Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation

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1 Home Search Coections ournas About Contact us My IOPscience Inverse scattering transform for the integrabe nonoca noninear Schrödinger equation This content has been downoaded from IOPscience Pease scro down to see the fu tet 6 Noninearity 9 95 ( View the tabe of contents for this issue or go to the ourna homepage for more Downoad detais: IP Address: This content was downoaded on 4//6 at 9:48 Pease note that terms and conditions appy

2 Noninearity 9 (6) London Mathematica Society Noninearity doi:88/95-775/9/3/95 Inverse scattering transform for the integrabe nonoca noninear Schrödinger equation Mark Abowitz and Ziad H Mussimani Department of Appied Mathematics University of Coorado Campus Bo 56 Bouder CO USA Department of Mathematics Forida State University Taahassee FL USA E-mai: mussimani@mathfsuedu Received une 5 revised December 5 Accepted for pubication 3 December 5 Pubished 4 February 6 Recommended by Dr Beatrice Peoni Abstract A nonoca noninear Schrödinger (NLS) equation was recenty introduced and shown to be an integrabe infinite dimensiona Hamitonian evoution equation In this paper a detaied study of the inverse scattering transform of this nonoca NLS equation is carried out The direct and inverse scattering probems are anayzed Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained The inverse scattering theory is deveoped by using a nove eft right Riemann Hibert probem The Cauchy probem for the nonoca NLS equation is formuated and methods to find pure soiton soutions are presented; this eads to epicit time-periodic one and two soiton soutions A detaied comparison with the cassica NLS equation is given and brief remarks about nonoca versions of the modified Korteweg de Vries and sine-gordon equations are made Keywords: integrabe nonoca NLS equation eft-right Riemann-Hibert probem PT symmetry Mathematics Subect Cassification numbers: 37K5 37Q55 35Q5 Introduction Eacty sovabe modes and integrabe evoution equations are ubiquitous in noninear science and pay an essentia roe in many branches of physics Generay speaking many of these systems can be derived from basic principes and arise as universa modes in diverse physica phenomena For eampe the Korteweg de Vries (KdV) and modified Korteweg de /6/395+3$33 6 IOP Pubishing Ltd & London Mathematica Society Printed in the UK 95

3 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani Vries (mkdv) equations describe the evoution of weaky dispersive and sma ampitude waves in quadratic and cubic noninear media respectivey Physicay speaking the KdV equation is we known in terms of its appication to shaow water waves [5 9] The integrabe cubic noninear Schrödinger (NLS) equation [8] is aso a universa mode It describes the evoution of weaky noninear and quasi-monochromatic wave trains in media with cubic noninearities In the contet of noninear optics the NLS equation is a key mode describing optica wave propagation in Kerr media [9 8] Moreover in the imit of deep water the NLS equation can be derived from the cassica irrotationa and inviscid water waves equations [5] The KdV mkdv and NLS equations share the mathematica property that a are integrabe and eacty sovabe evoution equations There are aso numerous continuous and discrete eacty integrabe evoution equations that are physicay reevant and appy to diverse probems in fuid mechanics eectromagnetics gravitationa waves easticity fundamenta physics and attice dynamics to name but a few [3 5 6] Recenty integrabe continuous and discrete nonoca noninear Schrödinger equations describing wave propagation in noninear PT symmetric media were aso found [ ] Interestingy there are muti-dimensiona etensions of many of these equations The best known integrabe muti-dimsiona equation is the Kadomtsev Petviashvii (KP) equation [7] It is too a universa mode describing the evoution of weaky dispersive and sma ampitude waves with additiona weak transverse variation It arises in water waves pasma physics and interna waves [3 5] Remarkaby muti-ine soiton soutions of the KP equation are observed virtuay daiy on shaow fat beaches [8] Generay speaking integrabiity is estabished once an infinite number of constants of motion or an infinite number of conservation aws are obtained However consideraby more information about the soution can be obtained if the inverse scattering transform (IST) can be carried out [4] Corresponding to rapidy decaying initia data IST provides a inearization and a cass of epicit soutions eg soitons The method associates a compatibe pair of inear equations (ie a La pair) with the integrabe noninear equation One of the equations termed the scattering probem is used to determine suitaby anaytic eigenfunctions and provides a mechanism to transform the initia data to appropriate scattering data The other inear equation serves to determine the evoution of the scattering data Using the anaytic behavior of the eigenfunctions an inverse scattering probem is deveoped It is convenient to transform the inverse scattering system to a generaized inear Riemann Hibert (RH) probem to sove for the underying meromorphic functions With the time dependence of the scattering data one can find the soution of the noninear evoution equation from the inverse or RH probem The IST method is viewed as an etension of Fourier transforms to a cass of noninear systems [ ] Here we investigate in detai the foowing nonoca noninear Schrödinger (NLS) equation i q( t) q ( t ) ± q ( tq ) ( t ) t 96 () where denotes compe conugate This equation was first introduced in [] but due to size imitations many detais had to be omitted In the above equation we see that the nonocaity occurs in a remarkaby simpe way; namey one of the noninear terms has the dependent variabe evauated at ie note the term q * ( t) We aso note that integrabe mutidimensiona etensions of the nonoca NLS equation have been found and studied [5] Another way to write the above equation is i qt( t) q ( t ) + Vtqt ( ) ( ) () where V( tq ; ) ± q( tq ) ( t) In this atter form the equation is viewed as a Schrödinger equation with a noninear PT symmetric potentia []: V( t) V * ( t) It is remarkabe

4 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani that such a simpe nonoca noninear Schrödinger equation turns out to be integrabe We aso note that this equation is Gaiean invariant PT-symmetric equations have been etensivey studied in recent years [ ] This is discussed further in section There are other nonoca integrabe systems which have been anayzed; two that are weknown are the Benamin-Ono and intermediate ong wave equations [3 5] These equations were found in the study of ong interna waves with a deep bottom ayer The IST for these systems has been known since the 98s We aso remark that the foowing nonoca evoution equations t q( t) + q ( t ) ± 6 qtq ( ) ( t) q ( t) q C : compe nonoca mkdv q( t) + q ( t ) ± 6 qtq ( ) ( t) q ( t) q : rea nonoca mkdv t R and (3) (4) qt( t) + s( tqt ) ( ) s( t ) ( qtq ( ) ( t)) t q R : rea nonoca sine Gordon (5) arise from a reduction of the AKNS scattering probem and are integrabe It is interesting that for this case (and odd fows in the hierarchy) the nonocaity shows up as sign inversions in both and t When q( t) q( t) a these equations reduce to their standard counterparts: mkdv and sine-gordon The IST formuation wi be discussed in a future communication In this paper we deveop the IST associated with nonoca noninear Schrödinger (NLS) equation () It turns out that the genera method of AKNS [5 6] can be appied but with a new symmetry reationship imposed This crucia symmetry reation has maor consequences; it eads to new symmetries amongst the eigenfunctions scattering data and a nove inverse probem which can be reated to a Riemann Hibert probem formuated via eigenfunctions defined at both pus and minus infinity; here we refer to the method as a eft/right Riemann Hibert approach Epicity this new symmetry condition eads to reationships between eigenfunctions defined at both and (hence the term eft and right) In the standard NLS system the Riemann Hibert probem eads to a cosed set of uncouped integra equations at either or In the nonoca NLS equation discussed in this paper a cosed set of integra equations can again be determined But to do this requires using both sets of eigenfunctions at and which in turn are couped The symmetry conditions reating the eigenfunctions at both and are fundamenta To our knowedge this is the first time that such symmetries between eft and right Riemann Hibert probems have been empoyed in the IST associated with noninear wave probems The paper is organized as foows In section we empoy the AKNS procedure to find a couped NLS type system in terms of two potentias: q( t) and r( t) With the symmetry r( t ) σq ( t) σ (6) the nonoca noninear Schrödinger (NLS) equation () resuts In section 3 we provide a method to derive an infinite number of oca and goba conservation aws associated with the nonoca noninear Schrödinger equation () This estabishes its integrabiity as an infinite dimensiona Hamitonian dynamica system In section 4 certain key asymptotic properties of the eigenfunctions and scattering data are discussed; this is foowed by section 5 where the symmetries of the eigenfunctions as we as of the scattering data are estabished The basic inverse scattering probem is deveoped in section 6 aong with a detaied anaysis of the 97

5 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani eft right Riemann Hibert probem The reconstruction formua of the potentias is presented in section 7 The time periodic one and two soiton soutions and some of their properties are given in section 9 Comparisons with the cassica NLS equation is in section ; the cacuation for the scattering data associated with equation () for specia bo initia condition is given in section We concude in section Linear pair and compatibiity condition The nonoca noninear Schrödinger equation We begin our discussion by considering the AKNS scattering probem [5 6] v Xv () T where v v ( t ) is a two-component vector ie v ( t ) ( v( t) v ( t )) and q( t) r( t) are (in genera) compe vaued functions that vanish rapidy as and k is a compe spectra parameter The matri X depends on the functions q and r as we as on the spectra parameter k ( ) X i k q t () rt ( ) ik The time evoution of the eigenfunctions v is given by where vt Tv (3) T A C B A (4) and A Band C are scaar functions of q( t) r( t) given by A ik + i q( trt ) ( ) (5) B kq( t ) i q ( t) (6) C kr( t ) + i r( t) (7) The compatibiity condition of system () and (3) ie v v i q( t) q ( t ) rtq ( ) ( t ) t t t yieds (8) i rt ( ) r ( t ) qtr ( ) ( t ) (9) t Under the symmetry reduction r( t ) σq ( t) σ () system (8) and (9) are compatibe and eads to the nonoca noninear Schrödinger equation first introduced in [] and mentioned above in the introduction We write the resuting equation () again for the convenience of the reader: i q( t) q ( t ) ± q ( tq ) ( t ) t 98

6 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani where again denotes compe conugation and q( t) is a compe vaued function of the rea variabes and t with corresponding La pairs given by i k q( t ) X () σq ( t) ik ik + i σq( tq ) ( t) kqt ( ) i q ( ) t T σkq ( t ) i σq ( t) ik i σqtq ( ) ( t ) () The crucia symmetry reduction () was first noted in [] and eads to a nove cass of nonoca integrabe noninear evoution equations incuding a nonoca NLS hierarchy This is a specia and remarkaby simpe reduction of the more genera AKNS system [4] which has not been previousy found A ist of few important properties of equation () are shown beow: Time-reversa symmetry: If q( t) is a soution so is q * ( t) Invariance under the transformation : If q( t) is a soution so is q( t) i Gauge invariance: If q( t) is a soution so is e θ qt ( ) with rea and constant θ Compe transation invariance: If q( t) is a soution so is q ( + i t) for any constant and rea Equation () is a Hamitonian dynamica system and is obtained using the variationa formuation where δq is given by δh i qt ( t) (3) δq ( t) δh ( t) is the variationa derivative of the Hamitonian with respect to q * ( t) and H [ q( tq ) ( t) σq ( tq ) ( t)] d σ (4) PT symmetry: If q( t) is a soution so is q * ( t) As mentioned in the Introduction caing the quantity V( t ) ± qtq ( ) ( t ) (5) which in cassica optics is referred to as a sef-induced potentia impies that V * ( t) V( t); and with this at hand equation () takes the form i qt( t) q ( t ) + Vtqt ( ) ( ) (6) This equivaent formuation aows one to connect equation () with PT symmetric optics for which V( t) represents a waveguide and the resuting equation remains invariant under the oint transformation of t t and a compe conugate Thus the nonoca equation () is PT symmetric [] We remark that wave propagation in PT symmetric couped waveguides or photonic attices has been observed in eperiments in cassica optics [ ] Gaiean invariance: If q( t) soves equation () with initia condition q( ) then ξ iξ t qt ( ) q( + i ξt t) e e (7) 99

7 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani ξ aso soves to equation () with corresponding initia condition q ( ) q ( ) e for any rea constant ξ If one restricts the cass of soutions to equation () to rapidy decaying soutions at then this woud put some restriction on the parameter ξ to guarantee that qt ( ) itsef remains within that cass We point out that the genera q r system (8) and (9) is aso Gaiean invariant: If q( t) r( t) sove system (8) and (9) so does K iω t qt ( ) q( i Vtt ) e e (8) K iω t r ( t ) r ( i Vt t) e e (9) with K K and ω ω a being rea parameters satisfying the reations V K and ω K An infinite number of oca and goba conservation aws associated with equation () can be derived hence it is an integrabe evoution equation Using a eft right Riemann Hibert formuation the inverse scattering transform is carried out and genera soution to equation () corresponding to rapidy decaying initia data is obtained incuding pure one and two soitons soutions Key important properties of equation () are aso contrasted with the cassica NLS equation where the nonoca noninear term q * ( t) is repaced by q * ( t) In particuar the symmetries of the eigenfunctions and scattering data associated with the cassica NLS equations with even initia data are shown to coincide with symmetries of () under the symmetry reduction () The nonoca rea and compe mkdv equations In the case where the functions AB and C are third order poynomia in the spectra parameter k [3 4] the compatibiity condition of system () and (3) yieds qt( t) + q( t ) 6 qtrtq ( ) ( ) ( t) () rt ( t ) + r( t) 6 q( trtr ) ( ) ( t ) () Under the symmetry reduction r( t ) σq ( t) σ () system () and () are compatibe and eads to the nonoca compe mkdv equation qt( t) + q( t ) 6 σqtq ( ) ( t) q( t) (3) where again denotes compe conugation and q( t) is a compe vaued function of the rea variabes and t When the function q( t) is assumed to be rea vaued then equation (3) reduces back to the rea mkdv equation (4) We point out that under space and time even initia conditions the nonoca mkdv equation reduces back to its cassica (oca) counterpart 3 The nonoca sine-gordon equation If in equation (4) we take A A/ k B B / k and C C / k then the compatibiity condition vt vt resuts in the foowing set of equation for the potents q and r qt ( t) + s( tqt ) ( ) (4) 9

8 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani rt ( t ) + strt ( ) ( ) (5) s( t) + ( q( trt )( )) t (6) where we have defined A s i/ Under the reduction r( t ) q( t) (7) the system (4) and (5) are compatibe and give rise to the nonoca sine-gordon equation qt ( t) + s( tqt ) ( ) s( t) st ( ) (8) 3 Infinite number of conserved quantities and conservation aws 3 Goba conservation aws The infinite number of conserved quantities of () is derived as foows We assume that q( t) decays rapidy at infinity As a resut soutions of the scattering probem () can be defined subect to the foowing boundary conditions () φ k () e k im () ψ k () e k im im φ( k ) ( ) i ik e i ik im ψ( k ) ( ) e (3) (3) Throughout the paper φ is not the compe conugate of φ We sha instead use φ to denote T the compe conugate of φ If φ( t ) ( φ( t ) φ ( t )) is the soution to () that satisfies the boundary conditions (3) then for Im k the scattering data ak ( ) φ( t ) e k i is anaytic in the upper haf compe k pane and approaches as ± Substituting φ ( t ) ep [ i k + ϕ( t )] into () we find (after eiminating φ ) that the function µ ( t ) ϕ ( t ) satisfies the Riccati equation µ q + µ qr ikµ (33) q Since for Imk> im k ϕ( we substitute the asymptotic epansion µ ( k ) n in equation (33) and equate powers of k to find and for any integer n µ n( t ) n+ (34) ( i µ ( t ) qtrt ( ) ( ) (35) µ ( t ) qtr ( ) ( t ) (36) µ n+ µ n n q + µ mµ n m (37) q m ik From the boundary conditions (3) it foows im φ ( k ) e and im ϕ ( k ) Combining this resut together with the definition of μ we have 9

9 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani n ak ( ) where we have defined n Cn ( i n+ (38) Cn µ n ( t ) d (39) Since for a k with Im k> a( is time independent then it foows that C n is aso time independent Beow is a ist of the first few conserved quantities under the symmetry reduction r( t ) σq ( t) σ : C σ qtq ( ) ( t ) d (3) σ C [ q( tq ) ( t ) + qtq ( ) ( t )] d (3) C σ [ q ( tq ) ( t ) σq ( tq ) ( t )] d σ C3 { q( tq ) ( t ) + qtq ( ) ( t ) σqtq ( ) ( t )[ 4 qtq ( ) ( t ) q ( tq ) ( t )]} d (3) (33) C σ { q ( tq ) ( t ) 6 σqtq ( ) ( tq ) ( tq ) ( t ) σ[ ( ) ( ) ( ) ( )] + ( ) 3 q tq t qtq t q tq ( t )} d (34) 3 Loca conservation aws In this section we epain how to derive an infinite number of oca conservation aws We start with the time-dependent probem (3) φ Aφ + Bφ (35) t Substituting the epression for μ and ϕ into (35) and taking the derivative of the resuting equations we find µ ( t ) B nonoc tµ ( t ) Anonoc + (36) qt ( ) Reca that the nonoca noninear Schrödinger equation () is obtained when A ik + i σq( tq ) ( t) (37) nonoc B kq( t ) i q ( t) nonoc (38) nonoc (39) C kσq ( t ) i σq ( t) 9

10 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani Substituting the power series epansion for μ from (34) and (37) (38) into (36) we obtain µ ( ) ( ) n t q t t i σqtq ( ) ( t ) + ik n+ ( i qt ( ) n The coefficients of ( ik ) n n are trivia for n For n we find µ ( ) n t n+ (3) ( i q ( ) t tµ n( t ) + i µ n( t ) µ n+ ( t ) n 3 (3) qt ( ) We can write the conservation aws in the form T X i (3) t where T and X are the so-caed densities and fues respectivey The first few conservation aws are given by T qtq ( ) ( t ) X qtq ( ) ( t ) + q ( tq ) ( t ) ( ) ( ) ( ) ( ) + ( ) ( ) σ ( ) T qtq t X q tqt qtq t q tq ( t ) 3 T qtq ( ) ( t ) σq ( tq ) ( t ) X q ( tq ) ( t ) + qtq ( ) ( t ) 4 σq ( tq ) ( tq ) ( t) 4 Direct scattering probem In this section we study the scattering probem () subect to the boundary conditions (3) It is epedient to reformuate the scattering probem in terms of eigenfunctions having constant boundary conditions (the so-caed ost functions ) defined by (note: hereafter for simpicity of notation we often suppress the time dependence) ik ( ) φ( ) ( ) i k Mk e k M k e φ( k ) (4) ik i k Nk ( ) e ψ( N( k ) e ψ( k ) (4) With this in mind one can show that the ost functions satisfy a inear impicit integra equation that in turn is used to estabish the foowing important resuts: In the space of absoutey integrabe functions L ( R) defined by f d < one can show that M( N( are R anaytic functions in the upper haf compe k pane whereas M ( k ) N( are anaytic functions in the ower haf compe k pane [6] From the integra representation one can aso derive the arge k asymptotics of the ost functions used in the inverse probem (cf [6]) rzqz ()() dz ( ) ik M k + Ok ( ) r ( ) ik (43) 93

11 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani N( k ) ik M ( k ) + ik q ( ) ik + Ok ( ) rzqz ()() dz q ( ) ik + Ok ( ) rzqz ()() dz (44) (45) + ( ) ik N k rzqz ()() dz + Ok ( ) r ( ) ik (46) The soutions φ( k ) and φ ( k ) of the scattering probem () with the boundary conditions (3) are ineary independent for a k satisfying ak ( ) see beow This foows from the fact that the Wronskian W(u v) is given by Wuv ( ) uv uv (47) of any two soutions u and v to () is independent of Simiar arguments hod for ψ( k ) and ψ ( k ) Therefore because the scattering probem () is a second order inear ODE the pairs { φφ } and { ψψ } are ineary dependent and one can epress one basis set in terms of the other: φ( k ) ak ( ) ψ( k ) + bk ( ) ψ( k ) (48) φ( k ) ak ( ) ψ( k ) + bk ( ) ψ ( k ) (49) With this resut the scattering coefficients are thus given by ak ( ) W ( φψ ) (4) ak ( ) W ( ψ φ) (4) bk ( ) W ( ψ φ) (4) bk ( ) W ( φ ψ) (43) Note that the scattering data satisfy the reation akak ( ) ( ) bkbk ( ) ( ) (44) If the potentias qr as then from the anayticity properties of the ost functions it can be shown that a( is anaytic in the upper haf compe k pane whereas ak ( ) is anaytic in the ower haf compe k pane [6] In genera b( and b( k ) cannot be etended off the rea k ais Therefore equation (44) is defined ony for Imk The arge k asymptotics of the scattering data a( and ak ( ) is given by [6] 94

12 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani ak ( ) rzqz ()() d z+ O( k ) ik ak ( ) + rzqz ()() d z+ O( k ) ik (45) (46) 5 Symmetry reduction r( t ) σq ( t) σ : eigenfunctions and scattering data 5 Symmetry of the eigenfunctions In this section we estabish important symmetry properties of the eigenfunctions of the eigenvaue probem () vaid under the symmetry reduction r( t ) σq ( t) where σ To T do so et vk ( ) ( v( k ) v( ) be a soution to system () with r( t ) σq ( t) If we take the compe conugation of () and et k k one reaches the foowing concusion: v ( v( k ) If soves ( ) then soves ( ) v( (5) σv ( k ) Therefore because the soutions of the scattering probem () are uniquey determined by their respective boundary conditions (3) we obtain the important symmetry reations ψ( k ) σ φ ( k ) (5) ( ) ψ k φ ( k ) σ (53) With the definitions (4) and (4) one can readiy write down the corresponding symmetry conditions of the ost functions: ( ) σ Nk M ( k ) (54) ( ) ( ) σ N k M k (55) Thus we see that the symmetry r( t ) σq ( t) eads to symmetry between eigenfunctions defined at ± 5 Symmetry of the scattering data The symmetry in the eigenfunctions in turn imposes a very important symmetry in the scattering data ak ( ) ak ( ) and b( b( From the Wronskian representations for the scattering data and the above symmetry reations together with the fact that the Wronskian does not depend on it foows for both signs σ ak ( ) a ( k ) (56) 95

13 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani ak ( ) a ( k ) (57) bk ( ) σb ( k ) (58) It therefore foows from equation (56) that if k ξ + iη is a zero (eigenvaue) of a( in the upper haf k pane then k ξ + iη is a zero of a( in the upper haf k pane Simiary if k is a zero of ak ( ) in the ower haf k pane so is k in the upper haf k pane 6 Inverse scattering: a eft right Riemann Hibert approach The inverse probem consists of constructing the potentia functions r( t) and q( t) from the scattering data ie refection coefficients ρ( kt ) and ρ ( kt ) defined on Im k as we as the eigenvaues k k and norming constants (in ) C() t C () t Our approach is based on soving two Riemann Hibert probems associated to what we refer as eft and right scattering probems and use the symmetry conditions estabished in section 5 to reate between the two pieces In doing so we wi make frequent use of the so-caed proection operators defined as foows For any integrabe function f( k C that rapidy decays to zero as k we define the proection operators P ± as f ( ξ) Pf ± im π ε i ξ ( k ± iε ) One of the most important properties of the proection operators are (6) Pf ± ± ± f± P± f (6) where f + (z) and f (z) are anaytic functions in the upper and ower compe haf pane respectivey satisfying f± () z as z 6 Left scattering probem Our starting point is the eft scattering probem (48) and (49) Divide equation (48) by a(; (49) by ak ( ) and use the definition of the ost functions given in (4) and (4) gives the equivaent formuation Mk ( ) ik N( +ρ( e Nk ( ) ak ( ) Mk ( ) ik Nk ( ) + ρ ( e N( ak ( ) where we have defined the eft refection coefficients (63) (64) bk ( ) bk ( ) ρ( ρ ( (65) ak ( ) ak ( ) Taking into account the corresponding boundary conditions as we as the asymptotic behavior of a( at arge k we find k () Mk ( ) im ak ( ) (66) 96

14 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani The functions M( and a( are both anaytic functions in the upper haf compe k pane Mk ( ) Since a( has simpe zeros at k k then is not anaytic and has simpe poes at k k ak ( ) (note that M( has no zeros in the upper haf pane) Let k k be a simpe zero of a( Then around k the Laurent series epansion for the function Mk ( ) Mk ( ) + anaytic function ak ( ) a ( k )( k ) Mk ( ) ak ( ) is given by (67) Let k be an eigenvaue of a( in the upper haf compe k pane ie ak ( ) Then equation (48) gives φ( k ) bk ( ) ψ( k ) which in terms of the ost functions reads i k Mk ( ) b( k ) N( k ) e (68) We subtract from both sides of equation (63) the contributions from a poes and use (68) to find Mk ( ) ak ( ) i CN ( ) e () N( () k ik CN ( ) e k + ρ( ) ( ) k where we have defined the eft norming constant C as C ik k e N k (69) bk ( ) a ( k (6) ) The eft hand side of equation (69) is an anaytic function in the upper haf pane and goes to zero as k hence it forms a + function Aso the function N ( k ) () is an anaytic function in the ower haf pane and goes to zero as k therefore it forms an function Appy P on (69) to find () N( + k i ( ) iξ CN k e ρξ ( ) e N ( ξ) + π d ξ k i ξ ( k i) (6) Simiary the functions M ( k ) and ak ( ) are both anaytic functions in k in the ower haf M( pane Since ak ( ) has simpe zeros at k k then is not anaytic and has simpe poes ak ( ) at k k (the function M ( k ) has no zeros in the ower haf pane) Let k k be a simpe M( zero for ak ( ) Then around k the Laurent series epansion for the function is given by Mk ( ) Mk ( ) + anaytic function (6) ak ( ) a ( ( At an eigenvaue k we have ak ( ) Then equation (49) gives φ ( k ) bk ( ) ψ ( k ) which impy i k Mk ( ) b( N( k ) e (63) Subtract a the poes from (64) and use the reation (63) to find ik ik M( CN( ) e CN Nk ( ) e ik k e N k ak ( ) () ( ) k k () ( ) ( ) + ρ k (64) ak ( ) 97

15 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani where as before the eft norming constant is given by C bk ( ) a ( k (65) ) Note that the eft hand side of equation (64) is an anaytic function in the ower haf pane and goes to zero as k hence it forms a function Aso the function N( k ) () is an anaytic function in the upper haf pane and goes to zero as k therefore it forms an + function Appy P + on (64) to find () Nk ( ) + k i ( ) iξ CN k e ρξ ( ) e N( ξ) π d ξ k i ξ ( k + i) (66) 6 Time evoution of the scattering data: eft scattering probem The time evoution of the scattering data is derived from the evoution equation (3) and is given by see [3] akt ( ) a( k ) a( kt ) ak ( ) (67) 4ikt 4ikt bkt ( ) e b( k ) b( kt ) e bk ( ) (68) Equation (67) impies that the zeros of the scattering data k and k (the soiton eigenvaues) are time independent The time-evoution of the refection coefficients and norming constants foows from (67) and (68) and are respectivey given by 4ikt 4ikt ρ( kt ) e bk ( )/ ak ( ) ρ( kt ) e bk ( )/ ak ( ) 4ikt 4ikt C C() e C C() e (69) (6) 63 Right scattering probem To account for the symmetry conditions (55) and (54) we view the system (48) and (49) as a eft scattering probem and suppement it with the right scattering probem ψ( k ) α( φ( + β( φ( k ) (6) ψ( k ) α( φ( + β( φ ( k ) (6) where α( α( β( and β ( k ) are the right scattering data We can aso write system (6) and (6) in the matri form Ψ ( k ) SR ( Φ( k ) (63) where the right scattering matri is α( β( SR ( (64) β( α( 98

16 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani T T and Φ( k ) ( φ( k ) φ( k )) Ψ( k ) ( ψ( k ) ψ( k )) where superscript T denotes matri transpose Foowing the same steps as for the eft RH above we can formuate the corresponding RH probem on the right and find the foowing inear integra equations which govern the functions M( k ) Mk ( ): () Mk ( ) + k i ( ) iξ B M k e R( ξ) e M( ξ) + π d ξ k i ξ ( k i) (65) () Mk ( ) + k i ( ) iξ B M k e R( ξ) e M( ξ) π d ξ k i ξ ( k + i) (66) where R ( and R ( k ) are the right refection coefficients defined by β( β ( Rk ( ) Rk ( ) (67) α( α( and B Bare the right norming constants defined by B β( β ( B (68) α ( k ) α ( k ) 64 Time evoution of the scattering data: right scattering probem The derivation of the time evoution of the right scattering data foows cosey that of the eft case and are given by α( kt ) α( k ) α( k t) α( k ) (69) + 4ikt 4ikt β( kt ) e β( k ) β( k t) e β( k ) (63) The time evoution of the norming constants and refection coefficients foows from (69) and (63) + 4ikt 4ikt B B() e B B() e (63) 65 Reation between the refection coefficients The eft scattering probem (48) and (49) can be rewritten in the matri form Φ ( k ) SL ( Ψ( k ) (63) T T where Φ( k ) ( φ( k ) φ( k )) Ψ( k ) ( ψ( k ) ψ( k )) (here superscript T stands for matri transpose) and S L ( is the so-caed eft scattering matri ak ( ) bk ( ) SL ( bk ( ) ak ( ) (633) With this notation at hand the scattering matri (633) is reated to the right scattering matri (64) throughout the reation SR( SL ( which epicity gives 99

17 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani α( a( α( ak ( ) β( b( β( bk ( ) From the definition of the right refection coefficient R given in (67) we have (634) ( ) ( ) β ( ) ( ) ( ) α( ) ( ) σ k bk b k ( ) σρ Rk k (635) k ak a k In obtaining the resut (635) we used the definition of the eft refection coefficient ρ given in (65) as we as the symmetry reation between the scattering data a aand b given in (56) (58) respectivey Simiar symmetry resut hod between R and ρ ie Rk ( ) σρ ( k ) (636) With this at hand we have reduced the number of scattering data (refection coefficients) in haf Net we aim at achieving the same goa for the norming constants 66 Cosing the system To cose the system we substitute k k in equations (6) (65) and k k in equations (66) (66) then impose the symmetry condition r( ) σq ( ) where σ throughout the symmetry reation between the ost functions (54) and (55) as we as the refection coefficients to find () M ( ) k σ ( ) + M k () M( ) k + M ( () N( + N ( ik Ce N( + k k N( πi iξ ρξ ( ) e N( ξ) d ξ ξ ( k i) N( ξ) (637) ik ( ) ξ B e N k i ρ ( ξ) σ ( ξ) e N + σ ( ) π d ξ k k N k i ξ ( k i) N ( ξ) (638) ik C e M k i ( ) M ξ ρξ ( ) e ( ξ) d k k σm ( k ) πi ξ ξ ( k + i) σm( ξ) () N ( ) k σ ( ) + N k ik B ( ) e M k + k k M ( k ) πi (639) iξ σρ ( ξ) e M( ) ξ d ξ ξ ( k + i) M ( ξ) (64) Equations (637) (64) are integro-agebraic equations that (can in principe) sove the inverse probem As we sha see ater the above system wi be used to construct breathing one soiton soution Indeed the system can be reduced to two equations in two unknowns eg for N ( and N ( by appropriatey substituting M ( k ) M ( from equation (638) into equation (639) eaving ony one vector equation for N ( and N ( The question of soving systems such as (637) (64) or the reduced system (639) has been studied by Beas and Coifman in [] In this paper they give epicit conditions as to when the IST method 93

18 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani yieds a unique soution (possiby for finite time) to noninear evoution equations associated which contain the scattering probems with genera potentias q r However we note that soiton soutions to the nonoca NLS equation can admit simpe poe singuarities in time Hence in genera one can ony epect to find soutions for finite time A fu discussion of the soutions to the nonoca NLS equation obtained by IST is outside the scope of this paper and remains to be studied 67 Eigenvaues on the imaginary ais and norming constants When formuating the eft and right RH probem for the ost eigenfunctions we have subtracted ony the contribution from a poes the are ocated at k k the zeros of the scattering data a( and ak ( ) respectivey However it foows from the symmetry of the scattering data (56) and (57) that there are two more contribution to the poe arising from k k Thus in order for the above anaysis to be consistent we restrict ourseves to the case for which a eigenvaues are ocated on the imaginary ais ie k k k k (64) To obtain the symmetry condition that reates the eft norming constants C and C to the right norming constants B and B we restrict the equations (637) (64) to eigenvaues satisfying (64) and obtain after some agebra the foowing set of equations () M( ) k + M ( () M ( + M ( () N( + N ( () N( + N ( ik σ ( ) ξ C e N k i ρ ( ξ) σ ( ξ) e N + ( ) π d ξ k + k N k i ξ ( k i) N ( ξ) ik B e N k i ( ) ξ ρ ( ξ) e σn( ξ) + d k k σn ( πi ξ ξ ( k i) N ( ξ) ik ( ) ξ C e M k i ρξ ( ) ( ξ) e M σ ( ) π d ξ k k M k i ξ ( k + i) σm ( ξ) (64) (643) (644) ik B e M k i σ ( ) ξ ρξ ( ) e M( ξ) d k k M( πi ξ + ξ ( k + i) σm( ξ) (645) Comparing equation (64) with (643) one finds B σc (646) Simiary one obtains from (644) and (645) the resut B σc (647) It shoud be noted that both reations are vaid under the symmetry reduction r( ) σq ( ) Later in section 9 we wi show that further restrictions on the norming constants ony aow σ 93

19 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani 68 Additiona symmetry between the eigenfunctions At an eigenvaue k k N (a zero of a( in the upper haf compe k pane) the eigen functions φ( k ) and ψ( k ) are ineary dependent ie φ( k ) bk ( ) ψ( k ) which in terms of the ost functions it reads (see equation (68)) ik M( be N ( (648) ik M( be N ( (649) Equations (648) and (649) yieds the product reation M( N( k ) M( N ( k ) (65) between the ost functions N and M vaid at an eigenvaue k N Now use the symmetry condition (54) and (55) between the ost functions in (65) to find N( N ( k ) N( N ( k ) N (65) Simiar reation can be derived for the other set of the ost functions N and M vaid at an eigenvaue k N Indeed starting from the right scattering probem (63) one finds that at an eigenvaue k the ost functions N and M are ineary dependent and satisfy ik N( β e M ( k ) (65) ik N( β e M ( k ) (653) Eiminating β e i k from equations (65) (653) and make use of the symmetry condition (54) (55) between the ost functions gives M( M( k ) M( M ( k ) N (654) The above reations (65) and (654) are usefu in finding epicit soiton soutions The above reations (65) and (654) are usefu in order to find the norming constants C and C and their dependence on the soiton eigenvaues k and k 7 Recovery of the potentias Reca from equation (6 ) that () N( + k The arge k behavior of N ( k ) is thus given by i ( ) iξ CN e ρξ ( ) e N ( ξ) + π d ξ k i ξ ( k i) (7) ik iξ N( CN ( k ) e ( ) ( ) π ρξe N ξ d ξ (7) k k k i From (46) we have r ( ) N ( k ) (73) ik 93

20 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani therefore we have the resut: ik iξ r ( ) i CN ( k ) e ( ) ( ) π ρξe N ξ d ξ (74) i We use the symmetries from (55) to find for r( ) q ( ) M( ± N ( k ) (75) together with the asymptotic reation (45) q ( ) M ( k ) (76) ik to find that From (7) we have q ( ) ± i kn ( k ) (77) ik ( ) ( ) ξ + ( ) ( ) π ρ ξ i N k CN k e e N ξ d ξ k (78) k k i ik ( ) ( ) ξ ( ) ( ) π ρ i N ξ k CN k e e N ξ d ξ (79) k k k i Therefore comparing (79) with (77) we find (σ ) σ ik ( ) σ ( ) + ρ iξ q i CN ( ξ) k e e N ( ξ) d ξ (7) π 8 Characterization of scattering data: genera trace formuae The functions a( and ak ( ) are anaytic in the upper and ower haf compe k pane respectivey > and tend to unity as k We assume a( and ak ( ) have the simpe zeros { kimk } and { kimk } respectivey and define < ak ( ) k k m k m α( a ( α ( k m m m m (8) Thus α( α ( are anaytic in the upper and ower haf compe pane respectivey tend to unity for k and have no zeros in their respective haf panes Therefore we have og α( πi og α ( πi og αξ ( ) og αξ ( ) d ξ ξ π dξ k i ξ k Im k > og αξ ( ) og αξ ( ) d ξ ξ π dξ k i ξ k Im k < Adding/subtracting the above equations in each haf pane respectivey yieds 933

21 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani og α( πi og α ( πi Then from equation (8) one finds ogak ( ) m og αξαξ ( ) ( ) dξ ξ k Im k > og αξαξ ( ) ( ) dξ ξ k Im k < k m og( αξαξ ( ) ( )) og + π d ξ Im k > k i ξ k m (8) ogak ( ) m k m og( αξαξ ( ) ( )) og π d ξ Im k < k i ξ k m (83) Since from equation (8) we have α( ( α akak ( ) ( ) equations (8) (83) together with the unitarity condition akak ( ) ( ) bkbk ( ) ( ) yied ogak ( ) m k og k m m + πi og( + bkbk ( ) ( )) d ξ Im k > ξ k ogak ( ) m k m og( + bkbk ( ) ( )) og π d ξ Im k < k i ξ k From the symmetry conditions (58) m b ( b ( k ) we arrive at ogak ( ) m k og k m m + πi og( b( ξ) b ( ξ)))) d ξ Im k > ξ k (84) ogak ( ) m k m og( + b( ξ)) b ( ξ))) og d Im k k k πi ξ < ξ k m (85) Thus we can reconstruct ak ( ) ak ( ) in terms of the eigenvaues (zero s) and ony one function b( And in the inverse probem we do not need bk ( ) bk ( ) ρ( ρ ( ak ( ) ak ( ) independenty We ony require one function b( since ρ ( b ( ak ( ) and we can determine ak ( ) ak ( ) from ony b( using equations (84) (85) To find the norming constants for the soitons b b C C a a 934

22 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani we need a a when r() q * ( ) These derivatives in genera position are found to be Simiary for a ( k ): πi ( ) ( ) m k a k km e ( k k ) m πi ( ) ( ) m k a k km e ( k k ) m Pure soitons have ρ ( : Simiary for a ( k ): Thus we have: m m ( ) ( ) m k a k km ( k k ) m m ( ) ( ) m k a k km ( k k ) iθ m e e C C a ( k ) a ( k ) m iθ og( b( ξ) b ( ξ)) dξ ξ k og( b( ξ) b ( ξ)) dξ k ξ (86) (87) So in the pure soiton case a a depend ony on the zero s k k But in the genera case a a depend on k k and b( (via b(b * ( ) as seen from the above formuae: (86) (87) 9 Soiton soutions 9 Norming constants Pure soiton soutions correspond to zero refection coefficients ie ρ( ξ ) and ρξ ( ) for a rea ξ In this case the system (637) (64) reduces to an agebraic equations that woud determine the functiona form of the soiton(s) Unike the case of the cassica NLS equation for which the two norming constants C and C are reated through a symmetry condition (simiary the soiton eigenvaues) here as we sha see the norming constants are determined either using the symmetry condition (65) and (654) which is tedious to appy or via a trace formua [6 ] that eads to a simpe formua reating the norming constants to the soiton eigenvaues We start the derivation from equation (54) whose first component satisfy (at an eigenvaue k ) N( k ) σm ( k ) (9) At an eigenvaue k k the eigenfunctions M( k ) and N( k ) are reated via equation (68) which in our case give M ( k ) be N ( k ) (9) ik where b bk ( ) Equation (9) combined with (9) gives ik N( σb e N ( k ) (93) 935

23 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani On the other hand the eigenfunctions N ( k ) and M ( k ) are reated through the symmetry condition (54) ie N( k ) M ( k ) (94) which together with (see equation (68)) one finds M( k ) be N( k ) (95) ik ik N( b e N ( k ) (96) By compe conugating equation (96); making the transformation k k and substituting the resuting epression into (93) we find the important reation σ b (97) i This resut impy that the phase of b is arbitrary and b e θ Moreover we see that these types of soiton soutions can ony be obtained when σ Simiar derivation hods for the functions M ( k ) and N ( k ) Indeed starting from the symmetry (55) and (63) one arrives after some cacuations to a simiar resut as the one shown above σ b (98) i where b bk ( ) The phase of the scattering data b is aso free and we thus write b e θ To find the norming constants b b C C a a (99) we need to compute a and a This is accompished with the hep of the trace formua [6 ] which for a genera N eigenvaues k k N takes the form N k ak ( ) k (9) By taking the derivative of a( with respect to k we find a ( N N k k k k k k (9) Evauating equation (9) at the poes k k n we find N ( k ) N ( a ( kn) im k k N n ( k ) ( k )( k ) (9) For the specia case for which a the eigenvaues reside on the imaginary ais we et k i η k iη for N with a η η being rea positive numbers In this case equation (9) simpifies to 936

24 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani ( η η) a ( kn) im η η N n ( η+ η) N N ( η+ η) i( η η )( η+ η ) (93) where we defined k i η with η > Simiar resuts can be obtained for the scattering data ak ( ) Indeed starting from N k ak ( ) k one can derive the reation a ( N N k k k k k k As before evauating equation (95) at the poes k k n we find (94) (95) N ( k ) N k a ( kn) im k k N n ( k ) ( ( Again when a the eigenvaues reside on the imaginary ais equation (96) simpifies to (96) a ( k ) n ( η η) N i( η+ η) im η N n ( η + η) ( η η )( η + η ) η N (97) where now k i η with η > and k i η k iη for N with a η η being positive rea numbers Beow we ist the norming constants associated with the one and two soiton soutions: One soiton soution Here we take N in equations (93) (97) and find i a ( a ( i( η + η ) η + η The norming constants are readiy obtained from (99) and are given by C η + η C η + η (98) (99) Two soiton soution Substituting N in equations (93) (97) we find C C ( η+ η)( η+ η) ( η+ η)( η+ η) C η η η η ( η+ η)( η+ η) ( η+ η)( η+ η) C η η η η (9) (9) 9 -Soiton soution In this section we give an epicit form for the one soiton soution by setting In this case system (638) and (64) give (σ ) 937

25 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani k N ( k ) C e i ( ) k k M k k N ( C e i ( ) k k M k M ( k ) k C e + i ( ) k k N k (9) (93) (94) M ( k C e k ) i ( ) k k N k (95) Now et k iη and k iη with η η both being rea and positive Substituting these quanti ties into system (9) (95) gives after engthy agebra N( iη ) N ( iη ) + C e CC e i( η + η ) ( η + η ) ( η + η ) CC e ( η + η ) η ( η η) (96) (97) M ( iη ) + η C e N ( i η) i( η + η ) (98) M( iη ) η C e N ( i η) i( η + η ) (99) The epicit form of the -soiton soution foows from (7) by setting the refection coefficient ρ to zero k i q ( ) i CN( k ) e (93) The norming constants C and C are given in equation (99) ie C ( η+ η) and C ( η+ η) with arbitrary phases Note that we woud have arrived at the same resuts had we used the symmetry conditions (65) and (654) between the ost functions After some agebra we obtain η ic ( η+ η) e q ( ) ( η + η ) ( η + η ) CCe (93) The time evoution of the norming constants C and C is found from equation (6) and is given by + 4iη t i ϕ+ π/ + 4iη t ( ) C C() e c e e c η + η 4iη t i ϕ+ π/ 4iη t ( ) C C() e c e e c η + η (93) (933) 938

26 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani Substituting (93) and (933) back in (93) we find after some agebra the most genera two parameter famiy of a breathing one soiton soution iϕ 4iη η ( η + η ) e e e qt ( ) (934) i( ϕ+ ϕ) 4i( η η ) t ( η + η ) + e e e To obtain the one soiton soution for the corresponding oca NLS equation we et η η and C C This impies ϕ + ϕ Net we show that the one soiton soution (934) is indeed Gaiean invariant To estabish this resut we use the transformation (7) and obtain iϕ 4iη η ( ξ ) ξ ξ ( η + η ) e e e i i e e qt ( ) (935) i( ϕ+ ϕ) 4i( η η ) t ( η + η )( + iξt) + e e e By defining the new soiton eigenvaues η and η via the transformation η η + ξ/ (936) t t + t t η η ξ/ (937) the one soiton soution then reads t ( iϕ 4iη η η + η ) e e e qt ( ) (938) ( ϕ+ ϕ) ( η η i 4i ) t ( η + η ) + e e e In order for the soiton qt ( ) the remain within the cass of rapidy decaying functions one requires ξ> η (939) where η is eacty the asymptotic decay rate of the soiton as One interesting feature of the one soiton soution (934) is the formation of singuarity at a finite time Indeed at the origin ( ) the soution (934) becomes singuar when ( n + ) π ( ϕ+ ϕ ) tn n Z (94) 4( η η ) It shoud be pointed out that not a members of the one-soiton famiy deveop a singuarity at finite time For eampe if one takes η η η then the one soiton soution (934) reads iϕ 4iη t η 4ηe e e qt ( ) (94) + i( ϕ+ ϕ) 4η e e Thus the soiton given in (94) deveops no singuarity so ong ϕ + ϕ ( n + ) π for any integer n 93 -Soiton soution In this section we construct a -soiton soution to the PT invariant noninear Schrödinger equation () with σ Such soution corresponds to soiton eigenvaues k i η k i η η > k i η k i η η > ; (94) 939

27 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani Substituting in equation (7) and set the refection coefficient ρ to zero we find η η q ( ) i σcn( k ) e + CN( k ) e (943) where C C are the norming constants (in ) whose time evoution is determined by 4iη t 4iη t C() t C() e C () t C () e (944) 4iη t 4iη t C() t C() e C () t C () e (945) and C C are given in (9) To determine the functiona form of the eigenfunctions N ( k ) and N ( k ) we sove the system of equations given in (637) and (64) with and After some engthy cacuations we arrive at the foowing set of agebraic equations satisfied by the soiton eigenfunctions ( ) γ η η N k e M( k ) γ e M( k ) (946) ( ) δ η η N k e M( k ) δ e M( k ) (947) ( ) γ η η N k e M ( k ) γ e M ( k ) (948) η η N ( k ) δ e M ( k ) δ e M ( k ) (949) η η M ( k ) + α e N( k ) + α e N( k ) (95) η η M ( k ) + β e N( k ) + β e N( k ) (95) η η M ( k ) α e N ( k ) + α e N ( k ) (95) η η M ( k ) β e N ( k ) + β e N ( k ) (953) where the quantities α β γ and δ are time dependent and defined as α β 4iη t ic () e ic() e α η + η η + η 4iη t 4iη t ic() e ic() e β η + η η + η 4iη t (954) (955) γ 4iη t C () e i( η + η ) γ 4iη t C () e i( η + η ) (956) δ 4iη t C () e i( η + η ) δ 4iη t C () e i( η + η ) (957) Soving system (946) (953) we get 94

28 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani η η η α( δ γ) e + β( δ γ) e + e N( G ( ) G ( ) where we defined the functions: G ( ) γα ( η + η ) + γβ ( η + η ) + [ δα η + δβ η e e e e + e η ] and ( ) ( ) G ( ) [ δα η + η + δβ η + η ] γα η e e e + γβe η (958) (959) (96) The soution for N ( k ) is given by N ( k ) η η η α ( γ δ ) e + β ( γ δ ) e + e H ( ) H ( ) (96) where we defined the functions η η η H ( ) γα + γβ + [ δα ( η + η ) + δβ ( η + η e e e e e ) + ] and (96) η ( ) ( ) H ( ) [ δα + δβ η η + η η + η ] e e γαe + γβe (963) Substituting equations (958) (96) into equation (943) yieds the -soiton soution Comparison with the cassica NLS equation In this part we briefy contrast the properties of the nonoca NLS () with that of the cassica (oca) NLS equation: i q( t) q ( t ) ± qt ( ) q( t ) t () obtained from equation () by repacing the term q * ( t) with q * ( t) In tabe we summarize and highight the key differences between the cassica and nonoca NLS equations Three different scenarios wi be addressed a of which concerning equation (): genera initia conditions posed on the whoe rea ine even initia conditions posed on the whoe rea ine and 3 genera initia conditions on the semi-infinite interva ( ) In [8] it was shown that () is integrabe on the whoe rea ine Furthermore it was found that the symmetries of the eigenfunctions of the associated Zakharov Shabat scattering probem are such that the eigenfunctions in the upper haf compe pane are reated to those in the ower haf pane This is in sharp contrast to the nonoca case where the eigenfunctions at the upper/ower haf pane are not reated On the other hand if one restricts the cass of initia conditions to be even (in ) then one obtains etra symmetry conditions on the scattering data that resembes the one we find This eads us to the important concusion that soiton soutions to () wi have a cassica NLS imit so ong () admits an even soiton soution 94

29 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani Tabe Symmetry reations between various quantities reated to the cassica and nonoca integrabe NLS equations Here σ Symmetry property Cassica NLS equation i q( t) q ( t ) ± qt ( ) q( t ) t Nonoca NLS equation i q( t) q ( t ) ± q ( tq ) ( t ) t Scattering data: ak ( ) a ( k ) bk ( ) σb ( k ) Eigenvaues: k k ak ( ) a ( k ) ak ( ) a ( k ) bk ( ) σb ( k ) a and a are not reated If k k are eigenvaues then k k are aso eigenvaues No reations between k and k Eigenfunctions Norming constants () N ( k ) N( () σn ( k ) () σm ( k ) M( () M ( k ) C σc free parameters; independent of k and k Nk ( ) ΛM ( k ) N( Λ M ( k ) σ Λ C C depend on k and k via equations (99) (93) and (97) Gaiean invariance ζ ζ qt ( ) q( ζt t) e i e i t ξ ξ qt ( ) q( + i ξt t) e e Finay we point out that simiar symmetry resuts were obtained in [7] for the cassica NLS () on the semi-infinite interva i t Some specia potentias: bo initia conditions In the previous sections we obtained pure soiton soutions for the nonoca NLS equation corresponding to refectioness potentias However more genera choices of initia data q( ) give rise to non zero refection coefficients in which case it is not (generay speaking) sovabe by inverse scattering transform As an eampe we consider in this section few specia cases of a bo initia conditions and compute the eigenvaues and conserved quantities Singe bo function The first eampe we consider the bo initia condition for < < q ( ) h for < < L for > () where h L are both rea and positive The initia condition for r( ) is obtained from () by imposing the symmetry condition r( ) q * ( ) ie for < < L r( ) h for L< < () for > We use the scattering probem 94

30 Noninearity 9 (6) 95 M Abowitz and Z H Mussimani v i kv+ q( tv ) (3) v i kv+ r( tv ) (4) with the scattering functions satisfying the boundary conditions (3) and (3) Since this is a inear constant coefficient second order differentia equation soution to system (3) and (4) is thus given by h ( ) k k v k ce i i + ce ik for < < L (5) v( k ce i v( v( ce ik ce h + ce ik k i i k for L< < (6) Matching the eigenfunctions at and L using the boundary conditions (3) and (3) gives h ikl c c i k e (7) h ikl h ( ) + ikl c e c ( e ) ik ik (8) On the other hand for > L we have from (3) and (48) ik ak ( ) e φ ( t ) for > L (9) i k bk ( ) e Matching the eigenfunctions at L gives the formua for the scattering data h ( ) ikl h ( ) 4ikL ikl ak + e ( e e ) ik ik () bk ( ) he ikl sin( kl) k () The asymptotic behavior of the scattering coefficient a( for arge and sma k are readiy obtained from () as h ak ( ) as k () ( i ak ( ) hl + Ok ( ) as k (3) The zeros of the scattering data a( are impicity given by soutions to iz e ± iz hl (4) 943