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The results on the GBDT for the coupled NLS are derived in 37 , Sec. 3] Let us formulate some of them below. Each GBDT for system ( 1.1 ) is determined by the initial system itself and five parameter matrices: [MATH] [MATH] ) matrices [MATH] [MATH] , and [MATH] and [MATH] matrices [MATH] [MATH] such that |
[EQUATION] If ( 2.1 ) holds, then the following linear systems are compatible and (jointly with the initial values [MATH] [MATH] , and [MATH] ) determine matrix functions [MATH] [MATH] , and [MATH] respectively: |
[EQUATION] Although the point [MATH] [MATH] is chosen above as the initial point, it is easy to see that any other point may be chosen for this purpose as well. |
Consider [MATH] [MATH] , and [MATH] in some domain [MATH] , for instance, [EQUATION] such that [MATH] is well-defined in [MATH] and satisfies ( 1.1 ) and such that [MATH] Introduce (in the points of invertibility of [MATH] in [MATH] ) matrix functions |
[EQUATION] Proposition 2.1 Let [MATH] satisfy the coupled NLS ( 1.1 ). Then, in the points of invertibility of [MATH] , the matrix function |
[MATH] given by ( 2.9 ) satisfies the coupled NLS as well. Remark 2.2 Proposition 2.1 was proved as 37 , Proposition 3.1] earlier and checked recently using a program |
developed by D.R. Popovych and based on the NCAlgebra package. Remark 2.3 Relations ( 2.5 )–( 2.8 ) imply that the matrix identity |
[EQUATION] holds everywhere on [MATH] The so called Darboux matrix corresponding to the transformation [MATH] has (at each point [MATH] of invertibility of [MATH] ) the form of the Lev Sakhnovich’s transfer matrix function (see |
and references therein): [EQUATION] In other words, we have the following statement (see 37 , Sectons 2, 3] ). Proposition 2.4 Let [MATH] satisfy the auxiliary systems ( 2.2 ). Then, the function |
[EQUATION] satisfies the transformed system [EQUATION] where [EQUATION] and [MATH] is given by ( 2.9 ). GBDT for nonlocal NLS In this section, we consider the case when the condition ( 1.6 ) is valid, and so the coupled NLS is reduced to the nonlocal NLS ( 1.7 ). In view of the first equality in ( 1.4 ), relations ( 1.... |
[EQUATION] 1. Consider first the case [MATH] . Then, taking into account ( 2.3 ), ( 2.4 ), and ( 3.1 ) we have [EQUATION] Relations ( 2.6 ) and ( 3.2 ) imply that |
[EQUATION] In the same way, formulas ( 2.6 ), ( 3.3 ), and ( 3.4 ) yield the equation [EQUATION] Comparing ( 2.7 ) with ( 3.5 ), ( 3.6 ) we see that in the case |
[EQUATION] we may set [EQUATION] In view of ( 3.8 ), relations ( 2.8 ) take the form [EQUATION] Thus, under condition [EQUATION] |
we have [MATH] , and so [EQUATION] According to ( 2.9 ) and ( 3.8 ) we have [EQUATION] From ( 3.7 ), ( 3.11 ), and ( 3.12 ), it is immediate that |
[EQUATION] Recall that by virtue of Proposition 2.1 the matrix function [MATH] satisfies the coupled NLS. The additional property ( 3.13 ) means that the block |
[MATH] of [MATH] satisfies the nonlocal matrix NLS. In other words, we constructed a GBDT-transformed solution of the nonlocal matrix NLS (with [MATH] ). |
2. Let us formulate our result on GBDT for the nonlocal NLS for both cases [MATH] Theorem 3.1 Let an [MATH] matrix function [MATH] satisfy the nonlocal NLS ( 1.7 ), and assume that a triple of matrices [MATH] , such that |
[EQUATION] is given, where [MATH] and [MATH] are [MATH] matrices, [MATH] , and [MATH] is an [MATH] matrix. Introduce the matrix function [MATH] by the first equality in ( 1.4 ) and by the relations ( 1.6 ), and determine [MATH] and [MATH] by their values [MATH] and [MATH] , respectively, at [MATH] |
and by the equations [EQUATION] where the coefficients [MATH] and [MATH] are defined [MATH] via [MATH] in ( 2.3 and ( 2.4 ). Then, the matrix function |
[EQUATION] also satisfies [MATH] in the points of invertibility of [MATH] the nonlocal NLS. That is, the equality [EQUATION] holds. |
P r o o f . It is immediate that the right-hand side of ( 3.18 ) coincides with [MATH] , and so for [MATH] the statement of the theorem is already proved in paragraph 1 above. |
Now, we assume that [MATH] and prove our theorem in a similar way as for the case [MATH] Namely, in view of the equality [MATH] we have |
[EQUATION] (instead of the equalities ( 3.2 )–( 3.4 ) in the case [MATH] ). Hence, relations ( 3.5 ) and ( 3.6 ) are substituted with |
[EQUATION] Thus, we may set [EQUATION] and formulas ( 2.8 ) take the form [EQUATION] In particular, under assumption ( 3.10 ) relations ( 3.25 ) and ( 3.26 ) yield the equality [MATH] |
and ( 3.11 ). Finally, taking into account ( 2.9 ), ( 3.11 ), and ( 3.24 ) we derive [EQUATION] and so the block [MATH] of the solution [MATH] of ( 1.1 ) satisfies the nonlocal matrix NLS ( 1.7 with [MATH] |
[MATH] The following corollary is immediate from the theorem’s proof. Corollary 3.2 Under the conditions of Theorem 3.1 , the identity ( 2.10 ) takes the form |
[EQUATION] and the equality ( 3.11 ) always holds. Remark 3.3 According to ( 2.11 ), ( 3.8 ), and ( 3.24 ), the Darboux matrix for the nonlocal NLS has the form |
[EQUATION] Moreover, in the case of the nonlocal NLS, the inverse matrix function [MATH] admits [MATH] see, e.g., general formulas [MATH] 1.75 [MATH] and [MATH] 1.76 [MATH] in |
[MATH] the reduction [EQUATION] Taking into account Proposition 2.4 , we see that the wave function [MATH] i.e., the fundamental solution [MATH] |
[MATH] of the transformed system ( 2.13 ), where [MATH] and [MATH] are given by ( 2.14 ) and ( 2.15 ) with [EQUATION] has the form |
[EQUATION] Here [MATH] is given in ( 3.29 ) and [MATH] is the fundamental solution of the initial system ( 2.2 ). Explicit solutions |
For some special choices of the initial solution [MATH] of the nonlocal NLS, Theorem 3.1 allows us to construct wide families of other explicit solutions of the nonlocal NLS. Clearly, the trivial initial solution [MATH] is the most popular choice in the construction of explicit solutions via Bäcklund-Darboux transforma... |
the fundamental solution [MATH] of the transformed system ( 2.13 ) considered in Remark 3.3 takes the form [EQUATION] Choosing [MATH] , one may set [MATH] , that is, investigate [MATH] at all real values of [MATH] and [MATH] In Sections and , we assume [MATH] and study this case in greater detail. |
1. First, partition [MATH] into [MATH] and [MATH] blocks and set: [EQUATION] In view of ( 1.6 ), ( 2.4 ), and the first equality in ( 4.2 ), we have |
[EQUATION] and equations ( 3.15 take a simple form [EQUATION] Using partition ( 4.2 ) and relations ( 4.3 ), write down [MATH] in an explicit form |
[EQUATION] Hence, relations ( 3.16 ) and ( 3.17 ) take the form [EQUATION] Now, we see that the following corollary of Theorem 3.1 is valid. |
Corollary 4.1 To each triple of matrices [MATH] , such that [MATH] and [MATH] are [MATH] matrices, [MATH] [MATH] is an [MATH] matrix and ( 3.14 ) holds, corresponds an explicit solution of the nonlocal matrix NLS ( 1.7 [MATH] with [MATH] |
This solution has the form [EQUATION] where [MATH] and [MATH] are the blocks of [MATH] [MATH] see ( 4.2 [MATH] and the derivatives of [MATH] are given explicitly by ( 4.5 ) and ( 4.6 ). Thus, the matrix function [MATH] is recovered, for instance, by |
[EQUATION] In terms of the blocks [MATH] and [MATH] of [MATH] , the identity ( 3.14 ) may be rewritten in the form [EQUATION] 2. |
Let us introduce several block matrices: [MATH] [EQUATION] Using ( 4.10 ), formula ( 4.4 ) may be rewritten in the form [EQUATION] |
Hence, taking into account ( 3.16 ), one can further simplify the procedure of constructing [MATH] Proposition 4.2 Assume that [MATH] satisfies the identity |
[EQUATION] Then, we have [EQUATION] where [MATH] P r o o f . Clearly, the right-hand side of ( 4.13 ) equals [MATH] at [MATH] Moreover, in view of ( 3.16 ), ( 4.11 ), and ( 4.12 ) the derivative of the right-hand side of ( 4.13 ) (with respect to [MATH] ) equals [MATH] Thus, the proposition’s statement is immediate. |
[MATH] Examples 1. When [MATH] (where [MATH] stands for the spectrum of [MATH] ) the matrix function [MATH] is uniquely recovered from the identity ( 3.28 ). It is a convenient way to calculate some examples. |
Example 5.1 Assume that [MATH] and that [MATH] , that is, [MATH] [MATH] , and [MATH] are scalars, and [MATH] and [MATH] are scalar matrix functions. We set [MATH] and fix [MATH] [MATH] , and [MATH] such that |
[EQUATION] Then, ( 3.28 ) and ( 4.4 ) yield [EQUATION] Thus, formula ( 4.7 ) for the solutions [MATH] of the nonlocal NLS ( 3.19 takes in our case the form |
[EQUATION] where [MATH] is given in ( 5.2 ). Solutions given in ( 5.3 ) are determined by four real valued parameters: [MATH] [MATH] [MATH] and [MATH] |
Functions [MATH] above look similar to the interesting one-soliton solutions of nonlocal NLS studied in . However, there is also an essential difference because the one-soliton solutions in |
are periodic with respect to [MATH] Instead of this property, we have the periodicity of [MATH] and of the denominator in ( 5.3 ) with respect to [MATH] Solutions of the form ( 5.3 ) appear, for instance, in |
(see also some further references therein). Remark 5.2 When [MATH] , the singularities [MATH] or blow ups [MATH] of [MATH] [MATH] i.e., zeros of [MATH] appear at one and only one value of [MATH] . Namely, they appear when [MATH] . For this [MATH] , the singularities appear with the periodicity [MATH] with respect to [M... |
2. When we take non-diagonalisable matrices [MATH] , factors polynomial in [MATH] and [MATH] appear (in addition to the exponents) in the expressions for the constructed solutions |
. Rational solutions are also constructed in this way The so called “multipole” solutions are constructed using matrices [MATH] with Jordan cells of order more than one as well (see, e.g., |
). For nonlocal NLS, we consider a simple particular case [EQUATION] Example 5.3 Assume that [MATH] [MATH] , and that [MATH] [MATH] , and [MATH] are given by ( 5.4 ). Here the solution [MATH] is again a scalar function but [MATH] is a [MATH] matrix function. The following relations for [MATH] are immediate from the ide... |
[EQUATION] After some simple calculations, using repeatedly ( 5.5 ) we derive [EQUATION] Next, in view of ( 4.4 ) and ( 5.4 ), we see that |
[EQUATION] Here we used the equalities [MATH] and [MATH] Relations ( 5.6 ) and ( 5.8 ) imply that [EQUATION] Finally, ( 5.7 ), ( 5.9 ), and ( 5.10 ) yield |
[EQUATION] Note that the polynomial terms in the expression for [MATH] make the study of zeros of [MATH] [MATH] that is, singularities of [MATH] much more complicated than in Example 5.1 |
Similar to the derivation of ( 5.8 ), we rewrite ( 4.7 [MATH] in our case [MATH] in the form [EQUATION] where [MATH] is given in ( 5.11 ). The expressions for [MATH] and for other terms on the right hand side of 5.12 ) will look more compact if we introduce the polynomial |
[EQUATION] Then, relations ( 5.11 ) and ( 5.10 ) may be rewritten as [EQUATION] In a similar way, taking into account ( 5.6 ) and ( 5.8 ), we construct other entries of [MATH] |
[EQUATION] Furthermore, relations ( 5.5 ) imply that [EQUATION] In view of ( 5.15 )–( 5.19 ), after some simple calculations we rewrite ( 5.12 ) as |
[EQUATION] A related family of solutions depending on one complex parameter is also constructed in We note that if we choose [MATH] instead of [MATH] |
given in ( 5.4 ), the solutions ( 5.3 ) appear again in the case [MATH] [MATH] i.e., one should avoid the simplest choice of [MATH] [MATH] in order to construct a new class of solutions [MATH] |
Finally, in the example below we construct the simplest family of multicomponent solutions of the nonlocal NLS. Example 5.4 Assume that [MATH] [MATH] , and [MATH] . Then, the solutions [MATH] of ( 3.19 ) are [MATH] vector functions and [MATH] are scalar functions. Introduce the parameters [MATH] and [MATH] |
by the equalities [EQUATION] Then, ( 3.28 ) and ( 4.4 ) yield [EQUATION] Hence, using again formula ( 4.7 [MATH] and slightly modifying the results of Example 5.1 [MATH] |
we derive [EQUATION] Compare ( 5.2 ) and ( 5.22 ) to see that the behaviour of the singularities of [MATH] from Example 5.1 which we discussed in Remark 5.2 , coincides with the behaviour of the singularities of [MATH] in the present example. |
Remark 5.5 We note that formulas ( 5.2 ), ( 5.13 )–( 5.19 ), and ( 5.22 ) provide precise expressions for [MATH] in the Examples 5.1 5.3 , and 5.4 , respectively. Thus, in view of the important formulas 4.1 ) and ( 3.29 ) as well as relations ( 4.4 ) and ( 5.8 ), the wave functions [MATH] are constructed for these exam... |
Algebro-geometric solutions In this section we discuss the algebro-geometric solutions for the nonlocal NLS in the scalar case [MATH] The coupled NLS ( 1.5 ) in the scalar case is given by |
[EQUATION] where the matrix polynomials [MATH] [MATH] in ( 6.1 ) defined by ( 2.2 )–( 2.4 ) now read [EQUATION] System ( 6.1 ) is also known as the AKNS system, which was introduced by Ablowitz, Kaup, Newell, and Segur in 1974. Algebro-geometric solutions are well known for the AKNS hierarchy, see for example Gesztesy ... |
and references therein. By definition, algebro-geometric AKNS solutions (or potentials) are the set of solutions of the stationary AKNS system |
[EQUATION] with [MATH] ranging in [MATH] . More details can be found in Appendix We call solutions of the stationary nonlocal NLS equation |
[EQUATION] with [MATH] ranging in [MATH] , algebro-geometric nonlocal NLS solutions. Note that the plus sign in ( 6.4 ), denoted by [MATH] , corresponds to the defocusing case, while the minus sign in ( 6.4 ), denoted by [MATH] corresponds to the focusing case. Such solutions can be recast as a particular case of algeb... |
Lemma 6.1 Let [MATH] . If [MATH] satisfies [MATH] with [MATH] then [MATH] and [MATH] defined by [EQUATION] satisfy [MATH] with constants [MATH] given by |
[EQUATION] The converse statement is also true. In a natural manner one can associate a hyperelliptic Riemann surface with ( 6.3 ), as described in ( A.18 ). The modified symmetry reduction ( 6.5 ) now implies certain constraints on the branch points of this surface, namely, the set of zeros [MATH] of [MATH] can either... |
[EQUATION] This relation implies (either one of) the following constraints on the set of zeros of [MATH] after possible relabeling: |
[EQUATION] Theorem 6.2 Assume either [MATH] [MATH] , or [MATH] and choose the homology basis [MATH] according to Theorem A.2 . Moreover, assume that |
[MATH] in ( A.30 ) satisfies [EQUATION] Then [MATH] represents a stationary nonlocal NLS solution if and only if [MATH] in ( A.29 ) satisfies the constraint |
[EQUATION] P r o o f First assume [MATH] . Given [MATH] [MATH] , and [MATH] , the constants [MATH] and [MATH] are uniquely determined by ( A.19 ) and ( A.24 ). Define the antiholomorphic involution [MATH] |
as in 19 , Example A.35 (i)] . One infers that the symmetric Riemann surface [MATH] is of dividing type (compare 19 , Def. A.33] ) and hence |
[EQUATION] Thus [MATH] defined in ( A.30 ) is purely imaginary, [MATH] So if [MATH] satisfies [MATH] , then by Lemma 6.1 and Theorem A.1 , the functions [MATH] and [MATH] admit representations ( A.27 ) and ( A.28 ). Applying ( 6.5 ) yields |
[EQUATION] Equation ( 6.9 ) is equivalent to [EQUATION] for some [MATH] and arbitrary [MATH] , and hence [EQUATION] and [MATH] . Similarly, one obtains |
[EQUATION] for some [MATH] and arbitrary [MATH] , and hence [EQUATION] Replacing [MATH] by [MATH] with [MATH] then yields ( 6.8 ) and ( 6.7 ). In case [MATH] |
[MATH] is again of dividing type and, in particular, [MATH] For [MATH] [MATH] is of nondividing type and [MATH] follows from 19 , (C.37), (C.39), (C.33)] . Hence the same arguments as before yield ( 6.7 ) and ( 6.8 ). |
[MATH] Remark 6.3 Given [MATH] as in [MATH] [MATH] , we do not know if we get a solution of [MATH] or [MATH] by the constraints on [MATH] . This has to be determined a priori, that is, there should be a correspondence between the location of the [MATH] ’s and the defocusing/focusing nNLS equation. For comparison, the s... |
Acknowledgments. This research was supported by the Austrian Science Fund (FWF) under Grants No. P29177 and V120. Appendix A Algebro-geometric AKNS solutions |
Following , we give a brief introduction to algebro-geometric AKNS solutions and their underlying Riemann surface and describe the theta function representation of such solutions which we need for Lemma 6.2 The analog of these formulas for the nonlinear Schrödinger equation was first published by Its and Kotlyarov |
in 1976. Since then, many authors presented slightly varying approaches to algebro-geometric solutions of the nonlinear Schrödinger and AKNS equations, see for instance Belokolos and Enol’skii |
, Gesztesy and Ratnaseelan , or Previato The stationary AKNS system ( 6.3 ) is equivalent to the stationary zero-curvature equation |
[EQUATION] where [EQUATION] The polynomials [MATH] [MATH] , and [MATH] incorporate the constants [MATH] [EQUATION] The stationary zero-curvature equation in ( A.11 ) yields that |
[EQUATION] and hence [MATH] is [MATH] -independent, implying [MATH] where the integration constant [MATH] is a monic polynomial of degree [MATH] . If [MATH] denote its zeros, then |
[EQUATION] In this manner we can associate a hyperelliptic curve [MATH] of genus [MATH] with ( 6.3 ) defined by [EQUATION] The curve [MATH] is compactified by joining two points at infinity, [MATH] [MATH] we denote the compactification again by [MATH] . Points [MATH] on [MATH] |
are represented as pairs [MATH] , where [MATH] is the meromorphic function on [MATH] satisfying [MATH] The complex structure on [MATH] is then defined in the usual way (see for example 19 , App. C] ). Hence |
[MATH] becomes a two-sheeted hyperelliptic Riemann surface of genus [MATH] We emphasize that by fixing the curve [MATH] (i.e., by fixing [MATH] ), the integration constants |
[MATH] [MATH] in ( 6.3 ) are uniquely determined, [EQUATION] Let [MATH] and [MATH] denote the zeros of [MATH] and [MATH] in ( A.14 ) and ( A.15 ), |
[EQUATION] We lift [MATH] and [MATH] to [MATH] by defining [EQUATION] Choose a homology basis [MATH] on [MATH] and denote by [MATH] the corresponding normalized holomorphic differential, that is, |
[EQUATION] Note that [MATH] Let [MATH] be the Riemann constant. The Riemann theta function associated with [MATH] is given by [EQUATION] |
Without loss of generality we choose the branch point [MATH] as a base point. Let [MATH] be a normalized differential of the second kind satisfying |
[EQUATION] where [MATH] denotes the local coordinate [MATH] for [MATH] near [MATH] Then [EQUATION] In addition, we denote the [MATH] -period of this difference by |
[EQUATION] Finally, we turn to divisors, the Jacobi variety, and the Abel map for divisors in our setting. A divisor [MATH] on [MATH] |
is a map [MATH] , where [MATH] for only finitely many [MATH] We define the positive divisor [MATH] by [EQUATION] and denote the set of all divisors on [MATH] by [MATH] The Jacobi variety [MATH] of [MATH] is defined by [MATH] , where |
[MATH] is the period lattice [MATH] . The Abel map for divisors is then defined by [EQUATION] With these quantities at hand, the algebro-geometric AKNS solutions admit the following representation in terms of Riemann theta functions, compare 19 , Thm. 3.11] |
Theorem A.1 Suppose that [MATH] are nonzero and satisfy the stationary AKNS system 6.3 ) on [MATH] . In addition, assume the affine part of [MATH] to be nonsingular and let [MATH] , where [MATH] is an open interval. Then |
[EQUATION] where [EQUATION] The constants [MATH] and [MATH] are uniquely determined by [MATH] (and its homology basis), the constant |
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