he ime behavior of he ampliudes, velociies and oher parameers of radiaing solions. For () wih boh he TOD and FOD and he cubicquinic nonlinear erms, an ordinary solion moving wih consan velociy V (in applicaion o he nonlinear opical pulses having consan self-frequency shif) has he form Ψ = φ( V )exp(iη( V )+σ +ϕ ]), () where V is a solion velociy, η a solion wave-number and σ a nonlinear frequency shif. Arbirary consans and ϕ will be omied below for breviy. In paricular, a soluion wih an ampliude ha only depends on he ime and a phase only depending on he coordinae in he direcion of propagaion such as ] Ψ = λ anh( V )exp(iη), where λ,ν = /V and η are he parameers represening he ampliude, pulsewidh and wavevecor per uni lengh, respecively, is o be deermined as λ = 5 β (γ ), 5 γ β4 /γ V = 3 β + γ β4 /γ 5, η = 3( β + γ β4 /γ )(3β + γ β4 /γ ). 5 I is worhwhile o invesigae he numerical soluions (in paricular solion soluions) for he HDCNS equaion. In his paper, we aim o inroduce a reliable algorihm, he Adomian decomposiion mehod (ADM), o approach () wih iniial profile. Unil now, several sudies in he lieraure have been conduced o implemen ADM o he NLS equaion 7 3]. ADM is a numerical echnique for solving a wide class of linear or nonlinear, algebraic or ordinary/parial differenial equaions. The mehod, which is well addressed in 4 6], has a useful aracion in ha i provides he soluion as an infinie series in which each erm can be easily deermined. The series is quickly convergen owards an accurae soluion. I has been proved o be a compeiive alernaive o he Taylor series mehod and oher series echniques. Several papers deal wih he comparison of he ADM wih some exising echniques in solving differen ypes of problems. In 7], i was found ha, unlike oher series soluion mehods, he decomposiion mehod is easy o program in engineering problems, and provides immediae and visible soluion erms wihou lineariaion and discreiaion. Advanages of he ADM over he Picard s mehod have been proved by Rach 8]. He showed ha he wo mehods are no he same and he Picard s mehod works only if he equaion saisfies he Lipschi condiion. Edwards e al. 9] have inroduced heir comparison of he ADM and Runge-Kua mehods for approximae soluions of some predaor prey model equaions. Wawa inroduced a comparison beween he ADM and Taylor series mehod 3]; he showed ha he ADM minimies he compuaional difficulies of he Taylor series in ha he componens of he soluion are deermined eleganly by using simple inegrals, alhough he Taylor series mehod provides he same answer obained by ADM. In 3] he ADM and wavele-galerkin mehod is compared. From he compuaional viewpoin, he comparison shows ha he ADM is efficien and easy o use. In 3], a comparison of he numerical resuls is obained by using he B-spline finie elemen mehod and ADM. From he resuls, he ADM algorihm provides highly accurae numerical soluions wihou spaial discreiaions for he nonlinear parial differenial equaion. The illusraions show ha he ADM is numerically more accurae han he convenional numerical mehod of he finie elemen. Subsequen works in his direcion have demonsraed he power of he mehod for numerical evaluaions.. The Mehod of Soluion This secion is devoed o review he ADM for solving he HDCNS equaion wih he iniial condiion Ψ(, )= f (). Following he Adomian decomposiion analysis, we rewrie () in he following operaor form: L Ψ = i β L,Ψ + β 3 6 L 3,Ψ i 4 L 4,Ψ + γ Ψ Ψ + γ Ψ 4 Ψ. Similar o 7 9], we define for () he linear operaors L,L,,L 3, 3 3 (3) and L 4, 4 4. By defining he onefold righ-inverse operaor

3 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion 7 ( )d, we find ha Ψ(,)=Ψ(,)+ i β L,Ψ + β 3 6 L 3,Ψ Therefore i 4 L 4,Ψ + γ Ψ Ψ + γ Ψ 4 Ψ]. Ψ(,)= f ()+ i β L,Ψ + β 3 6 L 3,Ψ i 4 L 4,Ψ + γ G (Ψ)+γ G (Ψ)]. (4) (5) The decomposiion mehod suggess ha he linear erms Ψ(,) be decomposed by an infinie series of componens Ψ(,)= Ψ n (,), (6) where he componens Ψ,Ψ,Ψ,..., as will be seen laer, are o be deermined individually in an easy way hrough a recursive relaion ha involves simple inegrals. The nonlinear operaors G (,) and G (,) are defined by he infinie series G i (Ψ)= A i,n, i =,. (7) Tha means ha he nonlinear erms Ψ Ψ and Ψ 4 Ψ are represened series of A i,n, (i =,) which are called Adomian polynomials. In nex Ψ n (,), (n ) is he componen of Ψ(, ) ha will eleganly be deermined. Hence, upon subsiuing hese decomposiion series ino (5) yields Ψ n (,)= f () i β i 4 L, Ψ n (,)+ β 3 6 L 3, Ψ n (,) (8) L 4, Ψ n (,)+γ A,n + γ A,n ]. The mehod suggess ha he eroh componen Ψ is usually defined as he erms arising from iniial condiions. Then we obain he componens series soluion by he following recursive relaionship: Ψ (,)= f (), (9) Ψ n+ = i β L,Ψ n + β 3 6 L 3,Ψ n i 4 L 4,Ψ n + γ A,n + γ A,n ], () where n. The Adomian polynomials A i,n can be generaed for all forms of nonlineariy which are generaed according o he following algorihm: ( A i,n = d n )] n n! dα n G i α k Ψ k, n. () k= α= This formula is easy o be se in a compuer code o ge as many polynomials as we need in he calculaion. We can give he firs few Adomian polynomials of he A i,n as A, = Ψ Ψ, A, = Ψ Ψ +Ψ Ψ, A, = Ψ Ψ + Ψ Ψ + Ψ Ψ +Ψ Ψ, A,3 = Ψ Ψ 3 + Ψ Ψ Ψ + Ψ Ψ Ψ + Ψ Ψ + Ψ Ψ Ψ +Ψ Ψ 3, A,4 = Ψ Ψ 4 + Ψ Ψ Ψ 3 + Ψ Ψ + Ψ Ψ Ψ 3 + Ψ Ψ + Ψ Ψ +Ψ Ψ + Ψ Ψ Ψ 3 +Ψ Ψ 4, A, = Ψ 4 Ψ, A, = 3 Ψ 4 Ψ + Ψ Ψ Ψ, A, = 3 Ψ Ψ Ψ + 6 Ψ Ψ Ψ + 3 Ψ 4 Ψ +Ψ 3 Ψ + Ψ Ψ Ψ, A,3 = Ψ 3 Ψ + 6 Ψ Ψ Ψ + 6 Ψ Ψ Ψ Ψ + 3Ψ Ψ Ψ + 6 Ψ Ψ Ψ Ψ + 6 Ψ Ψ Ψ Ψ + 3 Ψ 4 Ψ 3 + Ψ 3 Ψ Ψ + Ψ Ψ Ψ 3, A,4 = 3Ψ Ψ Ψ Ψ Ψ + 3 Ψ Ψ Ψ + 6 Ψ Ψ Ψ Ψ 3 + 6Ψ Ψ Ψ + 6 Ψ Ψ Ψ 3 Ψ + 6 Ψ Ψ Ψ + 6 Ψ Ψ Ψ Ψ 3 + 3Ψ Ψ Ψ + Ψ Ψ Ψ + 3Ψ Ψ Ψ + 3 Ψ 4 Ψ 4 +Ψ 3 Ψ + Ψ 3 Ψ Ψ 3 + Ψ Ψ Ψ 4 + Ψ Ψ Ψ. ()

4 8 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion The res of he polynomials can be consruced in a similar manner. In case he nonlinear erms G (Ψ) = Ψ Ψ and G (Ψ)= Ψ 4 Ψ are real funcions hen he Adomian polynomials are evaluaed by firs wriing Ψ = Ψ H(Ψ) H( Ψ), (3) where H(u) is he Heaviside (sep) funcion. Hence, (3) yields G (Ψ)= Ψ Ψ = Ψ 3 H(Ψ) H( Ψ)], G (Ψ)= Ψ 4 Ψ = Ψ 5 H(Ψ) H( Ψ)] (4). Therefore, G (Ψ)= G (Ψ)= H(Ψ) H( Ψ)] A,n{Ψ 3 }, H(Ψ) H( Ψ)] A,n {Ψ 5 }, (5) where A,n {Ψ 3 } and A,n {Ψ 5 } are he Adomian polynomials given by A, = Ψ 3, A, = 3Ψ Ψ, A, = 3Ψ Ψ + 3Ψ Ψ, A,3 = 3Ψ Ψ 3 + 6Ψ Ψ Ψ +Ψ 3, A,4 = 3Ψ Ψ 4 + 6Ψ Ψ Ψ 3 + 3Ψ Ψ + 3Ψ Ψ, A, = Ψ 5, A, = 5Ψ 4 Ψ, A, = Ψ 3 Ψ + 5Ψ 4 Ψ, A,3 = Ψ 3 Ψ + Ψ 3 Ψ Ψ + 5Ψ 4 Ψ 3, A,4 = 5Ψ Ψ 4 + 3Ψ Ψ Ψ + Ψ 3 Ψ + Ψ 3 Ψ Ψ 3 + 5Ψ 4 Ψ 4. (6) The funcion G i (Ψ), (i =,) is piecewise-differeniablel wih a singulariy a he origin. Since H(Ψ) H( Ψ)] = for Ψ, i follows from (4) ha, if we avoid he origin, hen G (Ψ)=Ψ 3 = A,n and G (Ψ)=Ψ 5 = A,n. 3. Exemplificaion of he ADM We firs consider he applicaion of he decomposiion mehod o he HDCNS equaion wih he iniial condiion Ψ(,)= ( ) λ λ sech (5γ λ 6γ ) 3β3 + 6β exp( i β 3 ), (7) where β,β 3,,γ,γ and λ are arbirary consans. Applying he inverse operaor on boh sides of (3) and using he iniial condiion (7) and he decomposiion series (6) and (7) yields Ψ n (,)= ( λ sech i β i 4 ) λ (5γ λ 6γ ) 3β3 + 6β exp( i β 3 ) L, Ψ n (,)+ β 3 6 L 3, Ψ n (,) L 4, Ψ n (,)+γ A,n + γ A,n ]. (8) For simpleness, we ake ω = λ (5γ λ 6γ )/ (3β 3 + 6β )] / in he following. Proceeding as before, he Adomian decomposiion mehod gives he recurrence relaion Ψ = Ψ(,)=λsech( ω)exp( i β 3 ), (9) Ψ n+ (,)= i β L,Ψ n + β 3 6 L 3,Ψ n i β ] 4 4 L 4,Ψ n + γ A,n + γ A,n, () where n. The resuling componens are Ψ = λ sech(ω)exp( i β 3 ), Ψ = i β L,Ψ + β 3 6 L 3,Ψ i 4 L 4,Ψ + γ A, + γ A, ]= λ 4cosh(ω) β 3 4 i(3 3 +β β 3 4 ω 4 β ω β 3 4 6β 3 ω β 4 )cosh(ω)+(4β 3 ωβ 4 β +8β 3 3 ω )sinh(ω)]exp( i β 3 ),

5 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion 9 Ψ = i β L,Ψ + β 3 6 L 3,Ψ i 4 L λ 4,Ψ + γ A, + γ A, ]= 5cosh(ω) 3 β4 6 (β 8 4 ω 8 + β ω 4 β 3 β β 3 β 4 β 6 3 ω β 4 864β β 3 4 ω + 44β ω 4 β β β β ω β β 7 4 β ω 6 + ω 6 β 3 β ω β 8 3 )cosh(ω) + i(576β ω 3 β 3 β β 3 3 ω 5 β β ω 5 β 3 β β β 3 3 ω β 5 3 ω 3 β β 7 3 ω 336β β 5 3 ωβ 4 576β ωβ 3 3 β 3 4 )sinh(ω)cosh(ω)+5β β 3 4 ω + 8β 6 3 ω β β ω β 3 4 ]exp( i β 3 ), Ψ 3 = i β L,Ψ + β 3 6 L 3,Ψ i 4 L 3 λ 4,Ψ + γ A, + γ A, ]= 8944cosh(ω) 4 β4 9 i( 78β 3 ω 6 β β 3 6 ω 6 β β3 99β3 4 β4 8 ω β3 8 β β β3 8 ω 4 β4 4 59β3 4 ω β4 5 β 3 49β3 6 ω β4 4 β + 34β 3 4 ω 4 β4 6 β + 34β3 β 738β3 ω β4 + 78β 3 β3 6 β ω 8 β3 β4 9 β 43β ω 8 β4 + 59β 3ω 4 β4 7 β β 3 4 ω 6 β4 7 β + 59β 6 3ω 4 β4 5 β 36β ω β4 ω 8ω β3 β4 738β 3 8 ω β4 3 β )cosh(ω) 3 +(78β 3 ω 7 β4 9 β + 368β 3 ω 5 β4 8 β β 3 3 ω 5 β4 7 β + 368β 5 3ωβ4 4 β β 3 5 ω 5 β4 6 β + 44β3 3 ω 7 β4 8 β + 88β3 5 ω 7 β β3 7 ω 5 β β 9 3ω 3 β β3 ω + 864β3 7 ωβ4 3 β + 7β 3 ω 9 β4 β 3688β 5 3ω 3 β4 5 β 384β 7 3ω 3 β4 4 β 3456β3 3 ω 3 β4 6 β β 3 9 ω β + 4β3 3 ω 9 β4 9 )sinh(ω)cosh(ω) + i( 34 3 ω 6 β 7 4 β 34β 8 3 ω β 3 ω 6 β 8 4 β ω β 5 4 β ω 4 β 6 4 β + 5β 3 ω β 4 384β 6 3 ω 6 β β 6 3ω 4 β 447β 3ω 4 β 7 4 β 3 + 5β 8 3ω β 3 4 β 843β 6 3 ω4 β 5 4 β )cosh(ω)+(8944β 5 3 ω3 β 5 4 β β 7 3ω 3 4 β + 37β 9 3 ω3 β β3 3 ω 3 β4 6 β 3 )sinh(ω)]exp( i β 3 ),... () The oher componens of he decomposiion series (6) can be deermined in a similar way. Subsiuing () ino he decomposiion series (6), which is a Taylor series, we obain he closed form soluion β3 ω(3β + β3 Ψ = Ψ +Ψ +Ψ +Ψ 3 +Ψ = λ sech ) ] 3β4 ω exp i(k β ] 3 ), () or equivalenly Ψ = λ exp(k β 3/ )i +(β 3 ω(3β + β 3 )/3β 4 ω)] + expβ 3 ω(3β + β 3 )/3β 4 ω], (3) wih λ ω = (5γ λ 6γ ) 3β3 + 6β, K = 3β 3 4 6ω β3 + β β3 β ω β4 3 4ω4 4β4 3, (4) where β,β 3,,γ,γ and λ are arbirary consans. From hese values of he pulse parameers, i is simple o see ha boh he ampliude and he widh of he solion are uniquely deermined from he characerisics of he nonlinear medium, i.e. he second- and fourhorder dispersion coefficiens and he wo nonlinear coefficiens.

6 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion For comparison, we consider anoher iniial condiion of () as Ψ(,)= γ JacobiSN( ϖ,m)+jacobicn( ϖ,m)]exp( i β 3 ) 5γ (5) wih ϖ = (β + 6β3 )/( 5 m ), where JacobiSN( ϖ,m) and JacobiCN( ϖ,m) are he Jacobian ellipic sine funcion and cosine funcion. They are periodic wih period K(m), where K(m), K(m) = π/ dx/ m sin x, is he complee ellipic inegral of he firs kind, β,β 3 and are arbirary consans, and he coefficiens γ,γ mus have opposie signs (γ γ < ). Rewriing () for iniial condiion (5) in a operaor form as (3), hen using (9) and () wih (), one can consruc he erms of he decomposiion series. Some of he erms of he series are as follows: Ψ = γ 5γ JacobiSN( ϖ,m)+jacobicn( ϖ,m)]exp( i β 3 ), Ψ (,)= i β L,Ψ + β 3 6 L 3,Ψ i 4 L 4,Ψ + γ A, + γ A, ],..., Ψ n+ (,)= i β L,Ψ n + β 3 6 L 3,Ψ n i 4 L 4,Ψ n + γ A,n + γ A,n ], (6) where n, and he Adomian polynomials A i,n, (i =,) are he same as in he formulae (). Performing he calculaions in (6) wih () using Maple and subsiuing hem ino (6) give he exac soluion wih Ψ(,)= γ 5γ JacobiSN(k ϖ,m)+jacobicn(k ϖ,m)]expi(k Ω)] (7) Ω = β 3, k = β 3ϖ(3β + β3 ) 3β4 (8) and β + 6β3 ϖ = β4 5, K = 3β ϖ (9β4 4ϖ m 4 + β3 5β 3 m + 9β4 4ϖ 9β4 4ϖ m ) m 4β4 3, (9) where β, β 3,, γ and γ are arbirary consans. When m =, he soluion (7) degeneraes o Ψ(,)= wih (8) and γ 5γ sech(k ϖ)+anh(k ϖ)]exp(ik iω) (3) K = ϖ 4 5ϖ β 3 β 4 4β 3 4, ϖ = β + 6β3 5β4. (3) In fac, we also obain he resul (3) from he iniial condiion Ψ(,)= γ exp(ϖ) exp(ϖ)] exp( i β 3 β + 6β3 ),ϖ = 5γ exp(ϖx)+] 5β4. (3) I is obvious from he above expression ha he coefficiens mus be saisfy he resricion β 3 β.

7 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion (a) N=5,= (b) N=,= (c) 4 N=,= (d) 4 N=3,= (e) Fig.. (a) (d) The module graphs of he exac (poin) resul () Ψ and approximae (line) resul ϕ n wih he iniial condiion (7) a (a) N = 5, = ; (b) N =, = ; (c) N =, = ; (d) N = 3, =. (e) The surface shows he module of numerical resul ϕ 3 for and Numerical Resuls of he ADM For numerical comparisons purposes, we consruc he soluion Ψ(, ) as Ψ(,)= lim ϕ N, (33) N where ϕ N (,), he N-erm approximaion for Ψ(,), is a finie series defined as N ϕ N (,)= Ψ n (,), N (34) and he recurrence relaion is given as in () wih (). I is worh poining ou ha he advanage of he decomposiion mehodology is he fas convergence of Table. Numerical resuls (in direcion) for modules of exac resul Ψ(,), approximae resul ϕ 3 (,), and absolue error Ψ(,) ϕ 3 (,), where Ψ(,) =sech( 4 3 )expi( 6 )] for (). ( i, i ) Exac Approximae Absolue soluion Ψ soluion ϕ 3 error Ψ ϕ 3 (.,.) E (.,.) E 9 (.,.3) E (.3,.4) E (.5,.5) E (.4,.) E (.5,.) E (.4,.3) E (.5,.4) E 9 (.5,.5) E

8 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion (a) N=4,= (b) N=7,= (c) N=,= (d) Fig.. (a) (c) The module graphs of he exac (poin) resul (7) Ψ and approximae (line) resul ϕ n wih he iniial condiion (5) a (a) N = 4, = ; (b) N = 7, = ; (c) N =, =, respecively, where m =. (d) The surface shows he module of numerical resul ϕ wih he iniial condiion (5) and m = for 6 6 and. (a).4. N=,=. (b) Fig. 3. (a) The module graph of he exac (poin) resul (3) Ψ and approximae (line) resul ϕ wih he iniial condiion (3), where =.. (b) The surface shows he module of numerical soluion ϕ of () for and.7.7. he soluions in real physical problems 4]. The heoreical reamen of convergence of he decomposiion mehod has been considered in he lieraure 4 6]. The obained resuls abou he speed of convergence of

9 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion 3 Table. Numerical ( resuls (in direcion) for modules of absolue error Ψ(,) ϕ 3 (,), where Ψ(,) = ) ( ( ) JacobiSN 4, + JacobiCN 4, )]exp i 3 5 for (). i i e.e.e 9.E 9.E+..E 9.E.E 9 7.E.E+.3 5.E 7.E 6.E 4.E.E E 4.E 7.E.E.E.5 5.E 3.E.E+ 3.E.E+ his mehod were enabling us o solve linear and nonlinear funcional equaions. In a recen work Ngarhasa e al. 33] have proposed a new approach of convergence of he decomposiion series. The auhors have given a new condiion for obaining convergence of he decomposiion series o he classical presenaion of he ADM. Moreover, as he decomposiion mehod does no require discreiaion of he variables ime and space, i is no effeced by compuaion round off errors and one is no faced wih he necessiy of large compuer memory and ime. The accuracy of he decomposiion mehod for () is conrollable and absolue errors Ψ ϕ N are very small wih he presen choice of and which are given in Tables and. In mos cases ϕ N is accurae for quie low values of N. For iniial condiion (5), we achieve a very good approximaion o he parial exac soluion by using only erms of he decomposiion series, which shows ha he speed of convergence of his mehod is very fas. I is eviden ha he overall errors can be made smaller by adding new erms of he decomposiion series. Boh he exac resuls and he approximae soluions obained by using he formulae (33) wih (34) are ploed in Figs., and 3 for () wih differen iniial condiions (7), (5) and (3), respecively. I is eviden ha when compuing more erms for he decomposiion series he numerical resuls are geing much more closer o he corresponding exac soluions wih he iniial condiions of (). 5. Conclusion In his paper, we considered a numerical reamen for he soluion of he HDCNS equaion using he ADM. To he bes of our knowledge, his is he firs resul on he applicaion of he ADM o his equaion. This mehod ransforms () ino a recursive relaion. The obained numerical resuls compared wih he analyical soluion show ha he mehod provides remarkable accuracy especially for small values of he space. Generally speaking, he ADM provides analyic, verifiable, rapidly convergen approximaion which yields insigh ino he characer and he behavior of he soluion jus as in he closed form soluion. I solves nonlinear problems wihou requiring lineariaion, perurbaion, or unjusified assumpions which may change he problem being solved. The mehod can also easyly be exended o oher similar physical equaions, wih he aid of Maple (or Malab, Mahemaica, ec.), he course of solving nonlinear evaluaion equaions can be carried ou in a compuer. As we known, alhough he decomposiion series (6) obained by using ADM is infinie, we ofen replace he exac soluion wih a finie series N ϕ N (,)= Ψ n (,), which is quickly convergen owards he accurae soluion for quie low values of N. On his accoun, here is a common phenomenon in he relaed lieraure 4 3]. I can easily be noed, ha no maer wheher he examples are from he relaed lieraure or from his paper, he space or ime variables in he picures are all aken in small scales. Since he Taylor series mehod provides he same answer obained by he ADM, we can proceed from he naure of he Taylor series 34, 35] o sudy his phenomenon. The Taylor series expansion of he funcion Ψ(,) abou = is given by Ψ(,)= or, equivalenly Ψ(,)= N Ψ (n) (, ) ( ) n, (35) n! Ψ (n) (, ) ( ) n +R N, N. (36) n! Here, R N is a remainder erm known as he Lagrange

10 4 X.-J. Lai e al. ADM for he High-Order Dispersive Cubic-Quinic Nonlinear Schrödinger Equaion (a) (b) (c) N=3,=.5 N=3,= N=3,= Fig. 4. The module graph of he exac (poin) resul () Ψ and approximae (line) resul ϕ 3 wih he iniial condiion (7) of () a differen values of. (a) (b) (c) N=5,= N=5,=5 e 6.8e 6.6e 6.4e 6.e 6 e 6 8e 7 6e 7 4e 7 e e 6e 4e e N=5,= Fig. 5. The module graphs of he exac (poin) resul () Ψ and approximae (line) resul ϕ 5 wih he iniial condiion (7) a differen values of. remainder, which is given by R N =... Ψ (N) (,)(d) N } {{ } N = ( ) N Ψ (N) ( ),,]. (N)! (37) As we known, he decomposiion series (6) is exacly a Taylor series of exac soluion Ψ abou a poin =, ha is N ϕ N = Ψ n (,)= N Ψ (n) (,) n, N. n! Then he remainder erm R N, i.e. he error beween analyical and approximae soluions, is R N = Ψ N Ψ (n) (,) n = Ψ ϕ N n! =... } {{ } N Ψ (N) (,)(d) N = ()N N! Ψ (N) ( ),,], N (38) which can be obained by using he mean-value heorem. From he formula above, we know ha he greaer he feching value of ( is farher and farher from he poin = ) is, he greaer is he error R N (see Fig. 4), alhough he approximaed soluion can be calculaed for any and. Nearing ero for, he approximae soluion is almos according o he exac soluion a any value of ime (see Fig. 5). From Figs. 5 one migh find ou ha boh he erm number N and he value of influence he approximaion precision of he numerical soluion (34) for he corresponding exac soluion of he HDCNS equaion, whereas ime has only a lile effec on his. ] D. Anderson and M. Lisak, Phys. Rev. A 7, 393 (983). ] G. P. Agrawal and C. Headley, Phys. Rev. A 46, 573 (99). 3] D. Mihalache, N. Trua, and L. C. Crasovan, Phys. Rev. E 56, 3955 (996). 4] M. Gedalin, T. C. Sco, and Y. B. Band, Phys. Rev. Le. 78, 448 (997).

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