Dressing for a novel integrable generalization of the nonlinear Schr"odinger equation

DRESSING F OR A NO VEL INTEGRABLE GENERALIZA TION OF THE NONLINEAR SCHR ¨ ODINGER EQUA TION JONA T AN LENELLS Abstract. W e implemen t the dressing metho d for a no vel int egr able gen- eralization of the nonlinear Sc hr¨ odinger equation. As an application, explicit formulas for the N -soliton s olutions are derived. As a b y-pr o duct of the analy- sis, we find a simplificatio n of the formulas for the N -solitons of the deriv ative nonlinear Schr¨ odinger equation given b y Huang and Chen. AMS Subj ect Class ifica tion (2000) : 35Q55, 37K15 . Keywords : In tegrable system, inv erse sp ectral theory , dressing metho d, solitons. 1. Introduction W e consider the followin g in tegrable generalization of the no nlinear Sc hr¨ odinger (NLS) equation, whic h w as firs t deriv ed in [2] by means of bi-Hamiltonian meth- o ds: (1.1) iu t − ν u tx + γ u xx + σ | u | 2 ( u + iν u x ) = 0 , σ = ± 1 , where γ and ν are nonzero real para meters and u ( x, t ) is a complex-v alued function. Equation (1.1) reduces to the NLS equation when ν = 0. Ho w ev er, the limit ν → 0 is in man y resp ects singular, a nd t he analysis of (1.1) is rather differen t fro m that o f NLS. Equation (1.1) app ears a s a mo del for nonlinear pulse propagation in o pt ical fib ers pro vided that o ne re tains certain terms of the next asymptotic o rder b ey ond tho se necessary for t he NLS equation [2, 5]. F rom a bi-Hamiltonian p oin t of view, equation (1.1) is related to NLS in the same wa y that the celebrated Camassa-Holm equation [1] is related to KdV [2]. Being in tegrable, (1.1) admits a Lax pair form ulation and the initial-v alue problem on the line can b e analyzed by means of the in vers e scattering transform (IST) [6]. The sp ectral a nalysis of (1.1) is close ly related to that of the deriv ativ e NLS equation. In fact, equation (1.1) is r elated b y a g auge transformatio n to the first negativ e mem b er of the in tegrable hierarc hy asso ciated with the deriv ative NLS equation [5]. Th e initial-b oundary v alue problem for equation (1.1) on the half-line w as studied in [7]. Here, w e implemen t the dressing metho d for eq uation (1.1) and deriv e, as an a pplicatio n of the general approach, explicit form ulas for the N - solito n so- lutions. The dressing metho d is a tec hnique whic h generates new solutions of an inte gra ble equation f rom an a lready kno wn seed solution cf. [8]. In our implemen tation of the dressing metho d to equation (1.1), w e start with a seed solution u 0 together with a corresp onding eigenfunction ψ 0 of the L ax pair. A new eigenfunction ψ = Gψ 0 is constructed b y m ultiplying ψ 0 b y a 2 × 2-matrix G whic h con ta ins an ev en n umber o f p oles in the spectral parameter. It is sho wn that ψ satisfies the same Lax p air as ψ 0 but with the p o t ential u 0 replaced with 1 2 JONA T AN LENELLS a new p otential u ( x, t ). The compatibilit y of the La x pair equations then implies that u ( x, t ) a lso is a solution of (1.1). An express ion fo r the one-soliton solution of (1.1) w as found in [6] b y solving directly the R iemann-Hilb ert pro blem asso ciated with the in vers e problem in the presence of tw o p o les and no j ump (eac h soliton contributes two p o les to the R iemann-Hilb ert pro blem b ecause t he p oles ar e of necessit y symmetrically distributed with resp ect to the o rigin). Applying the dressing approach to the particular seed solution u 0 = 0 w e recov er the one-soliton solution and also find an explicit expression f or the N -soliton solution f o r an y N . A con v enien t L ax pair fo r (1.1) is introduced in Section 2, while the implemen- tation of the dressing metho d is describ ed in detail in Section 3. In Section 4 we deriv e explicit expressions for the N -solitons. Finally , in Section 5 , motiv ated b y the analysis of (1.1), w e find a simplification of the formula for the N -soliton solution give n b y Huang and Chen [3 ] of the deriv ative nonlinear Schr¨ odinger (DNLS) equation (1.2) iq t + q xx + i  | q | 2 q  x = 0 , x ∈ R , t ≥ 0 . The connection b et w een equations ( 1 .1) a nd ( 1.2) a rises b ecause the x -parts of the corresp onding Lax pairs are iden tical up on iden tification of u x with q . 2. A Lax p air It w as noted in [5] that the nonzero v alues of the pa r a meters γ , ν, σ in (1.1 ) can b e a r bitr arily assigned via a c hange of v a r ia bles. W e will therefore, for simplicit y , henceforth assume that γ = ν = σ = 1. Then the transformatio n u → e ix u conv erts (1.1) in to the equation (2.1) u tx + u − 2 iu x − u xx − i | u | 2 u x = 0 , whic h admits the follow ing Lax pair [6]: (2.2)    ψ x + i 1 ζ 2 σ 3 ψ = 1 ζ U x ψ , ψ t + i  1 ζ − ζ 2  2 σ 3 ψ =  1 ζ U x − i 2 σ 3 U 2 + iζ 2 σ 3 U  ψ , where ψ ( x, t, ζ ) is a 2 × 2-matrix v alued eigenfunction, ζ ∈ ˆ C = C ∪ {∞} is a sp ectral parameter, and (2.3) U ( x, t ) =  0 u ( x, t ) − ¯ u ( x, t ) 0  , σ 3 =  1 0 0 − 1  . W e will assume that ψ o b eys the symmetries σ 3 ψ ( x, t, − ζ ) σ 3 = ψ ( x, t, ζ ) , ψ − 1 ( x, t, ζ ) = ψ † ( x, t, ¯ ζ ) . (2.4) A solution ψ o f (2.2) satisfying (2.4) can b e constructe d as follo ws: Assuming that u has s ufficien t smoothness and deca y , there exists a solutio n of (2.2) whic h tends to exp( − iθ σ 3 ) at infinity , where (2.5) θ := θ ( x, t, ζ ) = 1 ζ 2 x +  1 ζ − ζ 2  2 t. DRESSIN G FOR A N OVEL INTEGRABLE GENER ALIZA TION 3 This suggests the transformation ψ = Ψ e − iθ σ 3 and then Ψ solve s (2.6)    Ψ x + i 1 ζ 2 [ σ 3 , Ψ] = 1 ζ U x Ψ , Ψ t + i  1 ζ − ζ 2  2 [ σ 3 , Ψ] =  1 ζ U x − i 2 σ 3 U 2 + iζ 2 σ 3 U  Ψ . W e define tw o solutions Ψ 1 and Ψ 2 of (2.6) via the linear V olterra integral equations Ψ 1 ( x, t, ζ ) = I + Z x −∞ e iζ − 2 ( x ′ − x ) ˆ σ 3 ζ − 1 ( U x Ψ 1 )( x ′ , t, ζ ) dx ′ , (2.7a) Ψ 2 ( x, t, ζ ) = I − Z ∞ x e iζ − 2 ( x ′ − x ) ˆ σ 3 ζ − 1 ( U x Ψ 2 )( x ′ , t, ζ ) dx ′ , (2.7b) where ˆ σ 3 acts on a 2 × 2 matrix A b y ˆ σ 3 A = [ σ 3 , A ]. The second columns of these equations inv olve s exp[2 iζ − 2 ( x ′ − x )]. It follows that t he second column v ectors of Ψ 1 and Ψ 2 are w ell- defined and analytic for ζ ∈ ˆ C suc h that ζ 2 lies in the upp er and low er half- planes, resp ectiv ely . Moreov er, these column v ectors ha v e con tin uous extensions to ζ ∈ R ∪ i R \ { 0 , ∞} . Similar remarks apply to the first column ve ctors. W e define a solution ψ of (2.2) by (2.8) ψ ( x, t, ζ ) = (  [Ψ 2 ] 1 , [Ψ 1 ] 2  e − iθ σ 3 , Im ζ 2 ≥ 0 ,  [Ψ 1 ] 1 , [Ψ 2 ] 2  e − iθ σ 3 , Im ζ 2 < 0 , where  [Ψ 2 ] 1 , [Ψ 1 ] 2  denotes the matrix consisting of the first column of Ψ 2 together with the second column of Ψ 1 etc. Then ψ admits the symmetries in (2.4). Indeed, uniqueness of solution of (2.2) implies that t here exist matrices S 1 ( ζ ) and S 2 ( ζ ) indep endent of x, t suc h that σ 3 ψ ( x, t, − ζ ) σ 3 = ψ ( x, t, ζ ) S 1 ( ζ ) , ψ − 1 ( x, t, ζ ) = S 2 ( ζ ) ψ † ( x, t, ¯ ζ ) , and ev aluation of these equations as x → ±∞ us ing (2.7) sho ws tha t S 1 = S 2 = I . As a function of ζ , the eigenfunction ψ defined b y (2.8) is singular for ζ ∈ { 0 , ∞} and has a jump across the con tour R ∪ i R of the form ψ − = ψ + J ( ζ ), where ψ ± denote the v alues o f ψ on the left and right sides of the con tour and J ( ζ ) is a 2 × 2 j ump matrix indep enden t of x and t . How ev er, these singularities are inconsequen tial fo r the argumen ts b elow (whic h use the com binations ψ x ψ − 1 and ψ t ψ − 1 ). In fa ct, Liouville’s theorem implies tha t ψ , in general, cannot b e analytic ov er the whole Riemann ζ -sphere. 3. The dressing method Starting from a seed solution u 0 of (2.1) and a corresp onding eigenfunction ψ 0 ob eying the symmetries (2.4), 1 w e seek a ‘dressed’ eigenfunction ψ of the form (3.1) ψ = Gψ 0 , 1 As o utlined in the previous section, such a ψ 0 can be cons tructed fro m u 0 using only linear op erations. 4 JONA T AN LENELLS where the 2 × 2 - matrix v alued function G has the fo rm (3.2) G ( x, t, ζ ) = I + N X j =1  A j ( x, t ) ζ − ζ j − σ 3 A j ( x, t ) σ 3 ζ + ζ j  . Here { ζ j , − ζ j } N 1 ⊂ ˆ C \ { 0 , ∞} is a collection of simple p oles with corresp onding residues { A j ( x, t ) , − σ 3 A j ( x, t ) σ 3 } N 1 . The form (3 .2 ) of G is motiv ated by the condition tha t G → I as ζ → ∞ and b y the symmetry (3.3) σ 3 G ( x, t, − ζ ) σ 3 = G ( x, t, ζ ) , whic h asce rtains that the first symmetry in (2.4) is preserv ed b y (3.1 ). Moreov er, in order for the second symmetry in (2.4) to b e preserv ed b y (3.1), w e require that (3.4) G − 1 ( x, t, ζ ) = G † ( x, t, ¯ ζ ) , i.e. (3.5) G − 1 ( x, t, ζ ) = I + N X j =1 " A † j ( x, t ) ζ − ¯ ζ j − σ 3 A † j ( x, t ) σ 3 ζ + ¯ ζ j # . It follows from ( 3 .2) and (3.5) that G and G − 1 are analytic at b oth ζ = 0 and ζ = ∞ . The g o al no w is to sho w that ψ satisfies (2 .2) for some matrix U of the form (2.3). The compatibilit y of the Lax pair equations will then sho w t hat u := U 12 is a solution of equation (2.1). Differen tiation of (3.1) with resp ect to x and t follow ed by m ultiplication by ψ − 1 = ψ − 1 0 G − 1 leads to (3.6a) ψ x ψ − 1 = G x G − 1 + Gψ 0 x ψ − 1 0 G − 1 and (3.6b) ψ t ψ − 1 = G t G − 1 + Gψ 0 t ψ − 1 0 G − 1 , resp ectiv ely . 3.1. Analysis at ζ = 0 . W e r eplace ψ 0 x ψ − 1 0 and ψ 0 t ψ − 1 0 in (3.6) with the fol- lo wing expressions whic h follo w fro m the Lax pair equations satisfied by ψ 0 : (3.7)    ψ 0 x ψ − 1 0 = − i 1 ζ 2 σ 3 + 1 ζ U 0 x , ψ 0 t ψ − 1 0 = − i  1 ζ − ζ 2  2 σ 3 + 1 ζ U 0 x − i 2 σ 3 U 2 0 + iζ 2 σ 3 U 0 . Substituting in to the resulting equations the expansions G = G 0 + G 1 ζ + O ( ζ 2 ) , G − 1 = G − 1 0 − G − 1 0 G 1 G − 1 0 ζ + O ( ζ 2 ) , ζ → 0 , where G 0 ( x, t ) and G 1 ( x, t ) are indep enden t of ζ , it follows that ψ x ψ − 1 and ψ t ψ − 1 ha v e double p oles at ζ = 0; iden tification of terms of O (1 /ζ 2 ) a nd O (1 /ζ ) yields ψ x ψ − 1 = Q − 2 ζ 2 + Q − 1 ζ + O (1 ) , ζ → 0 , (3.8) ψ t ψ − 1 = Q − 2 ζ 2 + Q − 1 ζ + O (1 ) , ζ → 0 , DRESSIN G FOR A N OVEL INTEGRABLE GENER ALIZA TION 5 where Q − 2 = − iG 0 σ 3 G − 1 0 , (3.9) Q − 1 = iG 0 σ 3 G − 1 0 G 1 G − 1 0 − iG 1 σ 3 G − 1 0 + G 0 U 0 x G − 1 0 . On the other hand, from (3.2) we find the follow ing expressions for G 0 and G 1 : G 0 = I − N X j =1 A j + σ 3 A j σ 3 ζ j , G 1 = − N X j =1 A j − σ 3 A j σ 3 ζ 2 j , so that the matrices G 0 and G 1 are diagonal and off- dia gonal, respectiv ely . T hus the expressions for Q − 2 and Q − 1 in (3.9 ) simplify to (3.10) Q − 2 = − iσ 3 , Q − 1 = − 2 iG 1 G − 1 0 σ 3 + G 0 U 0 x G − 1 0 . In particular, Q − 1 is an off-diagonal matrix. 3.2. Analysis at ζ = ∞ . W e a g ain replace ψ 0 x ψ − 1 0 and ψ 0 t ψ − 1 0 in (3.6) with the expressions in (3.7). Substituting into the resulting equations the expansions G = I + G − 1 ζ + G − 2 ζ 2 + O (1 / ζ 3 ) , ζ → ∞ , G − 1 = I − G − 1 ζ + G 2 − 1 − G − 2 ζ 2 + O (1 / ζ 3 ) , ζ → ∞ , (3.11) where G − 1 and G − 2 are indep enden t of ζ , it follow s that ψ x ψ − 1 → 0 as ζ → ∞ whereas ψ t ψ − 1 has a double p ole at ζ = ∞ ; identific atio n of terms of O ( ζ n ), n = 0 , 1 , 2, yields ψ t ψ − 1 = − i 4 σ 3 ζ 2 + Q 1 ζ + Q 0 + O (1 / ζ ) , ζ → ∞ , where Q 1 = i 4 [ σ 3 , G − 1 ] + i 2 σ 3 U 0 , (3.12) Q 0 = i 4 [ σ 3 , G − 2 ] − i 4 σ 3 G 2 − 1 + i 4 G − 1 σ 3 G − 1 + i 2 G − 1 σ 3 U 0 − i 2 σ 3 U 0 G − 1 + iσ 3 − i 2 σ 3 U 2 0 . On the o ther hand, from (3.2) we find t he follo wing expressions fo r G − 1 and G − 2 : G − 1 = N X j =1 ( A j − σ 3 A j σ 3 ) , G − 2 = N X j =1 ζ j ( A j + σ 3 A j σ 3 ) , (3.13) so that the matr ices G − 1 and G − 2 are off- diagonal and diagonal, resp ectiv ely . Th us the expressions for Q 1 and Q 0 in (3.12) simplify to Q 1 = i 2 σ 3 U, Q 0 = − i 2 U 2 σ 3 + iσ 3 , (3.14) where we ha v e defined (3.15) U := G − 1 + U 0 . 6 JONA T AN LENELLS Equations ( 3 .5) and (3 .1 1) yield the fo llowing alternativ e expression for G − 1 : G − 1 = − N X j =1  A † j − σ 3 A † j σ 3  . Th us ( G − 1 ) 21 = − ( G − 1 ) 12 , so that U is of the fo rm (2.3) w ith u := ( G − 1 ) 12 + u 0 . 3.3. The generated solution. It follo ws from the previous t w o subsections that the functions (3.16a) ψ x ψ − 1 −  Q − 2 ζ 2 + Q − 1 ζ  and (3.16b) ψ t ψ − 1 −  Q − 2 ζ 2 + Q − 1 ζ + Q 0 + Q 1 ζ − i 4 σ 3 ζ 2  are analytic near ζ = 0 and ζ = ∞ . Thus , in view of (3.6) and t he expressions (3.2) and (3.5) for G and G − 1 , they are analytic on the whole Riemann ζ - sphere except p ossibly at p oints in the set {± ζ j , ± ¯ ζ j } N 1 . W e claim that the A j ’s in (3.2) can b e c hosen so that the functions in (3.1 6) are analytic ev erywhere. Assume that suc h a choice has b een made. Then, since the functions in (3.16) tend to zero as ζ → ∞ , Liouville’s theorem implies that they b oth v anish iden tically . T ogether with the expressions (3.10) and (3.14) for the co efficien ts Q − 2 , Q 0 , Q 1 , this implies that ψ satisfies the follo wing pair of equations: (3.17)    ψ x + i 1 ζ 2 σ 3 ψ = 1 ζ Q − 1 ψ , ψ t + i  1 ζ − ζ 2  2 σ 3 ψ =  1 ζ Q − 1 − i 2 σ 3 U 2 + iζ 2 σ 3 U  ψ , where U has the form (2.3). Using that Q − 1 is off-diagonal and iden tifying terms of O ( ζ ) in the compatibility equation ψ xt = ψ tx , w e find that (3.18) Q − 1 = U x . Consequen tly , (3 .1 7) ha s exactly the form of the Lax pair ( 2 .2) a nd its compati- bilit y implies that the function u = U 12 satisfies (2.1). W e infer from (3.13) and (3.15) tha t the g enerated solution u ( x, t ) is g iven b y (3.19) u ( x, t ) = 2 N X j =1 ( A j ( x, t )) 12 + u 0 ( x, t ) . 3.4. The dressing transformation. In order to complete the construction of the map u 0 7→ u , it only r emains to determine the structure of the A j ’s in (3.19) so that the functions ψ x ψ − 1 and ψ t ψ − 1 are regular a t the p o in ts {± ζ j , ± ¯ ζ j } N 1 . T o this end w e consider the f ollo wing series of N consecutiv e dressing tr ansfor- mations, each of whic h a dds tw o p oles: (3.20) G = D N D N − 1 · · · D 1 , ψ j = D j ψ j − 1 , j = 1 , . . . , N , DRESSIN G FOR A N OVEL INTEGRABLE GENER ALIZA TION 7 where, fo r j = 1 , . . . , N , D j ( x, t, ζ ) = I + B j ( x, t ) ζ − ζ j − σ 3 B j ( x, t ) σ 3 ζ + ζ j , (3.21a) D − 1 j ( x, t, ζ ) = I + B † j ( x, t ) ζ − ¯ ζ j − σ 3 B † j ( x, t ) σ 3 ζ + ¯ ζ j , (3.21b) and { B j ( x, t ) } N 1 are a set of 2 × 2-matrix v alued functions indep enden t o f ζ . Lemma 3.1. L e t { b j } N 1 b e a set of N nonzer o c omplex c onstants. Define the functions B j ( x, t ) , j = 1 , . . . , N , inductively by (3.22) B j ( x, t ) = | z j ( x, t ) ih y j ( x, t ) | , wher e the c olumn ve ctor | z j i = h z j | † and the r ow ve ctor h y j | = | y j i † ar e defi n e d in terms o f the ( j − 1) th eigenfunction ψ j − 1 by 2 h y j | =  b j b − 1 j  ψ − 1 j − 1 ( ζ j ) , (3.23) | z j i = ζ 2 j − ¯ ζ 2 j 2  α j 0 0 ¯ α j  | y j i , wher e α − 1 j = h y j |  ζ j 0 0 ¯ ζ j  | y j i . (3.24) Define ψ := ψ N by (3.20)-(3.24). Then the functions ψ x ψ − 1 and ψ t ψ − 1 ar e analytic at the p oints in the set {± ζ j , ± ¯ ζ j } N 1 . Pr o of. The pr o of pro ceeds b y induction. Supp o se that ψ j − 1 has been defined by (3.20)-(3.24) and that the func tions ψ j − 1 , x ψ − 1 j − 1 and ψ j − 1 , t ψ − 1 j − 1 are analytic at ζ = ± ζ i and ζ = ± ¯ ζ i for i = 1 , . . . , j − 1. W e will sho w that ψ j x ψ − 1 j and ψ j t ψ − 1 j , where ψ j = D j ψ j − 1 , are analytic at ζ = ± ζ i and ζ = ± ¯ ζ i for i = 1 , . . . , j . Differen tiation of ψ j = D j ψ j − 1 yields (3.25) ψ j x ψ − 1 j = D j x D − 1 j + D j ψ j − 1 , x ψ − 1 j − 1 D − 1 j . The assump tion on ψ j − 1 together with (3.2 1) implies that the righ t-hand side is analytic at the p oin ts {± ζ i , ± ¯ ζ i } j − 1 1 . T o show that ψ j x ψ − 1 j is regular at ζ j , we note that (3.25) implies that ψ j x ψ − 1 j has at most a simple p o le at ζ j , and that (3.26) Res ζ j  ψ j x ψ − 1 j  = ( B j ψ j − 1 ( ζ j )) x ψ − 1 j − 1 ( ζ j ) D − 1 j ( ζ j ) . Using (3.22 ) and (3.23), we can write (3.26) as Res ζ j  ψ j x ψ − 1 j  = | z j i x h y j | D − 1 j ( ζ j ) . W e claim that (3.27) h y j | D − 1 j ( ζ j ) = 0 , so that the residue v a nishes. Indeed, b y (3.21b), (3.28) h y j | D − 1 j ( ζ j ) = h y j | + h y j | y j i ζ j − ¯ ζ j h z j | − h y j | σ 3 | y j i ζ j + ¯ ζ j h z j | σ 3 , and the righ t- ha nd side of this equation v anishes b y virtue of (3.24) . Since the symmetry prop erties (2.4) are preserv ed b y (3.20 ) , w e deduce that ψ j x ψ − 1 j 2 Here and in so me eq uations b elow the ( x , t )-dep endence is suppressed. 8 JONA T AN LENELLS is analytic also at − ζ j and ± ¯ ζ j . Similar argumen ts establish the regularit y of ψ j t ψ − 1 j . ✷ By (3.2) and (3.20 ) , (3.29) A j = Res ζ j G ( ζ ) = D N ( ζ j ) . . . D j +1 ( ζ j ) B j D j − 1 ( ζ j ) . . . D 1 ( ζ j ) . On the other hand, it follo ws from Lemma 3.1 that (3.30) σ 2 B T j ( x, t ) σ 2 = ζ 2 j − ¯ ζ 2 j 2 ζ j D − 1 j ( x, t, ζ j ) , σ 2 =  0 − i i 0  , and (3.31) σ 2 D j ( x, t, ζ ) T σ 2 = ζ 2 − ¯ ζ 2 j ζ 2 − ζ 2 j D − 1 j ( x, t, ζ ) . Using (3.30 ) and (3.31), the expression (3.2 9) for A j can b e written a s (3.32) A j ( x, t ) = 1 a j σ 2 G − 1 ( x, t, ζ j ) T σ 2 where the complex constan ts { a j } N 1 are defined b y (3.33) a j = 2 ζ j ζ 2 j − ¯ ζ 2 j Y k 6 = j ζ 2 j − ζ 2 k ζ 2 j − ¯ ζ 2 k . Substitution of (3.32) in to the expression (3.2) for G yields (3.34) G ( x, t, ζ ) = I + N X j =1 1 a j  σ 2 G − 1 ( x, t, ζ j ) T σ 2 ζ − ζ j − σ 3 σ 2 G − 1 ( x, t, ζ j ) T σ 2 σ 3 ζ + ζ j  , Eliminating A j from (3.29) and (3.32), and using Lemma 3.1 together with the fact that ψ − 1 j − 1 D j − 1 · · · D 1 = ψ − 1 0 in the resulting equation, we find that (3.35) σ 2 G − 1 ( x, t, ζ j ) T σ 2 =  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) for some functions r j ( x, t ) , s j ( x, t ), j = 1 , . . . , N . Ev aluat ing equation (3.34) at ζ = ¯ ζ k , and using (3.35) w e find G ( x, t, ¯ ζ k ) = I + N X j =1 1 a j  1 ¯ ζ k − ζ j  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) (3.36) − 1 ¯ ζ k + ζ j σ 3  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) σ 3  . But the symmetry (3.4 ) together with (3.35 ) sho ws that (3.37) G ( x, t, ¯ ζ k ) = σ 2  r k ( x, t ) s k ( x, t )   ¯ b k ¯ b − 1 k  ψ − 1 0 ( x, t, ζ k ) σ 2 , DRESSIN G FOR A N OVEL INTEGRABLE GENER ALIZA TION 9 so that m ultiplying equation (3.36) by σ 2 ψ 0 ( x, t, ζ k )  ¯ b − 1 k − ¯ b k  T from the right, the left-ha nd side disapp ears and w e are left with 0 = σ 2 ψ 0 ( x, t, ζ k )  ¯ b − 1 k − ¯ b k  + N X j =1 1 a j  1 ¯ ζ k − ζ j  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) (3.38) − 1 ¯ ζ k + ζ j σ 3  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) σ 3  σ 2 ψ 0 ( x, t, ζ k )  ¯ b − 1 k − ¯ b k  . These are 2 N alg ebraic scalar equations fo r the 2 N unknown s { r j ( x, t ) , s j ( x, t ) } N 1 . Giv en the seed eigenfunction ψ 0 , the solution of (3.38) expresses the r j ’s and the s j ’s in terms of { ζ j , b j } N 1 . Th us, b y (3.32) and (3.35), the A j ’s are determined in terms of the 2 N complex constan ts { ζ j , b j } N 1 . This yields the follo wing result. Prop osition 3.2. L et N b e a p ositive inte ger and let { ζ j , b j } N j =1 b e non z er o c om plex c onstants such t hat ζ j 6 = ζ k for j 6 = k . Assume that u 0 ( x, t ) satisfies e quation (2.1) and let ψ 0 ( x, t, ζ ) b e an asso ciate d eigenfunction ob eying the sym- metries (2. 4). Then the fol lowing function u ( x, t ) is also a solution of e quation (2.1): (3.39) u ( x, t ) = 2 N X j =1 ( A j ( x, t )) 12 + u 0 ( x, t ) , wher e the 2 × 2 -matrix va l ue d function A j ( x, t ) is give n by (3.40) A j ( x, t ) = 1 a j  r j ( x, t ) s j ( x, t )   b j b − 1 j  ψ − 1 0 ( x, t, ζ j ) , the c ons tants { a j } N 1 ar e given by (3.33), and { r j ( x, t ) , s j ( x, t ) } N j =1 ar e determine d by the line a r algebr a i c system of e quations (3.38). 4. Solitons In this section w e a pply the dress ing method describ ed in the previous section to the part icular seed solutio n u 0 = 0 with corresp onding eigenfunction ψ 0 = exp( − iθ σ 3 ). The addition o f 2 N p oles to ψ 0 yields the N -soliton solution of (2 .1). Since in this case the solution of the algebraic system (3.38) can b e e xpressed in a simple form, w e find simple explicit form ulas for the solitons. Recall that θ = θ ( x, t, ζ ) w as defined in (2 .5). Substituting u 0 = 0 and ψ 0 = exp( − iθ σ 3 ) in to the equations ( 3.39) and (3.40), we find (4.1) u ( x, t ) = 2 N X j =1 r j a j f j , where f j := f j ( x, t ) = b j e iθ ( x, t,ζ j ) , j = 1 , . . . , N . On the o t her hand, for this c hoice of ψ 0 , the top entry of equation (3.38) yields (4.2) 2 N X j =1 r j K j k a j f j = ¯ f 2 k , k = 1 , . . . , N , 10 JONA T AN LENELLS where the en tries of the N × N -matrix K = K ( x, t ) are defined b y (4.3) K j k = 1 ζ 2 j − ¯ ζ 2 k ( ζ j f 2 j ¯ f 2 k + ¯ ζ k ) , j, k = 1 , . . . , N . Com bining (4.1 ) and (4.2), w e find the f ollo wing expression for the N - soliton solution. Prop osition 4.1. The N -soliton solution u N ( x, t ) of the gener aliz e d NLS e qua- tion (2.1) is given explic itly by (4.4) u N ( x, t ) = N X k ,j = 1 ¯ f 2 k ( K − 1 ) k j , wher e the N × N -matrix K = K ( x, t ) is defi ne d by (4.3), the c o effici e nts { f j } N 1 ar e given by (4.5) f j := f j ( x, t ) = b j exp i 1 ζ 2 j x + i  1 ζ j − ζ j 2  2 t ! , j = 1 , . . . , N , and the s o lution dep ends on the 2 N c omplex p ar ameters { b j , ζ j } N 1 . 5. D NLS solitons The deriv at iv e nonlinear Schr¨ odinger equation (1 .2 ) admits the La x pair [4 ] (5.1) ( ψ x + i 1 ζ 2 σ 3 ψ = 1 ζ Qψ , ψ t + 2 i 1 ζ 4 σ 3 ψ =  2 ζ 3 Q − i ζ 2 Q 2 σ 3 − 1 ζ ( iQ x σ 3 − Q 3 )  ψ , where the 2 × 2 - matrix v alued function Q ( x, t ) is giv en b y (5.2) Q =  0 q − ¯ q 0  . If w e iden tify Q with U x , t hen the x -part of this Lax pair coincides with the x -part of the Lax pair (2.2). A similar analysis that led to expression (4.4) for u N applied to equations (3.10) and (3.18) yields (cf. [3]) (5.3) u N x = − 2 i N X k ,j = 1 ¯ f 2 k ζ 2 j ( K − 1 ) k j ! 1 + N X k ,j = 1 1 ¯ ζ k ( K − 1 ) k j ! , where K and the f j ’s are defined as in Prop osition 4.1. In view of the iden ti- fication of Q with U x , it is natural to exp ect the N - soliton solution q N of the DNLS equation to b e giv en by an expression similar to the righ t-hand side of (5.3). In fact, it w as sho wn in [3 ] that q N ( x, t ) is giv en b y the rig ht-hand s ide of (5.3) with f j ( x, t ) replaced by (5.4) f j ( x, t ) = b j e i ( ζ − 2 j x +2 ζ − 4 j t ) . No w equations (4.4) and (5.3) imply that the right-hand side of (5.3) is t he x -deriv ative o f the right-hand side of (4.4) when f j is giv en by (4.5). Since all the time-dependence in these equations lies in the exp onen ts of f j and ¯ f j and is irrelev ant for the differentiation with res p ect to x , this prop ert y holds also when DRESSIN G FOR A N OVEL INTEGRABLE GENER ALIZA TION 11 the f j ’s are give n b y (5.4). This yields the following simple expressions f o r the DNLS N - solitons. Prop osition 5.1. The N -so liton solution q N ( x, t ) o f the D NLS e quation (1.2) is give n explicitly by (5.5) q N ( x, t ) = ∂ ∂ x N X k ,j = 1 ¯ f 2 k ( K − 1 ) k j ! , wher e the N × N -matrix K = K ( x, t ) is defi ne d by (4.3), the c o effici e nts { f j } N 1 ar e define d by (5.4), and the solution dep ends on the 2 N c om plex p ar ameters { b j , ζ j } N 1 . It can be c hec k ed explicitly for small v alues of N that the solitons u N and q N giv en b y (4 .4) and (5.5) indeed satisfy the g eneralized NLS equation (2 .1) and the DNLS equation (1.2), resp ectiv ely . 6. Conclusions W e hav e implemen ted t he dressing metho d to equation (2.1), whic h is equiv- alen t to the generalized NLS equation (1.1). This prov ides a w a y of generating new solutions from a lr eady kno wn ones. In part icular, starting with the trivial solution u 0 ≡ 0, w e arriv ed at the explicit ex pression (4.4) for the N -soliton solution. Since (1.1) is related by a gauge transformation to a mem b er of the DNLS hierarch y , the construction b ears similarities with the implemen tatio n of the dres sing metho d to DNLS [3] (see a lso [10]). The prese nce of additional sin- gularities in the t -part of the Lax pair mak es the argumen t for (2.1) somewhat more complicated. The link betw een equations (1.1) and (1.2) allo w ed us to find a simplifica- tion (see equation (5.5)) of the form ulas for the DNLS N - solitons presen ted in [3]. The fact that the righ t-hand side of (5.5) c an b e expressed as a partia l x -deriv ative is related to the fact tha t the DNLS hierarc h y can b e reformulated in terms of the p oten tial u = ∂ − 1 x q . Consequen tly , the solitons of mem b ers of this hierarch y can b e express ed as x -deriv ativ es of elemen tary functions cf. [9]. Ac knowle dgemen t The author thanks the r efer e es for valuable suggestions, A. S. F okas for helpful disc ussi o ns, and T. Tsuchida for bringing the r efer enc e [9] to his attention. Reference s [1] R. Camassa and D. D. Holm, An in tegrable shallo w wa ter equation with p eake d solitons, Phys. R ev . L ett. 71 ( 1993), 1661–16 64. [2] A. S. F ok as, On a class of physically im p ortan t inte grable equations, Phys. D 87 (1995 ), 145–15 0. [3] N. N. Huang and Z. Y. Chen, Alfv ´ en solito ns, J. Phys. A 23 (1990), 439–453 . [4] D. J. Kaup and A. C. New ell, An exact solution for a deriv ativ e nonlin ear Sc hr¨ odinger e quation, J. M ath. Phys. 19 (197 8), 798–8 01. [5] J. Lenells, Exact ly solv able mo del for nonlin ear pulse propagation in optical fib ers, Stud. Appl. Math. 123 (2 009), 215–232. 12 JONA T AN LENELLS [6] J. Lenells and A. S . F ok as, On a n o v el in tegrable generalizatio n of the n onlinear Sc hr¨ odinger e quation, Nonline arity 22 (2 009), 11–27. [7] J. Lenells and A. S. F ok as, An int egrable generalization of the nonlinear Sc hr¨ odinger equation on the half-line and solitons, Inverse pr oblems 25 (2009), 11500 6. [8] S. No viko v, S. V. Manak o v, L. P . Pitae vskii, and V. E . Zakh arov, Theory of solitons. The inv erse s cattering metho d. C on s ultan ts Bureau [Plenum], New Y ork, 1984. [9] T. Ts uc hida, New redu ctions of integrable matrix PDEs: S p ( m )-inv arian t sys- tems, J. Math. P hys. , to app ear. [10] Y, Xiao, Note on the Darb oux transformation for the deriv ativ e nonlinear Sc hr¨ odinger e quation, J. P hys. A 24 (199 1), 363– 366. Institut f ¨ ur Angew andte Ma thema tik, Leibniz Univ ersit ¨ at Hannover, Welfen- gar ten 1, 3016 7 Hannover, G ermany E-mail a ddr ess : l enells@ ifam.u ni-hannover.de