On the Multi-Component Nonlinear Schr"odinger Equation with Constant Boundary Conditions

On the m ulti-comp onen t nonlinear Sc hr¨ odinger equation with constan t b oundary conditions V. A. A tanasov, V. S. Gerdjik o v Institute of Nucle ar R ese ar ch and Nucle ar Ener gy, Bulgarian A c ademy of Scienc es, Sofia 1784, Bulgaria e-mail: victor@inrne.bas.bg e-mail: gerjiko v@inrne.bas.bg Abstract The m ulti-comp onen t nonlinear Schr¨ odinger equation related to C . I ' S p (2 p ) /U ( p ) and D . I II ' S O (2 p ) /U ( p )-type symmetric spaces with non-v anishing b oundary conditions is solv- able with the inv erse scattering metho d (ISM). As Lax op erator L we use the generalized Zakharo v-Shabat operator. W e show that the ISM for the Lax op erator L ( x, λ ) is a nonlinear analog of the F ourier-transform metho d. As appropriate generalizations of the usual F ourier- exp onen tial functions we use the so-called ”squared solutions”, whic h are constructed in terms of the fundamental analytic solutions (F AS) χ ± ( x, λ ) of L ( x, λ ) and the Cartan-W eyl basis of the Lie algebra, relev an t to the symmetric space. W e derive the completeness relation for the ”squared solutions” which turns out to provide sp ectral decomp osition of the recursion (generating) op erators Λ ± , a natural generalizations of 1 i d dx in the case of nonlinear evolution equations (NLEE). 1 In tro duction The in tegrability of the scalar nonlinear Sc hr¨ odinger equation (NLS) with v anishing b oundary conditions (v.b.c.): iq t + q xx + 2 | q ( x, t ) | 2 q ( x, t ) = 0 (1.1) w as discov ered b y Zakharov and Shabat in their pioneer work [21]. Soon after [22] Zakharo v and Shabat pro ved the in tegrability and the ph ysical imp ortance of the NLS with constan t b oundary conditions (c.b.c.): iq t + 2 q xx − 2( | q ( x, t ) | 2 − ρ 2 ) q ( x, t ) = 0 , lim x →±∞ q ( x, t ) = q ± , (1.2) where the asymptotic v alues q ± satisfy | q ± | 2 = ρ 2 . Notice the sign difference in the cubic nonlin- earit y as well as the additional term with the chemical p oten tial ρ . Both versions of NLS equation serv ed as mo dels on which generalizations were made. The simplest non-trivial m ulticomp onen t generalization of NLS is the vector NLS kno wn as the Manako v mo del [16]: i − → q t + − → q xx + 2( − → q † − → q ( x, t )) − → q ( x, t ) = 0 , (1.3) where − → q ( x, t ) is an n -component complex-v alued v ector v anishing fast enough for x → ±∞ . The c.b.c. version of v ector NLS i − → q t + − → q xx − 2  ( − → q † − → q ( x, t )) − ρ 2  − → q ( x, t ) = 0 , (1.4) 1 where lim x →±∞ − → q ( x, t ) = − → q ± and − → q − = U 0 − → q + where U 0 is constant unitary matrix also finds applications. Here ρ 2 = − → q † ± − → q ± . Equations (1.1) and (1.3) are particular cases of matrix NLS whic h is obtained from the system: i q t + q xx + 2 qrq ( x, t ) = 0 , − i r t + r xx + 2 rqr ( x, t ) = 0 , (1.5) after imp osing appropriate inv olution (reduction) compatible with the ev olution of (1.5). Here q and r are n × m matrix-v alued functions of x and t . One such in volution is: r = B − q † B − 1 + , B ± = diag (  ± 1 , ...,  ± m ) , (  ± 1 ) 2 = 1 , (1.6) and the corresponding MNLS acquires the form: i q t + q xx + 2 q B − q † B − 1 + q = 0 , (1.7) F or n = m = 1 and r = q ∗ the system goes into the scalar NLS (1.1); for m = 1 and n > 1 and with appropriate choice of in volution (1.6) the system is transformed into the Manak ov model (1.3). All these versions are solv able with the ISM. The ISM is applicable to nonlinear evolution equations (NLEE) if they can b e represen ted as compatibilit y condition of t wo linear problems [18, 20, 1, 2]: [ L ( λ ) , M ( λ )] = 0 , (1.8) whic h holds identically with resp ect to the sp ectral parameter λ . The t wo linear op erators L ( λ ) and M ( λ ) in the Zakharov-Shabat system (Z-Sh) for the MNLS on symmetric spaces asso ciated with the simple Lie algebra g ' C r and g ' D r with (v.b.c.) are: Lψ =  i ∂ ∂ x + Q ( x, t ) − λ σ 3  ψ ( x, t, λ ) = 0 , (1.9) M ψ =  i ∂ ∂ t + V 2 ( x, t ) + λV 1 ( x, t ) − 2 λ 2 σ 3  ψ ( x, t, λ ) = 0 , (1.10) Q ( x, t ) =  0 q ( x, t ) r ( x, t ) 0  , σ 3 =  1 1 0 0 − 1 1  , (1.11) where Q ( x, t ) and σ 3 are 2 r × 2 r matrices with compatible blo c k structure. Here V 1 ( x, t ) = 2 Q ( x, t ) , V 2 ( x, t ) = [ad − 1 σ 3 Q, Q ] + 2 i ad − 1 σ 3 Q x ( x, t ) (1.12) and ad − 1 σ 3 is the in v erse of the adjoint action ad σ 3 with respect to the element σ 3 : ad σ 3 Y = [ σ 3 , Y ]. An effective to ol to obtain new versions of MNLS is the reduction group in tro duced by Mikhailo v [17]. It allo ws one to imp ose algebraic constrain ts on the p oten tial Q ( x, t ) which are automatically compatible with the evolution. F or example, the inv olution (1.6), whic h leads to MNLS with v.b.c. (1.7) is kno wn as Z 2 -reduction and can b e written as [10]: B U † ( x, t, λ ∗ ) B − 1 = U ( x, t, λ ) (1.13) where B is an automorphism of g matrix such that B 2 = 1 1 , [ σ 3 , B ] = 0 , and U ( x, t, λ ) = Q ( x, t ) − λ σ 3 . (1.14) Belo w w e analyze the multi-component nonlinear Sch¨ odinger equation (MNLS): i q t + q xx − 2 qq † q + q µ + µ q = 0 , (1.15) 2 with c onstant b oundary conditions (c.b.c.) at x → ±∞ : lim x →±∞ q ( x, t ) = q ± , µ = q + q † + = q − q † − , µ = q † + q + = q † − q − , (1.16) where q ( x, t ) is n × r matrix-v alued function, related to A . II I , C . I or D . I I I -t yp e symmetric spaces. The case n 6 = r can b e related to A . I I I symmetric spaces only and has b een solved with the ISM in [11]. Therefore we concen trate on the MNLS (1.15) related to C . I or D . I I I -t yp e symmetric spaces, whic h means in particular that n = r . Its Lax pair is obtained from (1.9)–(1.12) b y replacing V 2 ( x, t ) with: V 2 ( x, t ) = [ad − 1 σ 3 Q, Q ] + 2 i ad − 1 σ 3 Q x ( x, t ) − σ 3 Q 2 ± . (1.17) Here w e ha ve also imposed the additional condition Q 2 + = Q 2 − . It ensures that the t wo asymptotic Lax operators L ± = i d dx + Q ± − λσ 3 ha ve the same spectrum. It also ensures that the p oten tials V 1 ( x, t ) and V 2 ( x, t ) in the second operator M ( λ ) v anish for x → ±∞ . As a result the solutions of the MNLS (1.15) q ( x, t ) do not undergo strong oscillations with resp ect to time, see [11, 9]. Lax op erators of the form (1.9) can b e asso ciated with each of the symmetric spaces listed b elo w (for the definition see [12] and the App endix). They are defined by sp ecifying the simple Lie algebra g , ha ving typical representation in 2 r × 2 r matrices and the Cartan subalgebra element σ 3 : • C . I : g ' C r ' sp (2 r ), σ 3 = H ~ a , where the v ector ~ a in the ro ot space E r dual to σ 3 is given b y ~ a = P r k =1 e k • D . II I : g ' D r ' so (2 r ), σ 3 = H ~ a , where the v ector ~ a in the ro ot space E r dual to σ 3 is giv en b y ~ a = P r k =1 e k Here the orthonormal vectors e k span the ro ot space E r of both types of algebras. The elemen t σ 3 b elongs to the Cartan subalgebra h and is dual to ~ a . Using σ 3 w e can split the set of positive ro ots into tw o tw o subsets 4 + = 4 + 0 ∪ 4 + 1 . These sets, for the algebras that we are w orking with, are comp osed of the following roots: 4 + 0 ≡ { e i − e j , 1 ≤ i < j ≤ r } , 4 + 1 ≡ { 2 e i , e i + e j , 1 ≤ i < j ≤ r } (1.18) for g ' sp (2 r ) and 4 + 0 ≡ { e i − e j , 1 ≤ i < j ≤ r } , 4 + 1 ≡ { e i + e j , 1 ≤ i < j ≤ r } (1.19) for g ' so (2 r ). The ro ot vectors of the algebra are denoted b y E α where α is the corresponding root. Let us introduce a pro jector P σ 3 = ad − 1 σ 3 ad σ 3 on to the co-adjoin t orbit O σ 3 of the element σ 3 . Here the inv erse of the adjoint action is ad − 1 σ 3 Y = 1 2 σ 3 Y . The generic element of X ∈ O σ 3 is the one that satisfies the relation X = P σ 3 X . Ob viously the potential of the Z-Sh system Q ( x, t ) and its v ariation δ Q ( x, t ) belong to O σ 3 . This pap er extends the results of [11, 9]. In Section 2 w e fo cus on the solutions of the direct scattering problem for the case of Lax op erator describing MNLS with c.b.c.. In Section 3 we deriv e the completeness relation for the ”squared solutions” of the Lax op erator generalizing the results of [14, 15]. Here we prov e that the ISM is equiv alent to a generalized F ourier transform also for Lax op erators with c.b.c. Th us we hav e sho wn that the nonlinear evolution of equation (1.15) transforms in to linear one in terms of the scattering data of L . 3 2 Solutions of the Lax op erator L The sp ectrum of the asymptotic operators L ± is purely con tinuous and is determined by the eigen v alues of Q ± whic h generically ma y be arbitrary complex n umbers. How ever, h ere w e consider only the case when L b ecomes self-adjoin t. As a result its p otential Q ( x, t ) acquires the form: Q ( x, t ) = − Q † ( x, t ) Q ( x, t ) =  0 q ( x, t ) − q † ( x, t ) 0  (2.1) F or simplicit y reasons w e will consider only the case when all of the eigenv alues of the asymptotic matrices Q ± are real and equal: m 1 = m 2 = ... = m r = m 6 = 0 m ∈ R (2.2) As a result we hav e the following condition on the eigenv alues of the asymptotic matrices [11]: q ± q † ± ( x, t ) = m 2 1 1 and the corresp ondence with the isotropic problem is obvious: µ = µ = m 2 1 1. The requirement that the potentials of the Z-Sh system b elong to g can b e formulated as a reduction condition [17, 5]: S − 1 0 U t ( x, t, λ ) S 0 = − U ( x, t, λ ) , S − 1 0 V t ( x, t, λ ) S 0 = − V ( x, t, λ ) , S − 1 0 σ 3 S 0 = − σ 3 , (2.3) whic h has trivial action on λ . The matrix S 0 is the one which realizes the definition of the algebras C r ' sp (2 r ) or D r ' so (2 r ) in the typical represen tation [5, 12] . In what follows we will define the Lie algebra g by: g ≡  X : X + S − 1 0 X t S 0 = 0  , (2.4) where S 0 = r X s =1 ( − 1) s +1 ( E ss − E ss ) for g ' sp (2 r ) and S 0 = r X s =1 ( − 1) s +1 ( E ss + E ss ) for g ' so (2 r ). Here s = 2 r − s + 1 and E ks are 2 r × 2 r matrices, defined b y ( E ks ) ij = δ ki δ sj . Note that S 2 0 =  0 1 1, where  0 = − 1 for sp (2 r ) and  0 = 1 for so (2 r ). Suc h reduction (2.3) imp oses restrictions only on the co efficien ts of Q ( x, t ) suc h that for C r ' sp (2 r ) w e can put: Q ( x, t ) = X i 0 , S 2 : Im j ( λ ) < 0 . (2.13) Half of the columns of the Jost solutions are analytic functions of λ on the first sheet and the other on the second sheet. ψ ( x, λ ) = ( | ψ − ( x, λ ) i , | ψ + ( x, λ ) i ) , φ ( x, λ ) = ( | φ + ( x, λ ) i , | φ − ( x, λ ) i ) , (2.14) where | ψ ± i and | ψ ± i denote a r × 2 r matrix composed of the corresponding r columns of the Jost solutions. The sup erscript ” + ” means analyticity on the first sheet and ” − ” - analyticity on the second sheet. Next, we can construct F AS on each of the sheets b y simply com bining the blo c ks of the Jost solutions with the same analyticit y properties. χ + ( x, λ ) ≡  | φ + i , | ψ + i  ( x, λ ) , χ − ( x, λ ) ≡  | ψ − i , | φ − i  ( x, λ ) (2.15) Let us write down the F AS χ + ( x, λ ), analytic on the sheet S 1 and χ − ( x, λ ), analytic on the sheet S 2 using appropriate decomp ositions of the scattering matrix (2.19), whic h consists of the same upp er (low er) blo c k-triangular functions S ± and T ± as they are in the v.b.c. case [8]: χ ± ( x, λ ) = ψ ( x, λ ) T ∓ = φ ( x, λ ) S ± (2.16) 5 These triangular factors are: S + =  1 1 d − 0 c +  , T − =  a + 0 b + 1 1  , S − =  c − 0 − d + 1 1  , T + =  1 1 − b − 0 a −  (2.17) and can be view ed also as generalized Gauss decomp ositions of the T ( λ ). T ( λ ) = T − ( λ ) b S + ( λ ) = T + ( λ ) b S − ( λ ) . (2.18) Here and after the hat b means taking the inv erse matrix. W e can use for the scattering matrix the same block-matrix structure as in v.b.c. case [11]: φ ( x, λ ) = ψ ( x, λ ) T ( λ ) , T ( λ ) =  a + ( λ ) − b − ( λ ) b + ( λ ) a − ( λ )  b T ( λ ) =  c − ( λ ) d − ( λ ) − d + ( λ ) c + ( λ )  (2.19) The elements of the in verse matrix are defined as follows: c − ( λ ) = b a + ( λ )( 1 1 + ρ − ρ + ) − 1 = ( 1 1 + τ + τ − ) − 1 b a + ( λ ) d − ( λ ) = b a + ( λ ) ρ − ( λ ) ( 1 1 + ρ + ρ − ) − 1 = ( 1 1 + τ + τ − ) − 1 τ + ( λ ) b a − ( λ ) c + ( λ ) = b a − ( λ )( 1 1 + ρ + ρ − ) − 1 = ( 1 1 + τ + τ − ) − 1 b a − ( λ ) d + ( λ ) = b a − ( λ ) ρ + ( λ ) ( 1 1 + ρ − ρ + ) − 1 = ( 1 1 + τ − τ + ) − 1 τ − ( λ ) b a + ( λ ) Here ρ ± ( λ ) = b ± ( λ ) b a ± ( λ ) = b c ± ( λ ) d ± ( λ ) and τ ± ( λ ) = b a ± ( λ ) b ∓ ( λ ) = d ∓ ( λ ) b c ± ( λ ) are the m ulticomp onen t generalizations of the reflection ρ ± , τ ± co efficien ts. (for the scalar case see [21, 22, 19]). Giv en the potential Q ( x ) one can obtain the Jost solutions uniquely . The Jost solutions in turn determine uniquely the scattering matrix T ( λ ) and its inv erse b T ( λ ). Q ( x ) con tains at most |4 1 + | indep enden t complex-v alued functions of x . Th us it is natural to exp ect that at most |4 + 1 | of the co efficien ts of T ( λ ) for λ ∈ R m , instead of (2 r ) 2 , will b e indep enden t. Here |4 + 1 | is the n umber of ro ots in 4 + 1 ,i.e. |4 + 1 | = r ( r + 1) / 2 for C r and |4 + 1 | = r ( r − 1) / 2 for D r . The contin uous spectrum R m = ( −∞ , − m ) ∪ ( m, ∞ ) is determined by the condition | λ | ≥ m . The set of indep enden t co efficien ts of T ( λ ) are known as the set of minimal scattering data T . They w ere in tro duced by Kaup for the Z-Sh system asso ciated with g ' sl (2) and v.b.c.. He pro ved that a ± ( λ ) can be recov ered from T using the analyticity properties, i.e. the so-called disp ersion relation. The same problem for the generalized Z-Sh system with c.b.c. is more difficult. Here w e just introduce T i = T i,c ∪ T i,d as the proper generalization of the minimal set of scattering data: T 1 ,c ≡ { ρ + α ( λ ) , ρ − α ( λ ) , λ ∈ R m } , T 1 ,d ≡ { ρ ± α ( λ ± j ) , λ ± j } N j =1 T 1 ,c ≡ { τ + α ( λ ) , τ − α ( λ ) , λ ∈ R m } , T 2 ,d ≡ { τ ± α ( λ ± j ) , λ ± j } N j =1 where α ∈ 4 + 1 . The reconstruction of the diagonal blocks a ± ( λ ) from their analyticit y properties requires a solution of r × r matrix-v alued Riemman-Hilb ert problem. Here λ ± j are discrete eigen- v alues of L . The sets T i,c c haracterizing the contin uous sp ectrum need to b e completed by the sets T i,d c haracterizing the discrete sp ectrum of L whic h in turn requires the kno wledge of the dressing factors. These problems will be addressed elsewhere. 3 W ronskian relations Let the class of allow ed p oten tials M b e a slice of O σ 3 determined by additional constrain ts: i.) an y generic elemen t F ( x ) = P σ 3 F ( x ) of M is matrix-v alued function which v anishes fast enough 6 for | x | → ∞ and ii.) the phase factor V which connect the asymptotic v alues of the potential Q + = V † Q − V is an integral of motion. The deriv ative of the p oten tial Q x ( x, t ) b elongs to the class of allow ed p otentials. The v ariation of the potential δ Q ( x, t ) is an allo wed p oten tial pro vided it satisfies the second additional condition. The mapping F : M → L betw een the class of allo wed p oten tials M and the scattering data L of L is analyzed by means of W ronskian relations [3, 4]. These relations allo w us to formulate the main result of this w ork, i.e. that the ISM is a generalized F ourier transform in the case of C.I and D.II I -type symmetric spaces. They also serve to introduce the skew-scalar pro duct h h A ( x ) , B ( x ) i i = 1 2 Z dx h A ( x ) , [ σ 3 , B ( x ) ] i (3.1) whic h is non-degenerate for A ( x ) , B ( x ) ∈ M and provides it with symplectic structure. W e start with the iden tity: h b χ ( Q ( x, t ) − λσ 3 ) χ ( x, λ ) , E ± α i | ∞ x = −∞ = − i Z ∞ −∞ dx  i 2 [ σ 3 , σ 3 Q x ] , P σ 3 χ E ± α b χ ( x, λ )  (3.2) where χ ( x, λ ) can b e any fundamental solution of L . F or conv enience we choose them to b e the F AS in tro duced ab o ve. The l.h.side of (3.2) can be calculated explicitly by using the asymptotics of F AS for x → ±∞ . It would b e expressed by the matrix elemen ts of the scattering matrix T ( λ ), i.e. by the scattering data of L as follo ws:   P σ 3 χ + ( x, λ ) E α b χ + , σ 3 Q x   = − j ( λ ) D b T − σ 3 T − , E α E = 2 j ( λ ) b + α , α ∈ 4 + 1   P σ 3 χ + ( x, λ ) E − α b χ + , σ 3 Q x   = j ( λ ) D b S + σ 3 S + , E − α E = 2 j ( λ ) d − − α , α ∈ 4 + 1   P σ 3 χ − ( x, λ ) E α b χ − , σ 3 Q x   = j ( λ ) D b S − σ 3 S − , E α E = 2 j ( λ ) d + α , α ∈ 4 + 1   P σ 3 χ − ( x, λ ) E − α b χ − , σ 3 Q x   = − j ( λ ) D b T + σ 3 T + , E − α E = 2 j ( λ ) b − − α , α ∈ 4 + 1 (3.3) The second set of W ronskian relations whic h we consider relate the v ariation of the p otential δ Q to the corresp onding v ariations of the scattering data δ ρ and δ τ . F or this purpose we use the iden tity: h b χ δ χ ( x, λ ) , E ± α i | ∞ x = −∞ = Z ∞ −∞ dx  i 2 [ σ 3 , σ 3 δ Q ] , P σ 3 χ ( x, λ ) E ± α b χ  (3.4) If we assume that the v ariation of the phase factor δ V v anishes w e arrive at:   P σ 3 χ + ( x, λ ) E α b χ + , σ 3 δ Q   = − i D b T − δ T − , E α E = i ( δ ρ + a + ) α , α ∈ 4 + 1   P σ 3 χ + ( x, λ ) E − α b χ + , σ 3 δ Q   = i D b S + δ S + , E − α E = i ( δ τ + c + ) − α , α ∈ 4 + 1   P σ 3 χ − ( x, λ ) E α b χ − , σ 3 δ Q   = i D b S − δ S − , E α E = i ( δ τ − c − ) α , α ∈ 4 + 1   P σ 3 χ − ( x, λ ) E − α b χ − , σ 3 δ Q   = − i D b T + δ T + , E − α E = i ( δ ρ − a − ) − α , α ∈ 4 + 1 (3.5) These relations are basic for the analysis of the related NLEE and their Hamiltonian structures. The abov e iden tities also allow us to in tro duce the prop er generalizations of the usual F ourier exp onen tial functions. Let us in tro duce the set of ”squared solutions”: Φ ± α ( x, λ ) = P σ 3 χ ± ( x, λ ) E ± α b χ ± ( x, λ ) , for α ∈ 4 + 1 (3.6) 7 Ψ ± α ( x, λ ) = P σ 3 χ ± ( x, λ ) E ∓ α b χ ± ( x, λ ) , for α ∈ 4 + 1 (3.7) Θ ± α ( x, λ ) = P σ 3 χ ± ( x, λ ) E ± α b χ ± ( x, λ ) , for α ∈ 4 + 0 (3.8) Ξ ± α ( x, λ ) = P σ 3 χ ± ( x, λ ) E ∓ α b χ ± ( x, λ ) , for α ∈ 4 + 0 (3.9) Υ ± k ( x, λ ) = P σ 3 χ ± ( x, λ ) H k b χ ± ( x, λ ) , for k = 1 , ..., r (3.10) These are the ”squared solutions” of the Lax op erator L connected with simple Lie algebra g . They are constructed by means of F AS χ ± ( x, λ ) and the Cartan-W eyl basis of the algebra and are analytic functions of λ on the corresp onding sheets of the sp ectral surface. The equations that Φ ± α an Ψ ± α satisfy are a direct consequence of the fact that F AS and their in verse satisfy the Z-Sh system system: i d Φ ± α dx +  Q ( x ) − λ σ 3 , Φ ± α ( x, λ )  = 0 , i d Ψ ± α dx +  Q ( x ) − λ σ 3 , Ψ ± α ( x, λ )  = 0 (3.11) The ”squared solutions” also serve as building blo c ks of the Green function for L [6, 7, 8]: G ± ( x, y , λ ) = G ± 1 ( x, y , λ ) θ ( y − x ) − G ± 2 ( x, y , λ ) θ ( x − y ) , (3.12) where G ± 1 ( x, y , λ ) = X α ∈4 + 1 Φ ± α ( x, λ ) ⊗ Ψ ± α ( y , λ ) (3.13) G ± 2 ( x, y , λ ) = X α ∈4 + 1 Ψ ± α ( x, λ ) ⊗ Φ ± α ( y , λ ) + X α ∈4 + 0 ∪4 + 1 Ξ ± α ( x, λ ) ⊗ Θ ± α ( y , λ ) + r X k =1 Υ ± k ( x, λ ) ⊗ Υ ± k ( y , λ ) (3.14) and θ ( x ) is the usual step function. 4 Completeness relation and ev olution The main result in this section is that the sets { Φ ± α } and { Ψ ± α } form complete sets of functions in M . The idea of the pro of is simple. Apply the contour integration metho d along a prop er contour (see figure.1) to a conv eniently c hosen Green function (3.12). F rom the Cauc hy theorem we ha ve 1 2 π i I C G + ( x, y , λ ) dλ = N X k =1 Res λ = λ ± k G + ( x, y , λ ) (4.1) In tegrating along the contours we treat separately the contribution from the infinite semi-arcs and the ones from the contin uous sp ectrum R m = C 1 ∪ C 2 whic h is comp osed of the cuts C 1 = ( −∞ , − m ) and C 2 = ( m, ∞ ). Sp ecial care must b e tak en for the end p oints λ = ± m of the sp ectrum. Assuming that the end points of the sp ectrum giv e no contribution lim ε → 0 Z ε + + Z ε − ! G + dλ = 0 (4.2) 8 Figure 1: The contour along which w e in tegrate lies completely on the first sheet of the 2-sheeted sp ectral surface asso ciated with the square ro ot j ( λ ) = √ λ 2 − m 2 where with index ε ± w e hav e denoted the integrals along the infinitesimal semi-arcs around the end p oin ts of the sp ectrum, we obtain the follo wing completeness relation δ ( x − y )Π σ 3 = 1 π X α ∈4 + 1 Z R m dλ  Φ + α ( x, λ ) ⊗ Ψ + α ( y , λ ) − Φ − α ( x, λ ) ⊗ Ψ − α ( y , λ )  − 2 i X α ∈4 + 1 N X k =1  d dλ Φ + α ( x, λ )     λ = λ + k ⊗ Ψ + α ( y , λ ) + Φ + α ( x, λ ) ⊗ d dλ Ψ + α ( y , λ )     λ = λ + k } Here Π σ 3 = P α ∈4 + 1 [ E α ⊗ E − α − E − α ⊗ E α ]. The assumption that w e hav e made is that λ + j are simple p oles of the ”squared solutions” Φ + α and Ψ + α . Using the completeness relation one can expand any generic elemen t of the phase space M ov er eac h of the complete sets of ”squared solutions” Ψ ± α and Φ ± α . This relation is utilized with the help of the following the trick − 1 2 tr 1 { ([ σ 3 , F ( x )] ⊗ 1 1)Π σ 3 } = 1 2 tr 2 { Π σ 3 ( 1 1 ⊗ [ σ 3 , F ( x )]) } = F ( x ) (4.3) where tr 1 (and tr 2 ) mean taking the trace of the elements in the first (or in the second) p osition of the tensor pro duct. The completeness relation (4.3) allo ws to establish one-to-one corresp ondence b et ween the elemen ts of M , such as Q x and Q t , and its expansion co efficients. It is also directly related to the sp ectral decomp ositions of the generating (recursion) op erators Λ ± . These operators are the ones whose eigenfunctions are the ”squared solutions”. Their deriv ation starts by in tro ducing the splitting of the ob ject e ± α = χ ± ( x, λ ) E ± α b χ ± ( x, λ ) into blo c k diagonal and blo c k off-diagonal parts: e ± α ( x, λ ) = e d, ± α ( x, λ ) + Φ ± α ( x, λ ) , e d, ± α ( x, λ ) = ( 1 1 − P σ 3 ) e ± α ( x, λ ) (4.4) 9 end use the equation it satisfies i d e ± α dx +  Q ( x ) − λ σ 3 , e ± α ( x, λ )  = 0 (4.5) Th us equation (4.5) splits into i d e d, ± α dx +  Q ( x ) , Φ ± α ( x, λ )  = 0 (4.6) i d Φ ± α dx +  Q ( x ) , e d, ± α ( x, λ )  = λ  σ 3 , Φ ± α ( x, λ )  (4.7) Equation (4.6) can b e integrated formally with the result e d, ± α ( x, λ ) = C d, ± α ;  ( λ ) + i Z x  ∞ dy  Q ( y ) , Φ ± α ( y , λ )  , (4.8) C d, ± α ;  ( λ ) = lim x →  ∞ e d, ± α ( x, λ ) ,  = ± 1 (4.9) Next insert (4.8) into (4.7) and act on b oth sides b y ad − 1 σ 3 . This giv es us: (Λ ± − λ ) Φ ± α ( x, λ ) = i  C d, ± α ;  ( λ ) , ad − 1 σ 3 Q ( x )  , (4.10) where the generating op erators Λ ± are given b y: Λ ± Ξ( x ) = ad − 1 σ 3  i d Ξ dx + i  Q ( x ) , Z x ±∞ dy [ Q ( y ) , Ξ( y )]  (4.11) Th us Ψ ± α (resp. Φ ± α ) will be eigenfunctions of Λ + (resp. Λ − ) if and only if C d, ± α ;  ( λ ) = 0. Ev aluating the limit of (4.9) for all α in the specific case (4.2) w e find: (Λ + − λ ) Ψ ± α ( x, λ ) = 0  Λ + − λ ± j  Ψ ± α ( x, λ ± j ) = 0 , α ∈ 4 + 1 (4.12) (Λ − − λ ) Φ ± α ( x, λ ) = 0  Λ − − λ ± j  Φ ± α ( x, λ ± j ) = 0 , α ∈ 4 + 1 (4.13) This result can b e generalized for arbitrary f (Λ ± ): ( f (Λ + ) − f ( λ )) Ψ ± α ( x, λ ) = 0  f (Λ + ) − f ( λ ± j )  Ψ ± α ( x, λ ± j ) = 0 , α ∈ 4 + 1 (4.14) ( f (Λ − ) − f ( λ )) Φ ± α ( x, λ ) = 0  f (Λ − ) − f ( λ ± j )  Φ ± α ( x, λ ± j ) = 0 , α ∈ 4 + 1 (4.15) The class of higher MNLS on symmetric spaces of C.I and D.I I I -t yp e and with c.b.c. can be put do wn in terms of the deriv ativ e of the p oten tial Q t with respect to the ev olution parameter and the dispersion la w f ( λ ) = − 2 λ [18, 8] as follows: i ad − 1 σ 3 ∂ ∂ t Q + f (Λ)ad − 1 σ 3 Q x = 0 (4.16) Substituting the ob jects in this form ula with their expansions o v er the ”squared solutions” w e obtain equations for the evolution of the scattering data. The expansion co efficien ts of ad − 1 σ 3 Q t and ad − 1 σ 3 Q x on the contin uous spectrum turn out to b e exactly the minimal set of scattering data. The evolution for the reflection and transition co efficien ts is pro vided b y i ∂ ρ ± ∂ t ± f ( λ ) j ( λ ) ρ ± ( t, λ ) = 0 , i ∂ τ ± ∂ t ∓ f ( λ ) j ( λ ) τ ± ( t, λ ) = 0 λ ∈ R m . (4.17) 10 The observ ation that the scattering data evolv es trivially is visible from the equation depicting the ev olution of the scattering matrix T ( λ ). This equation is a result of the compatibility condition (1.8) and the fact that the tw o Jost solutions ψ and φ are solutions of the second op erator of the Lax pair in the Z-Sh system (1.9). Acting with i d dt on T ( λ ) (2.12), we get: i d dt T ( t, λ ) − 2 λj ( λ ) [ σ 3 , T ( λ )] = 0 , (4.18) where f ( λ ) = − 2 λ is the disp ersion la w for the MNLS with c.b.c.. This equation for MNLS can also b e derived from the explicit form of the Lax representation (1.10) b y ev aluating the limit lim x →±∞ M ψ = 0. F or the r × r blo c ks making up the scattering matrix we hav e: ∂ a ± ( t, λ ) ∂ t = 0 , i ∂ b ± ∂ t ∓ 2 λ j ( λ ) b ± ( t, λ ) = 0 (4.19) These equations ha ve obvious solutions. F rom the first equation is clear that the diagonal blo c ks are conserved and their inv arian ts - upp er (low er) principal minors as well as their determinan ts are generating functionals of the sp ecial series of local infinitely many in tegrals of motion I k : ln det a ± ( λ ) = ∞ X k =1 λ − k I k (4.20) This is the ma jor idea of the ISM - a one-to-one change of v ariables- from the multicomponent q ( x, t ), in terms of which the MNLS(1.15) is written, tow ards the scattering data which satisfy linear evolution equations. 5 Conclusion The result of this w ork is that the in terpretation of the ISM as a generalized F ourier transformation holds true in the case of Lax operators with constant boundary conditions on symmetric spaces connected with the Lie algebras C r ' sp (2 r ) and D r ' so (2 r ). The completeness relation of the ”squared solutions” of the generalized Z-Sh system in the case when the Lax op erator L b ecomes self-adjoint is derived. The ”squared solutions” turn out to b e generalizations of the usual F ourier exp onen tial function and eigenfunctions of the recursion op erators Λ ± . This result allo ws one to pro v e that the corresponding NLEE results in linear evolution for the scattering data. The recursion op erators Λ ± op en the path tow ards the construction of action-angle v ariables for the NLEE solv able with this generalization of the Z-Sh system and from there the Hamiltonian form ulation of these equations and their hierarc hies connected with Λ ± . The physical applications of the NLS eq. b oth with v anishing and non-v anishing b oundary conditions is well known; the same holds true for the Manako v system as w ell as for the sp (4) MNLS with v.b.c., see [13]. It will be interesting to find physical applications also for the MNLS with c.b.c. Ac kno wledgements V.S.G wishes to ackno wledge partial supp ort from NSF of Bulgaria under con tract No.1410 and V.A.A. would like to express his gratitude to the organizers of GAS05 for their hospitality and financial supp ort. 11 App endix The ab o ve definition of g (2.4) satisfies the requirement that the Cartan subalgebra h will be made up of diagonal matrices. The Cartan generators H k , dual to e k , are giv en b y: H k = E kk − E k k (5.1) The element σ 3 = P r k =1 H k , b elongs to h and is dual to ~ a . The ro ot vectors in the t ypical represen tation are giv en b y E e i − e j = E ij − ( − 1) i + j E j i E e i + e j = E i j −  0 ( − 1) i + j E j i (5.2) where 1 ≤ i < j ≤ r and  0 = ± 1. Since  0 = 1 for g ' so (2 r ) equation (5.2) gives v anishing result for i = j which is compatible with the fact that 2 e i are not ro ots of so (2 r ); for g ' sp (2 r )  0 = − 1 and equation (5.2) by putting i = j provides also an expression for E 2 e i . Ho wev er this expression is not normed with resp ect to the Killing form h E α , E − α i = 2 . The W eyl generators asso ciated with the root 2 e i that we will use are given b y [12]: E 2 e i = √ 2 E i i (5.3) References [1] Ablo witz M., Kaup D., New ell A. and Segur H., The Inverse Sc attering T r ansform - F ourier A nalysis for Nonline ar Pr oblems , Stud. Appl. Math 53 249-315 (1974). [2] Ablo witz M. and Seegur H., Solitons and the Inverse Sc attering T r ansform. SIAM Studies in Applie d Mathematics Philadelphia: SIAM, 1981 [3] Calogero F.,Degasp eris A., Nonline ar Evolution Equations Solvable by the Inverse Sp e ctr al T r ansform I Nuov o Cim. 32B 201-242 (1976). [4] Calogero F.,Degasp eris A. Nonline ar Evolution Equations Solvable by the Inverse Sp e ctr al T r ansform II Nuov o Cim. 39B 1-54 (1976). 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