Transformation from the nonautonomous to standard NLS equations

EPJ man uscript No. (will be inserted b y the editor) T ransfo rmation from th e nonau tonomous to stan da rd NLS equations Dun Zhao 1,2 , Xu-Gang He 1 , and Hong-Gang Luo 2,3,4 1 School o f Mathematics and Statistics, Lanzhou Un ivers ity , La nzhou 730000 , China 2 Cen ter for Interdisciplinary Studies, Lanzhou U niversit y , Lanzhou 7300 00, China 3 Key Lab oratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou Universit y , Lanzhou 730000, China 4 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China Received: date / Revised version: date Abstract. In this pap er we show a systematical metho d to obtain exact solutions of the n onautonomous nonlinear Schr¨ odinger (NLS) equation. A n integrable condition is first obtained by the Painlev ´ e analysis, whic h is shown to b e consisten t with that obtained by the Lax pair metho d . Un der this condition, we present a general transformation, whic h can directly con vert all allo we d exact solutions of the standard NLS equation i nto the corres p onding exact solutions of th e nonautonomous NL S equation. The method is quite p o werf ul since the stand ard NLS equation h as b een w ell studied in the past decades and its ex act solutions are v ast in th e literature. The result provides an effective wa y to control the soliton dynamics. Finally , th e fundamental b right and dark solitons are taken as examples to demonstrate its explicit applications. P A CS. 05.4 5.Yv Solitons – 42.65.T g Optical solitons; nonlinear guided wa ves – 03.75 .Lm T unneling, Josephson effect, BoseCEinstein conden sates in p eriodic p otentials, solitons, vortices, and top ological ex - citations 1 In tro duction The standard nonlinear Schr¨ odinger (NLS) eq ua tion i ∂ ∂ T Q ( X , T ) + ε ∂ 2 ∂ X 2 Q ( X , T ) + δ | Q ( X , T ) | 2 Q ( X , T ) = 0 , (1) where ε and δ a re constants, is a fundamental nonlin- ear equation to g overn system dynamics in man y differ- ent fields such as Bo se-Einstein condensates (BEC)[1, 2, 3] and nonlinear optics[4, 5]. The nature of Eq. (1) ha s bee n extensively e x plored in past decades by ma ny dif- ferent w ays and its exact solutions including so liton [6] are v ast in the literature. F or a r e v iew, one can refer to Ref. [7]. When applied to different contexts, Eq. (1) ha s many different e x tensions. F or example, in BEC a n addi- tional har monic exter na l p otential is ne e ded, the resulted equation is well kno wn as Gross-Pita evskii equatio n. In addition, the concept of soliton management has been ex- tensively explored in r ecent years[8]. The go a l is to co ntrol effectively the so liton dy na mics. In BEC, the nonlinear in- teraction can b een easily tuned by an external mag ne tic field, namely the F eshb ach resonance management[9, 10]. On the other hand, in th e context of optical soliton com- m unication, the disper sion management has been explor ed extensively to improve the o ptical soliton communication [11, 12]. In b oth cases , the basic equa tion to describ e the system dynamics should be a generalized nonautonomo us NLS equa tion [13], which reads in one-dimensiona l(1 D) case i ∂ u ( x, t ) ∂ t + ε f ( x, t ) ∂ 2 u ( x, t ) ∂ x 2 + δ g ( x, t ) | u ( x, t ) | 2 u ( x, t ) + V ( t ) x 2 u ( x, t ) = 0 . (2) Here ε and δ are the sa me as those in E q. (1 ) and f ( x, t ) and g ( x, t ) are dimensionless control parameters of disp er- sion and nonlinear int eractio n, resp ectively . V ( t ) is time- depe ndent harmonic tra p potential in BEC, whic h is ab- sent in optical transmission line. Thes e co efficien ts are as- sumed usua lly to be real. Equation (2) a nd/or its similar versions are v ery dif- ficult to solve beca use of the time- and s pace-dep endent disp e rsion a nd nonlinear in teraction managements and the presence of the exter nal p o tent ial in BEC. Some sp ecia l solutions hav e b een obtained by , for example, the Lax pair metho d [13, 14, 15, 1 6, 17], the s imilarity transforma - tion [1 8, 19, 20, 21], and so o n. How ever, a gener al metho d to find solutions of E q. (2) ha s not yet b een o btained. Here we obtain a gener al transformation, which ca n con- vert all allowed exact solutions of the sta ndard NLS equa- tion (1) to the cor resp onding solutions of Eq. (2). T o our knowledge, the r e s ult is repor ted for the first time in the literature, which provides a s tr aightforw ard and systemat- ical wa y to find the exac t solution of the g eneralized NLS 2 Dun Zhao et al.: T ransformation from the nonautonomous to standard NLS equations equation (2), as shown b y tw o examples given in the fina l part of the pa pe r. Below we first show this tra ns formation is consistent with the Painlev ´ e integrability condition of Eq. (2). 2 The Painl ev´ e analysis and T ra nsfo rmation Motiv ated by the rela tionship betw een complete in tegra- bilit y and the Painlev´ e pro p er ty of partial differential equa- tions [22, 23], we p er form the WTC test [22] to study po ssible in tegra bilit y condition o f Eq. (2 ). F ollo wing the Krusk al ansatz [2 4] and the sta ndard pro c e dur e o f the Painlev ´ e analysis [2 3], we obtain a c o mpatibility condi- tion g t,t g − 2 g 2 t g 2 + f 2 t f 2 − f t,t f + g t g f t f + 4 εf V = 0 , (3) where the subscripts denote the time deriv atives. Her e we should men tion that the Painlev ´ e test requires that f ( x, t ) and g ( x, t ) must be space- indepe ndent , i.e., f ( x, t ) = f ( t ) and g ( x, t ) = g ( t ). I t is in teresting to note that this con- dition is completely consistent with the in tegra bilit y co n- dition obtained by Lax pair [13]. The complete int egra bilit y of Eq. (2) under the com- patibilit y condition ca n also be further confirmed through a tr ansformatio n that reduces Eq. (2) to the standard NLS equation (1). B elow we lo ok for suc h a transformation in a general fo rm of [25] u ( x, t ) = Q ( X ( x, t ) , T ( t )) e ia ( x,t )+ c ( t ) , (4) where X ( x, t ) , T ( t ) , a ( x, t ) and c ( t ) a re real functions to be determined by the requirement that u ( x, t ) and Q ( X , T ) are the solutions o f E qs. (2) a nd (1), resp ectively . Inser ting Eq. (4) into E q. (2) and compar ing w ith Eq. (1), we o btain a set of differential equations, which have so lutions under the condition Eq. (3), a ( x, t ) = 1 4 εf ( t )  d d t ln f ( t ) g ( t )  x 2 + C 1 g ( t ) f ( t ) x − C 2 1 ε Z g ( t ′ ) 2 f ( t ′ ) d t ′ + C 2 , (5) X ( x, t ) = g ( t ) f ( t ) x − 2 C 1 ε Z g ( t ′ ) 2 f ( t ′ ) dt ′ . (6) T ( t ) = Z g 2 ( t ′ ) f ( t ′ ) d t ′ + C 3 , (7) c ( t ) = 1 2 ln g ( t ) f ( t ) , (8) where C 1 , C 2 , and C 3 are constants rela ted to the sp ecial bo undary conditio ns and the initial state. Here for s im- plicit y , they ar e set to b e zero in the fo llowing discussions. The Painlev´ e integrability condition Eq. (3) is, in fact, a s ubtle balance c o ndition to keep the no na utonomous systems integrable. F rom the manag ement viewpoint of the solitons[8], Eq. (3) also pr ovides an effectively wa y to manipulate the soliton dyna mics. While any t wo par ame- ters among f ( t ) , g ( t ), and V ( t ) are set, the rema ining one can be tuned according to E q . (3 ) in o rder to control the coherent dyna mics o f solito ns. The applications of Eq. (3) hav e b een extensiv ely explor ed in Ref. [13]. How ever, the transformatio n Eqs. (5) - (8) have not been figur ed out by the Lax pair metho d. Such transforma tions are quite systematic in obtaining the e xact solutions of the nonau- tonomous NLS equatio n. F o r a given nonautonomous NLS equation, we first chec k if the co efficien ts satisfy the com- patibilit y condition Eq. (3). If it is true, then the nonau- tonomous NLS equation can b e r educed to the standar d NLS equatio n (1). All allowed exact so lutio ns, including the canonical solitons, of the standard NLS equation (1) can thus b e conv erted into the corresp o nding solutions of the nonautonomo us NLS equa tio n. In this sens e, a c a noni- cal soliton can be viewed as a “seed” of the cor resp onding soliton-like s o lutions of Eq. (2) under the co mpatibility condition Eq. (3). Some remar ks are in order . i) If f ( t ) = g ( t ) and V ( t ) = 0, the no nautonomous NLS equation Eq. (2) hav e the canonical so liton solutions (up to a phase) regardless of the explicit form of the time-dep endent no nlinea rity and disp e rsion. This is b ecause in this cas e the balance be- t ween nonlinearity a nd disp ersion is kept. In this s e ns e the soliton-like s o lution of Eq. (2) is a q uasi-cano nic a l soli- ton. ii) When g ( t ) 6 = f ( t ), the original bala nce b etw een nonlinearity and disp ersio n is br oken down. In this c a se the ca nonical so liton must deform itself to build new bal- ance b etw een nonlinearity and disp ersion. In this sense, the solito n-like so lution of Eq. (2) is a deformed ca nonical soliton. The amplitude o f the soliton will be sca led b y the factor of p g ( t ) /f ( t ), as sho wn by c ( t ). This clea rly indi- cates the influence of the disp er sion and nonlinear man- agements to the soliton b ehavior. iii) It is very interest- ing to no te that the confining harmonic e xternal p oten- tial is absent in the tr a nsformation eq uations. Ho wev e r, the pr esence of the p otential affects the balance betw e e n nonlinearity and dispersio n and builds a deep connection betw een the o ptical solitons and the ma tter -wa ve ones . iv) If V ( t ) = 0, the solito ns can b e quas i- canonical or de- formed dep ending on if f ( t ) is equal to g ( t ) or not, as men tioned above. On the contrary , if V ( t ) 6 = 0, Eq. (3) indicates tha t f ( t ) 6 = g ( t ). This mea ns that the a mplitude of the soliton m ust change b ecause o f Eq. (8 ). This leads to an impor tant obs erv a tio n that there do es not exist the canonical and e ven quasi-ca nonical matter -wa ve solitons under compatibility condition Eq. (3). It is also helpful to mention so me techniques to find the soliton-like solutions of the nonautonomous NLS equa- tion in the literature. The Lax pair a nalysis is very useful in discus sing integrability conditions [14, 13, 1 7]. A widely used metho d is the similarity tra nsformation [18, 20, 21 ], which introduces some explicit tra nsformation parame- ters. Thes e par ameters ar e determined b y a set of ordi- nary differential equa tions, which in gener al case can no t be so lved analytically , as emphasize d in Ref. [20]. Another similarity tr ansformation r e ducing the nona utomonous NLS equation to a stationary NLS one has also be e n intro- Dun Zhao et al.: T ransformation from the nonautonomous to standard NLS equations 3 duced [26]. Alternativ ely , b y the Lie p oin t s ym me- try group analysis [27], Eq. (2) or its similar v er- sion can b e classified into diffe rent classe s and eac h class can b e con v erted into the corresp onding rep- resen tativ e equation by some allow ed transforma- tions. As a res ult, the exact sol uti ons o f the rep- resen tativ e equation can b e transforme d i n to the corresp onding solutions of the equations in the same class. Ho w ev er, i t w as als o p oin ted out i n [ 2 7] that in most cases it i s diffi cult to o btain the exact solutions of these representativ e equations and the in tegrability of certain represen tativ e equations is unclear. Quite di fferen t from these tec hniques, the presen t work fo cuse s on the integrabilit y of Eq. (2) and builds a deep connection b et w een the no nau- tonomous N LS equation and its autonom o us coun- terpart, whic h pro vides a more systematical w a y to find sol utions of the nonautonomous NLS equa- tion. Moreo ver, the corresp ondi ng transformation form ulas are explicit and straigh tforw ard. In addi- tion, from the co ntrol viewp oint, our metho d also provides an effective wa y to control the soliton dyna mics, as men- tioned ab ov e. 3 App lications Although the tra nsformation obtained can b e a pplied to all allow ed e xact solutions o f the standard NLS equation, the further discus sion is purp osely restr ic ted to the fun- damental bright and dark soliton solutions of the nonau- tonomous NLS equa tion without the harmo nic external po tent ial, which is enoug h to show how the soliton dynam- ics is con trolled by cor r esp onding tunable pa rameters. In this case, Eq. (3) b ec o mes f ( t ) = g ( t ) exp  − α Z g ( t ′ ) d t ′  , (9) where α is a co nstant and the tr ansformatio n equatio ns (5)-(8) beco me a ( x, t ) = − α 4 exp( G α ( t )) x 2 , X ( x, t ) = ex p( G α ( t )) x, T ( t ) = Z dt ′ g ( t ′ ) exp( G α ( t ′ )) , and c ( t ) = (1 / 2 ) G α ( t ) , where G α ( t ) = α R t 0 g ( t ′ ) d t ′ . When ε = 1 / 2 and δ = 1 ( δ ε > 0), Eq. (1) has the fundamen tal canonical bright soliton solution Q ( X, T ) = sech ( X ) exp( iT / 2 ) and when ε = − 1 / 2 and δ = 1 ( δε < 0), the fundamental dark soliton so lution o f Eq . (1) has the form of Q ( X , T ) = tanh( X ) exp( iT ). Starting from these t wo solutions, we show the corres po nding soliton s o lutions of Eq. (2) for four different c ases: g ( t ) = 1 , exp( t ) , exp( − t ), - 5 0 5 0 1 2 - 5 0 5 0 2 4 - 5 0 5 0. 0 0. 5 1. 0 - 5 0 5 0 1 2 3 - 5 0 5 0. 0 0. 5 1. 0 ( e ) ( d) ( c ) ( b) | u ( x , t ) | 2 T X | Q ( X , T ) | 2 | u ( x , t ) | 2 C a noni c a l B r i ght S ol i t on ( a ) Fig. 1. (a) The canonical bright soliton of Eq . (1) with ε = 1 / 2 and δ = 1 and the corresponding brigh t soliton-like solutions of Eq. (2) for d ifferent nonlinearity mo du lations: ( b ) g ( t ) = exp( t ), (c) g ( t ) = exp( − t ), (d) g ( t ) = cos( t ), and (e) g ( t ) = 1. In all cases α = 1. and cos( t ), which repres ent consta nt , enhancement, sup- pression, and p erio dic mo dula tio ns of nonlinear it y , resp e c- tively . W e e mpha size that these mo dulations can be r ead- ily realized, for example, by the F eshbach resona nc e tech- nique in the Bose-E instein condensa te context. The corr e - sp onding disper sion modulatio ns fo llow Eq. (9). It is noted that α = 0 leads to G 0 ( t ) = 0, which is trivial up to a phase, as mentioned ab ov e. A nonzero α has no nt rivial results and without loss of genera lity w e take α = 1 be- low. In Fig. 1 and Fig. 2 we explicitly present the br ight soliton-like and the dark so liton-like so lutions for four dif- ferent no nlinearity modula tions, resp ectively . F or co mpar- ison, w e also plot the canonica l solitons , a s shown in Fig. 1(a) and Fig . 2 (a), resp ectively . Fig. 1(b) and (c) show that the cano nical bright soliton bec omes more and more either sharp er or br oader depend- ing on enhance ment or suppre s sion of the nonlinearity . When the nonlinearity k eeps unchanged and the disp er - sion is suppr e s sed, the canonica l bright s o liton also b e- comes more a nd more shar pe r , a s shown in Fig. 1(e). This can be under sto o d b y the fact that the soliton is due to the balance betw een disper sion and nonlinearity . Most inter- esting case is Fig. 1(d), where the nonlinearity modula tion is p erio dic. As a result, the canonical bright soliton is als o mo dulated p er io dically . All these re sults indica te that the bright soliton-like solution and its canonica l counterpart has a close relationship. F or different no nlinearity mo dula- tions, we hav e c heck ed the integration of R | u ( x, t ) | 2 dx and found it keeps unchange in time, which further shows the nature o f the bright so liton-like s o lutions o f Eq. (2). The 4 Dun Zhao et al.: T ransformation from the nonautonomous to standard NLS equations - 5 0 5 0 . 0 0 . 5 1 . 0 ( a ) - 5 0 5 0 . 0 0 . 5 1 . 0 | Q ( X ,T ) | 2 | u ( x , t ) | 2 / 2 t | u ( x , t ) | 2 / 2 t ( b ) - 5 0 5 0 . 0 0 . 5 1 . 0 ( c ) - 5 0 5 0 . 0 0 . 5 1 . 0 ( d ) - 5 0 5 0 . 0 0 . 5 1 . 0 ( e ) C a n o n i c a l D a r k S o l i t o n X T Fig. 2. The canonical dark soliton of Eq. (1) with ε = − 1 / 2 and δ = 1 and the correspondin g dark soliton-like solutions of Eq. (2) for different nonlinearity mo du lations same as t h ose in Fig. 2. F or clarity , the amp litud e of the dark soliton-like solutions is normalized by η t = exp  1 2 G 1 ( t )  . similar r esult is als o true to the dark so liton-like solutions, as shown in Fig. 2. These results shed light on the under- standing of the soliton dynamics a nd provide an exa ct w ay to make a disper sion and/o r nonlinearity manag ement of solitons. It is exp ected to hav e a re alistic application to the optical so lito n communication technologies and the matter-wa ve solito n dyna mics. Finally , it s hould b e p ointed o ut that the present anal- ysis can also be applied to all exact solutions o f Eq. (1), including the m ulti-soliton cases. 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