Integrable inhomogeneous NLS equations are equivalent to the standard NLS

In tegrable inhomogeneous NLS equations are eq uiv alen t to the standard NLS Anjan Kundu Theory Group, Saha Institute of Nuclear Ph ysics Calcutta, INDIA anjan.kundu@saha.ac.in Octob er 26, 2 018 Abstract A class of inhomogeneous nonlinear Schr¨ odinger equations (NLS), claiming to b e novel int eg rable systems with rich pr o p e rties contin ues app earing in PhysRev and P RL. All such equations are s hown to be not new but equiv alent to the standa rd NLS, which trivially explains their integrabilit y featur es. P A CS no: 02.3 0.Ik , 04.20 .Jb , 05.45. Yv , 02.30.Jr Time and again v arious forms of inhomogeneous n on lin ear S chr¨ odinger equations (IHNLS) along with their discrete v arian ts are app earing as cen tral r esult mostly in the pages of Phys. Rev, and PRL CLU76,RB87,PRA91,Kon93, PR L 00,PRL05,PRL07 ¸ , w h ic h are either susp ected to b e integ rab le due to the fin ding of p articular analytic or stable computer solutions, or assum ed to b e only P ainlev´ e in tegrable arxiv08 ¸ , or else claimed to b e completely new integ rab le systems. Apparently the solution of suc h inte grable systems needs generali zation of the in v erse scatte r ing metho d (ISM), in which the usu al isosp ectral app roac h in vo lving only constan t sp ectral parameter λ has to b e extended to nonisosp e ctr al flo w w ith time-depen den t λ ( t ). Moreo v er certain features of the soliton solutions of such inhomogeneous NLS, lik e the changing of the solitonic amplitude, shap e and vel o cit y with time were thought to b e new and su rprising disco very. W e sh o w here that all these IHNLS , though completely in tegrable are not new or indep en d en t in tegrable systems, and in fact are equiv alen t to the standard homogeneous NLS, link ed thr ou gh simple gauge, scaling and coord in ate trans formations. The standard NLS is a we ll kno wn integ r able system with known Lax pair, soliton solutions and us u al isosp ectral ISM n ls,ALM ¸ . As w e see b elo w, a simple time-dep endent gauge transformation of the standard isosp ectral system with constan t λ can create the illusion of h a ving complicated nonisosp ectralit y . Similarly , a time-dep endent scaling of the standard NLS field Q → q = ρ ( t ) Q would n aturally lead the constan t soliton amplitude to a time-dep endent one. In the same w a y a trivial co ordinate transformation x → X = ρ ( t ) x would c hange th e u sual constan t v elo cit y v of the NLS soliton to a time-v ariable quant ity v ( t ) = v ρ ( t ) and the inv arian t shap e of the standard solito n with constant extension Γ = 1 κ to a time-dep endent one with v ariable extension Γ( t ) = Γ ρ ( t ) (see Fig 1a a,b). Therefore all the ric h integrabilit y prop erties of the IHNLS, observ ed in earlier p ap ers, including more exotic and seemingly sur p rising features like n onisosp ectral fl o w, app earance of shap e c hanging and accele r ating soliton etc. can b e trivially explained from the time- dep end ent transformations of these IHNLS f rom the standard NLS and the corresp onding explicit result , namely the Lax pair, N-solit on solutions, infin ite conserv ed qu an tities etc. for the inhomogeneous NLS mo dels can b e deriv ed easily from their w ell known counterparts in the homogeneous NLS case through the same tr an s formations nls ¸ . 1 Let’s start from a recent v ersion of IHNLS PRL07 ¸ , whic h is generic in some sense: iQ t + 1 2 D Q xx + R | Q | 2 Q − (2 αx + Ω 2 x 2 ) Q = 0 , (1) where th e n onautonomous coefficien ts D ( t ) , R ( t ) of the d isp ersive and the nonlinear terms are arb itrary functions of t and the other time-dep endent functions are α ( t ) = s t ρ, Ω 2 ( t ) = 1 4 ( θ t − 1 2 D θ 2 ) , where ρ = R D , θ = ρ t ρD , (2) s ( t ) b eing another arbitrary fun ction. I t is easy to s ee th at a time-dep endent scaling of the field can c hange the co efficien t of the nonlinear term in (1) and at th e same time generate an additional term from iQ t , while a change in phase of the field inv olving x 2 w ould yield extra terms from Q xx . As a result transformin g Q → q = √ ρe i θ 4 x 2 Q . w e can r ewr ite IHNLS (1) in to another form iq t + D 2 q xx + D | q | 2 q − (2 αx + iD θ ) q − iD θ xq x = 0 . (3) In PRL 07 ¸ Eq (1) was declared to b e a new disco very and as a pro of of its inte grability a L ax pair asso ciated with Eq (3) was p resen ted, which we rewrite here in a compact and con v enient form by in tro d ucing a m atrix U (0) ( q ) = √ σ ( q σ + − q ∗ σ − ) as U ( λ ( t )) = − iλ ( t ) σ 3 + U (0) ( q ) , V ( λ ( t )) = D V 0 ( λ ( t )) − iαxσ 3 + D θ xU ( λ ( t ))) , (4) where V 0 ( λ ( t )) = − iλ 2 ( t ) σ 3 + λ ( t ) U (0) + i 2 σ 3 ( U (0) x − ( U (0) ) 2 ) . (5) W e can chec k from the ab o ve Lax pair that the flatness condition U t − V x + [ U, V ] = 0 yields the IHNLS (3) u nder the constrain t λ ( t ) t = α + D θ λ ( t ). Usin g relations (2) on e can resolv e this constrain t to get λ ( t ) = ρ ( t )( λ + s ( t )), which w as giv en in PRL07 ¸ . W e no w establish the equiv alence b et ween the Lax pair U ( λ ( t )) , V ( λ ( t )) (4, 5) for the IHNLS and the well kno wn Lax pair U nls ( λ ) , V nls ( λ ) of the standard NLS nls ¸ , sho wing exp licitly that the nonisosp ectral λ ( t ) is con vertible to constan t sp ectral parameter λ th rough simple transformations. F or th is it is interesting to notice first, that the structure of the NLS Lax pair is hidden already in the expression of the IHNLS Lax pair as U ( λ ( t ) = λ ) = U nls ( λ ) and V 0 ( λ ( t ) = λ ) = V nls ( λ ). Therefore th e aim sh ould b e to r emo v e the t -dep enden ce from λ ( t ) = ρ ( t )( λ + s ( t )) by absorbing the arbitrary fu nctions ρ ( t ) and s ( t ) in step by step mann er. Note that the Lax pair U, V , as evident fr om the associated linear problem ∂ ∂ x Φ = U Φ , ∂ ∂ t Φ = V Φ , corresp ond to in finitesimal generators in th e x and the t direction, resp ectiv ely and therefore a simple co ordinate c hange ( x, t ) → ( ˜ x = ρ ( t ) x, ˜ t = t ) resulting ∂ ∂ ˜ x = 1 ρ ∂ ∂ x , ∂ ∂ t = ∂ ∂ ˜ t + ρ t x ∂ ∂ ˜ x = ∂ ∂ ˜ t + D θ x ∂ ∂ x , wo u ld yield U ( x, t ) = ρU ( ˜ x, ˜ t ) and V ( x, t ) = V ( ˜ x, ˜ t ) + Dθ xU ( x, t ). Therefore us in g su c h a transformation and comparing w ith (4), w e can easily remo ve the ρ ( t ) factor from λ ( t ) in U ( x, t ), w hic h h ow ev er would scale the field as q → q ρ and at the same time eliminate from the transformed V ( ˜ x, ˜ t ) the nonstandard term D θ xU ( x, t ) app earing in V( x, t) (4). F or th e remov al of additiv e term ρ ( t ) s ( t ) fr om λ ( t ), present in U ( x, t ), one can p erform a gauge transformation Φ → ˜ Φ = g Φ with g = e iρsσ 3 , taking the Lax pair to a gauge equiv alen t pair ˜ U = g x g − 1 + g U g − 1 , ˜ V = g t g − 1 + g V g − 1 . (6) One notices that though the ab ov e transf ormations are enough to remov e explicit t dep enden ce from U d ue to its linear dep endence on λ ( t ), the remo v al of t f rom V ( λ ( t )) b ecomes a b it inv olv ed d ue to 2 the nonlinear entry of λ 2 ( t ) and λ ( t ) U (0) in it, whic h br ing in more time-dep endent terms lik e 2 Dρ 2 s and D ρ 2 s 2 . T h ese extra terms how ev er can b e exactly comp ensated for by extending sligh tly the ab o v e co ordinate and gauge transformations by in tro d ucing additional fun ctions f ( t ) , ˜ f ( t ) and c h o osing them as f t = 2 Dρ 2 s and ˜ f t = D ρ 2 s 2 . The m ultiplicativ e f actor D ρ 2 app earing in all terms in V ( λ ( t )) can b e absorb ed easily b y a further coord inate c hange t → T = Dρ 2 t . Therefore taking the ab o ve argument s in to accoun t on e finally solv es the problem completely through the follo w in g thr ee s teps of simple transformations: i ) Co or dinate tr ansformation : ( x, t ) → ( X , T ) , X = ρ ( t ) x + f ( t ) , T = D ρ 2 t, with f t = 2 Dρ 2 s, (7) ii ) Gauge tr ansformation (6) , wh ere g = e i ( ρ ( t ) s ( t ) x + ˜ f ( t )) σ 3 with ˜ f t = D ρ 2 s 2 , (8) iii ) Field tr ansformation : q → ψ , wh ere ψ = 1 ρ ( t ) q e 2 i ( ρ ( t ) s ( t ) x + ˜ f ( t )) . (9) The ab ov e transformations w ould tak e (4) directly to the standard NLS Lax pair U nls ( λ ) = − iλσ 3 + U (0) , wh er e U (0) = ψ σ + − ψ ∗ σ − , V nls ( λ ) = − iλ 2 σ 3 + λU (0) + i 2 σ 3 ( U (0) X − ( U (0) ) 2 ) . (10) whic h prov es the equiv alence of the Lax p air (4 ) for the IHNLS (3) and th e Lax pair (10) asso ciated with the stand ard NLS: iψ T + 1 2 ψ X X + | ψ | 2 ψ = 0 , (11) obtained as the fl atness condition of (10). One can also c hec k that under th e change of indep endent and dep end ent v ariables (7) and (9) the inh omogeneous NLS (3) is transf ormed directly to the h omogeneous NLS (11). Therefore w e remark that the inhomogeneous NLS (1) and (3) are equiv alen t to the homogeneous NLS (11), a w ell kno wn integ r ab le system. The corresp onding Lax pairs (4) and (10) are also gauge equiv alent to eac h other, which therefore trivially explains the complete integ r ab ility of the inho- mogeneous NLS . All signatures of the complete integ r ab ility lik e the Lax p air, N-soliton s olutions, infinite conserv ed quantiti es etc. for these HNLS can b e obtained easily from the corresp onding w ell kno wn expr essions for the NLS system (11 , 10) by inv erting the set of transform ations (7,8,9) as ( X, T ) → ( x, t ) , g → g − 1 , ψ → q . As a result, exp licit t -dep end ence obviously en ters in th e Lax op erators as well as in the amp litude, p hase and the x-dep end en ce of the field q ( x ) of the IHNLS system, r esulting the sp ectral p arameter λ → λ ( t ) and making the constant amplitude A , extension Γ and velocit y V of the soliton to b ecome t -dep endent. Fig. 1 demonstrates this situation, sho w- ing that the NLS soliton (mo dule) a) | ψ | = A sec h ξ , ξ = 1 Γ ( X − V T ) go es to IHNLS soliton b): | q | = A ( t )sec h ˜ ξ , ˜ ξ = 1 Γ( t ) ( x − v ( t )), where A ( t ) = Aρ ( t ), v ( t ) = D V ρ ( t ) t − f ( t ) ρ ( t ) , Γ( t ) = Γ ρ ( t ) under the tr an s formations inv erse to (7 ,8,9). Therefore even th ou gh I HNLS soliton (Fig. 1b) lo oks rather exotic and quite differen t f r om the standard NLS soliton (Fig. 1a), these solutions are related simply b y coord inate and scale transformations and b elong to equiv alent in tegrable systems. It is w orth ment ionin g that, th ou gh in all earlier pap ers only 1-solit on of the IHNLS was consid- ered, one can easily deriv e th e exact N-soliton for the IHNLS, thanks to its complete in tegrabilit y , by exploiting again its equiv alence with the in tegrable NLS , i.e. by simply mapp ing th e kn own N-soliton of the standard NLS through the same tr an s formations (7-9). By redefi ning the field f urther: Q → b ( t ) Qe ia ( t ) with arbitrary fu nctions a ( t ) , b ( t ), we can generate more inhomogeneous terms in (1) resulting a more general f orm of IHNLS iQ t + D 2 Q xx + R | Q | 2 Q − (2 αx + Ω 2 x 2 + a + iγ ) Q = 0 , γ ( t ) = b t b , (12) 3 equiv alent n atur ally to the integ r ab le NLS. The IHNLS (12) w as found to b e the maxim um in homo- geneous NLS system wh ic h can p ass the Painlev ´ e integrabilit y criteria PLA87 ¸ . A r ecently prop osed IHNLS arxiv08 ¸ , whic h is s imply a particular case of (12) at a = 0 and α = 0, is th er efore also equiv alen t to the standard NLS and hence, con trary to the assumption in arxiv08 ¸ that the system is only P ainlev ´ e in tegrable and not completely inte grable, th e equiv alence w ith NLS assures the complete inte grability , including the existence of infinite conserv ed qu an tities, N-soliton solutions etc. for this IHNLS arxiv08 ¸ W e now lo ok into other forms of integrable HNLS app eared earlier in Phys. Rev. RB87,PRA91,Kon93 ¸ and PRL CLU76,PRL00,PRL05 ¸ and sho w th eir equiv alence to the standard NLS, similar to as found ab o ve. The s im p lest form of inhomogeneit y to the NLS: 2 xq was pr op osed in CLU76 ¸ , wh ic h is clearly a particular case of (3) with α = 1 , Ω = 0, ensured b y the c hoice R = D = 1 , s = t , p ro ving th us its equiv alence with the NLS . A m ore general IHNLS with F ( x ) Q was considered in RB87 ¸ and sho w n fin ally that in tegrabilit y restricts the choic e only u pto F ( x ) = a + αx + µx 2 , whic h is consisten t with the general in tegrable IHNLS (12), sh own to b e equ iv alen t to the standard NLS (11). How ever for constructing su ch int egrable IHNLS, as shown here, x-dep endent sp ectral parameter considered in RB87 ¸ is not n eeded an d similarly the restriction on fu nction h ( t ) app earing in λ ( t ), fou n d b y the author apparently as a condition for the int egrabilit y , actually do es n ot app ear allo wing the function to b e arbitrary , as s ho wn here. In PRL00 ¸ a v ariant of IHNLS wa s considered, whic h was s u sp ected to b e integrable through com- puter simulatio n . It is easy to see ho wev er, th at this IHNLS can b e obtained as a p articular case from (12) at α = 0 , s = 1 , a = 0 , γ = 0 , Ω = 0, but with non trivial R ( t ) , D ( t ) , γ ( t ) ob eying certain constrain ts. Similarly IHNLS prop osed in PRL05 ¸ can b e seen to b e deriv able from (12 ) as a p articular case with D = 1 , R ( t ) = g 0 e ct , s = 1 , a = 0 , γ = 0 , giving α = 0, but Ω = − c 2 . Therefore b oth these inhomogeneous NLS PRL00,PRL05 ¸ are equiv alen t to the standard NLS and hence completely in tegrable. Some inte grable discrete v ersions of IHNLS, namely inh omogeneous Ablo witz-Ladik mo dels (ALM) w ere prop osed in PRA91,Kon93 ¸ , con taining in addition to the standard ALM ALM ¸ an in homogeneous term n ω ψ n , with ω = 1 PRA91 ¸ or ω ( t ) as an arbitrary fu nction Kon93 ¸ . W e fin d that in spite of the discrete case a sim ilar reasoning found here holds true and the p r op osed inhomogeneous ALM can b e sh o wn to b e gauge equiv alent to the standard ALM ALM ¸ , u nder d iscrete gauge transform ation: ˜ U n = g n +1 U n g − 1 n , ˜ V n = g n V n g − 1 n + ˙ g n g − 1 n , with g n = e − in Γ( t ) σ 3 and r ed efinition of th e field as q n → ψ n = q n e i (2 n +1) Γ( t ), wh ere Γ t ( t ) = ω ( t ) is an arbitrary function as found in Kon 93 ¸ . Based on the ab o ve result we therefore conclude that the general inhomogeneous NLS, if integrable, should b e of th e form (12). Other form s of inte grable IHNLS are only its particular cases. Ho wev er all these inhomogeneous NLS are neither n ew nor indep endent inte grable systems, but are equiv alen t to the stand ard homogeneous NLS, from wh ic h all their in tegrable structures lik e Lax p air, N-soliton solutions, in finite num b er of commuting conserv ed quan tities etc . can b e obtained easily thr ough simple mapping. The time-dep endent solito n amplitude, shap e and v elo cit y as w ell as the nonisosp ectral flo w in these inhomogeneous NLS are ju st an artifact of the time-dep endent co ordinate, gauge and field transformations, needed to get these systems from the standard NLS. T herefore b efore prop osing an y new in tegrable inhomogeneous NLS the authors sh ould chec k whether it can b e linked in any wa y to the general in tegrable IHNLS (12), w hose equiv alence with the w ell known NLS w e ha ve prov ed here. References [1] H. H. Chen and C. S. Liu, Phys. Rev. Lett. 37 , 693 (1976) [2] R. Bala kr ishnan, Ph ys. Rev. A 32 , 1144 (1985 ) [3] R. 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Lett A 125 , 456 (1987) 5 Figure 1: Exact soliton solutions (mo du le) f or the in tegrable a) h omogeneous NLS (11) and b) inho- mogeneous NLS (3) with A = 2 . 0 , V = 0 . 5. and the particular c hoice R = . 008 t 2 , D = 1 , s = 0. In spite of the significan t differences b et we en th e app earance and dynamics of these t wo solutions they are related b y s im p le transformations (7-9) and b elong to equiv alen t inte grable systems. 6