On the nonlinear Schrödinger equation with a time-dependent boundary condition
aa r X i v : . [ n li n . S I] A ug On the nonlinear Schr¨odinger equation with a time-dependentboundary condition
Baoqiang XiaSchool of Mathematics and Statistics, Jiangsu Normal University,Xuzhou, Jiangsu 221116, P. R. China,E-mail address: [email protected]
Abstract
We study the nonlinear Schr¨odinger equation on the half-line with a boundary conditionthat involves time derivative. This boundary condition was presented by Zambon [J. HighEnerg. Phys. 2014 (2014) 36]. We establish the integrability of such a boundary both byusing the Sklyanin’s formalism and by using the tool of B¨acklund transformations togetherwith a suitable reduction of reflection type. Moreover, we present a method to derive ex-plicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darbouxtransformation method in conjunction with a boundary dressing technique.
Keywords: integrable boundary conditions, B¨acklund transformation, Darboux transfor-mation, soliton solutions. We study the nonlinear Schr¨odinger (NLS) equation iu t + u xx + 2 u | u | = 0 , (1.1)posed on the positive x -axis with the following boundary condition [1] (cid:16) iu t − u x p b − | u | − a + b ) u + 2 u | u | (cid:17)(cid:12)(cid:12)(cid:12) x =0 = 0 , (1.2)where a and b are two arbitrary real constants.The boundary condition (1.2) was presented by Zambon in [1] via dressing a Dirichlet bound-ary with an integrable defect of type I for the NLS equation. In this paper we will not concentrateon the subject of integrable defect systems. We refer the reader to [2–9] and references thereinon this interesting subject. Due to the presence of the time derivative in the boundary term, wewill refer to (1.2) as a time-dependent boundary condition in this paper. It is worth mentioningthat an analogous boundary for an integrable discrete model, the Ablowitz-Ladik system, waspresented and investigated very recently in [11].The integrability of the boundary condition (1.2) has been established in [1] by using themodified Lax pair technique in conjunction with the classical r -matrix method (see e.g. [12]).It was shown that the boundary matrix (the solution of the reflection equation) associated with(1.2), unlike the constant one found in the Sklyanin’s formalism [13], is dynamical: it dependson the field of the NLS equation at the boundary location. In other words, one needs to considerthe dynamical generalization of the Sklyanin’s formalism in order to study the integrability ofthe boundary condition (1.2) via r -matrix method.One of the aims of the present paper is to describe a method, which is different from theone presented in [1], for investigating the integrability of the time-dependent boundary (1.2).This method is based on B¨acklund transformations (BTs) together with a suitable reduction ofreflection type. It is worth reminding that the idea to study boundary conditions compatiblewith integrability via BTs was initiated by Habibullin in [14], where the analysis is based on thespace-part of BTs. The main difference with respect to the method in [14] is that our argumentis based on the time-part of the BTs. We first present a connection between the Sklyanin’sformalism and the time-part of the BTs. Based on this observation we are able to constructtime-dependent boundary conditions directly from BTs combined with a reduction, withoutdressing a known integrable boundary. The advantage of the method is that it enables us toconstruct a generating function of the modified conserved quantities to support the integrabilityof the boundary (1.2) directly from the Lax pair formulation.Another aim of the present paper is to study soliton solutions of the NLS equation (1.1)together with the boundary condition (1.2). We present an approach for constructing solitonsolutions for such a time-dependent boundary problem, which is based on the tool of Darbouxtransformations (DTs) [15] in conjunction with a boundary dressing technique [16]. As a con-sequence, we obtain explicit formulae for multi-soliton solutions of the NLS equation in thepresence of the boundary condition (1.2).The paper is organized as follows. In section 2, we establish the integrability of the boundarycondition (1.2) both by using the Sklyanin’s formalism and by using the tool of B¨acklund trans-formations together with a reduction technique. In section 3, we show how to calculate solitonsolutions of the NLS equation with the integrable boundary condition (1.2). Some concludingremarks are drawn in section 4. For self-containedness, we start by a brief summary of the constructions of conservation lawsand the BTs for the NLS equation. The NLS equation (1.1) in the bulk admits the followingLax pair formulation φ x ( x, t, λ ) = U ( x, t, λ ) φ ( x, t, λ ) , (2.1a) φ t ( x, t, λ ) = V ( x, t, λ ) φ ( x, t, λ ) , (2.1b)where λ is a spectral parameter, φ = ( φ , φ ) T , and U = − iλ u − ¯ u iλ ! , V = − iλ + i | u | λu + iu x − λ ¯ u + i ¯ u x iλ − i | u | ! . (2.2)Here and in what follows the bar indicates complex conjugation. By using the Lax pair for-mulation, one can construct an infinite set of conservation laws for the NLS equation with thecondition of vanishing boundary or periodic boundary. Indeed, denoting Γ = φ φ , we find from(2.1) the following x -part and t -part Riccati equationsΓ x = 2 iλ Γ − ¯ u − u Γ , (2.3a)Γ t = − λ ¯ u + i ¯ u x − (cid:0) − iλ + i | u | (cid:1) Γ − (2 λu + iu x ) Γ , (2.3b)together with the following conservation equation( u Γ) t = (cid:0) i | u | + (2 λu + iu x ) Γ (cid:1) x . (2.4)Using the vanishing boundary condition or the periodic boundary condition, equation (2.4)implies that the function u Γ provides a generating function of the conservation densities. Bysubstituting the expansion Γ = P ∞ n =1 Γ n (2 iλ ) − n into (2.3a) and by equating the coefficients ofpowers of λ , we find explicit forms of conservation densities u Γ n withΓ = ¯ u, Γ = ¯ u x , Γ n +1 = (Γ n ) x + u n − X j =1 Γ j Γ n − j , n ≥ . (2.5)In order to study BTs for the NLS equation, we introduce another copy of the auxiliaryproblems for ˜ φ with Lax pair ˜ U , ˜ V defined as in (2.2) with the new potentials ˜ u , ˜ v , replacing u , v . We assume that the two systems are related by the gauge transformation, φ ( x, t, λ ) = B ( x, t, λ ) ˜ φ ( x, t, λ ) , (2.6)where the matrix B ( x, t, λ ) satisfies B x ( x, t, λ ) = U ( x, t, λ ) B ( x, t, λ ) − B ( x, t, λ ) ˜ U ( x, t, λ ) , (2.7a) B t ( x, t, λ ) = V ( x, t, λ ) B ( x, t, λ ) − B ( x, t, λ ) ˜ V ( x, t, λ ) . (2.7b)For the NLS equation (1.1), we take B = ( λ − a ) I + 12 − i Ω i (˜ u − u ) i (˜ u − u ) ∗ i Ω ! , Ω = p b − | u − ˜ u | , (2.8)Then equation (2.7) induces the following BT between the potentials u and ˜ u : u x − ˜ u x = − ia ( u − ˜ u ) − ( u + ˜ u ) Ω , (2.9a) u t − ˜ u t = 2 a ( u x − ˜ u x ) − i Ω (˜ u x + u x ) + i ( u − ˜ u ) (cid:0) | u | + | ˜ u | (cid:1) . (2.9b)Following [1], we now show how the boundary condition (1.2) can be derived. The deriva-tion is based on dressing a Dirichlet boundary with an integrable defect condition of the NLSequation. In order to describe the defect conditions, we denote by ˜ u the field on the negative x -axis and by u the field on the positive x -axis. For the NLS equation, the defect conditions oftype I are defined by the BT (2.9) frozen at the defect location x = 0 rather than on the wholeline (see, for example [3, 5], for details). We assume that the field ˜ u at the boundary x = 0satisfies the Dirichlet boundary condition, that is ˜ u | x =0 = 0 and ˜ u t | x =0 = 0. In this situation,the corresponding defect conditions become( u x − ˜ u x ) | x =0 = (cid:16) − iau − u p b − | u | (cid:17)(cid:12)(cid:12)(cid:12) x =0 , (2.10a) u t | x =0 = (cid:16) a ( u x − ˜ u x ) − i (˜ u x + u x ) p b − | u | + iu | u | (cid:17)(cid:12)(cid:12)(cid:12) x =0 , (2.10b)By using (2.10a) to eliminate ˜ u x in (2.10b) we obtain the time-dependent boundary condition(1.2). r -matrix We now discuss the integrability of the time-dependent boundary (1.2) via the classical r -matrixmethod. The analysis is based on an extension of the boundary K ( λ ) matrix in Sklyanin’sformalism from non-dynamical case to a dynamical (time-dependent) case. We note that mostof the arguments in this subsection are essentially the same as those in [1] (see also [10] for recentstudies on the dynamical consideration of Sklyanin’s formalism). The new results are that wefind a general solution of the dynamical reflection equation (see (2.22) below), and present acanonical realization of the associated boundary Poisson brackets (see (2.28) below).We first sketch out the standard Sklyanin’s formalism for studying integrable boundaryconditions. For the NLS equation, we consider the canonical Poisson brackets { u ( x, t ) , u ( y, t ) } = { ¯ u ( x, t ) , ¯ u ( y, t ) } = 0 , { u ( x, t ) , ¯ u ( y, t ) } = iδ ( x − y ) , (2.11)where δ ( x − y ) is the Dirac δ -function. It follows that the transition matrix, T ( x, y, λ ) = x exp Z xy U ( ξ, t, λ ) dξ, (2.12)satisfies the following well-known relation (see e.g. [12]) { T ( x, y, λ ) , T ( x, y, µ ) } = [ r ( λ − µ ) , T ( x, y, λ ) T ( x, y, µ )] , (2.13)where T ( x, y, λ ) = T ( x, y, λ ) ⊗ I , T ( x, y, µ ) = I ⊗ T ( x, y, µ ), and the classical r -matrix r is r ( λ ) = 12 λ . (2.14)In the study of integrable boundary conditions on the half-line, Sklyanin in his seminal paper [13]presented a generalization of the monodromy matrix T ( λ ), which is τ ( λ ) = T ( λ ) K ( λ ) T − ( − λ ) , (2.15)where T ( λ ) = T ( ∞ , , λ ). In the case of K ( λ ) being a constant matrix, it can, by using (2.13),be shown that if K ( λ ) satisfies the relation (called reflection equation)[ r ( λ − µ ) , K ( λ ) K ( µ )] + K ( λ ) r ( λ + µ ) K ( µ ) − K ( µ ) r ( λ + µ ) K ( λ ) = 0 , (2.16)then the quantities tr( τ ( λ )) are in involution for different values of the spectral parameter: { tr( τ ( λ )) , tr( τ ( µ )) } = 0 . (2.17)Moreover, if the boundary satisfies the condition V (0 , t, λ ) K ( λ ) = K ( λ ) V (0 , t, − λ ) , (2.18)then d tr( τ ( λ )) dt = 0. Thus tr( τ ( λ )) provides a generating function of Poisson commuting integralsof motion. The well-known Dirichlet boundary, Neumann boundary and Robin boundary condi-tions and their integrability can be concluded by choosing the matrix K ( λ ) that is proportionalto some constant diagonal matrices (see [13] for details).In order to study the integrability of the time-dependent boundary, we need to consider thedynamical generalisation of the K ( λ ) matrix. In this case, we can deduce that if K ( λ ) satisfiesrelations { K ( λ ) , K ( µ ) } = [ r ( λ − µ ) , K ( λ ) K ( µ )] + K ( λ ) r ( λ + µ ) K ( µ ) − K ( µ ) r ( λ + µ ) K ( λ ) , (2.19) { K ( λ ) , U ( x, t, µ ) } = 0 , (2.20)then the quantities tr( τ ( λ )) are in involution: { tr( τ ( λ )) , tr( τ ( µ )) } = 0. Moreover, if the K ( λ )matrix satisfies the equation dK ( λ ) dt = V (0 , t, λ ) K ( λ ) − K ( λ ) V (0 , t, − λ ) , (2.21)then d tr( τ ( λ )) dt = 0. We will restrict our attention to the case that the K ( λ ) matrix holds atthe boundary location. In this case, the Poisson bracket (2.20) is automatically zero. For thedynamical reflection equation (2.19), we find the following solution K ( λ ) = λ − cλ − λ − cλ − ! + S S S − S ! , (2.22)where c is an arbitrary constant, and the dynamical variables ( S , S , S ) obey to the followingPoisson brackets: { S , S } = S , { S , S } = − S , { S , S } = 2 S . (2.23)Inserting (2.22) into (2.21), we find S = − iu | x =0 , S = − i ¯ u | x =0 , S = − i p α − | u | (cid:12)(cid:12)(cid:12) x =0 , (2.24)and the following boundary condition (cid:18) i dudt − cu + 2 u | u | − u x p α − | u | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 , (2.25)where c and α are arbitrary real constants. By setting α = 4 b and c = a + b in (2.25), werecover the boundary condition (1.2). This implies the integrability of the boundary condition(1.2).As mentioned above, the boundary quantities ( S , S , S ), given by (2.24), should obey to thePoisson brackets (2.23), in order to guarantee the Poisson commutativity of integrals of motionassociated with the boundary condition (2.25). Next we show ( S , S , S ) admit a naturalcanonical realization. Indeed, we rewrite the field of the NLS equation as u ( x, t ) = i p ( x, t ) , ¯ u ( x, t ) = − i q ( x, t ) − iα p − ( x, t ) , (2.26)and denote the boundaries of p and q at x = 0 by p ( t ) = p (0 , t ) , q ( t ) = q (0 , t ) , (2.27)In this case, ( S , S , S ), defined by (2.24), become S = 12 p ( t ) , S = − q ( t ) − α p − ( t ) , S = 12 q ( t ) p ( t ) . (2.28)One can check directly that ( S , S , S ), given by (2.28), satisfy the algebra (2.23), if the variables q ( t ) and p ( t ) satisfy the canonical Poisson brackets, that is { q ( t ) , q ( t ) } = { p ( t ) , p ( t ) } = 0 , { q ( t ) , p ( t ) } = 1 . (2.29)Thus (2.28) does provide a canonical realization of the boundary algebra (2.23). In this subsection, we will describe a method for studying the time-dependent boundary condi-tions, which is different from the one presented above and which is based on the BTs.We first describe how the time-part of BTs is related to the Sklyanin’s formalism. The keypoint is the following observation: if there exists a compatible reduction on the fields u and ˜ u such that ˜ V (0 , t, λ ) = V (0 , t, − λ ) , (2.30)then the time-part of BT (2.7b) evaluated at x = 0 is equivalent to the boundary equation (2.21)appearing in Sklyanin’s formalism in the dynamical case. The odd reduction ˜ u ( x, t ) = − u ( − x, t )meets perfectly this requirement for the reduction on fields. Indeed, one can check directly thatthe NLS equation admits the odd reduction ˜ u ( x, t ) = − u ( − x, t ) and equation (2.30) becomes anidentity under this reduction. As a consequence, the time-part of BT (2.7b) with the reduction˜ u ( x, t ) = − u ( − x, t ) evaluated at x = 0 is equivalent to the boundary equation (2.21). The above analysis implies that we can derive the time-dependent boundary condition (1.2)directly via imposing the the odd reduction ˜ u ( x, t ) = − u ( − x, t ) on two fields u and ˜ u related byBT (2.9). Indeed, by inserting (2.9a) into (2.9b) we obtain u t − ˜ u t = − ia ( u − ˜ u ) − Ω (2 a ( u + ˜ u ) + i (˜ u x + u x )) + i ( u − ˜ u ) (cid:0) | u | + | ˜ u | (cid:1) . (2.31)By imposing the reduction ˜ u ( x, t ) = − u ( − x, t ) on (2.31) and evaluating the resulting equationat x = 0, we then recover the boundary condition (1.2) up to a slight scaling.In general, it is not easy to find a dynamical boundary K matrix that matches the boundaryequation (2.21), a guess work is usually employed. The analysis presented above provides us ahint to deduce such a boundary matrix, that is the K matrix can be derived from BTs togetherwith a reduction of reflection type. Indeed, by inserting the reduction ˜ u ( x, t ) = − u ( − x, t ) intothe B¨acklund matrix (2.8) evaluated at x = 0 and after a slight adjustment to the diagonalterm of the resulting matrix, we recover the boundary K matrix defined by (2.22) and (2.24) insection 2.1.Another advantage of the method is that we can construct explicitly the generating functionof the infinite set of modified conservation laws to support the integrability of the boundarydirectly from the Lax pair formulation. The result is as follows. Proposition 1
A generating function for the integrals of motion of the NLS with integrableboundary condition (1.2) is given by I ( λ ) = Z ∞ u ( x, t ) (Γ( x, t, λ ) − Γ( x, t, − λ )) dx + ln (cid:16) λ − ( a + b ) λ − − i p b − | u ( x, t ) | − iu ( x, t )Γ( − x, t, − λ ) (cid:17)(cid:12)(cid:12)(cid:12) x =0 , (2.32) where Γ( x, t, λ ) satisfies the Ricatti equation (2.3a) (see (2.5) for explicit forms of Γ ). Proof
Consider two copies of the auxiliary problem (2.1) related by (2.6). Following [5], wehave ddt (cid:18)Z −∞ ˜ u ˜Γ( x, t, λ ) dx + Z ∞ u Γ( x, t, λ ) dx + ln ( B (0 , t, λ ) + B Γ(0 , t, λ )) (cid:19) = 0 , (2.33)where B jk , j, k = 1 ,
2, is the jk -entry of the defect matrix B . For the boundary problem (1.2),under the reduction ˜ u ( x, t ) = − u ( − x, t ) we have˜Γ( x, t, λ ) = Γ( − x, t, − λ ) . (2.34)Using (2.34) in (2.33), we obtain the integrals of motion (2.32) after some algebra. (cid:3) By inserting (2.5) into (2.32), we immediately obtain explicit forms for the conservationdensities. For example, the first few members are I = − i Z ∞ | u | dx − i Ω( u ) | x =0 ,I = i Z ∞ (cid:0) | u | − | u x | (cid:1) dx + (cid:20) i | u | − i ( a − b (cid:21) Ω( u ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 ,I = − i Z ∞ u (cid:0) ¯ u xxxx + 6¯ u | u x | + ¯ u u xx + 5 u (¯ u x ) + | u | (6¯ u xx + 2¯ u | u | ) (cid:1) dx − (cid:20) i u ¯ u xxx + i | u | ¯ uu x + 3 i | u | u ¯ u x − i a + b ) u ¯ u x (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x =0 − i Ω( u ) (cid:18) u ¯ u xx + 38 | u | + ( a + b )( a + b − | u | ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 − (cid:20) i u ¯ u x (Ω( u )) − i ( a + b − | u | )(Ω( u )) + i u )) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , (2.35)where Ω( u ) = p b − | u ( x, t ) | . These formulae coincide with the ones found in [1]. In this section we will show how to derive soliton solutions of the NLS equation in the presenceof the boundary condition (1.2). Our method is based on the tool of DTs [15] in conjunctionwith a boundary dressing technique [16].The well-known DT for the NLS equation is constructed as follows [15]. If ˜ u and ( ˜ φ , ˜ φ ) T satisfy differential equations (2.1), then so does u = ˜ u − i ( ξ − ¯ ξ ) f ¯ f | f | + | f | , (3.1a)( φ , φ ) T = D ( x, t, λ )( ˜ φ , ˜ φ ) T , (3.1b)where ( f , f ) T is a special solution of the linear auxiliary system (2.1) with ˜ u and with λ = ξ ,and D ( x, t, λ ) = ( λ − ¯ ξ ) I + ¯ ξ − ξ | f | + | f | | f | f ¯ f ¯ f f | f | ! . We now show that u and ˜ u connected by (3.1a) also satisfy a BT of the same form as (2.9).Indeed, differentiating (3.1) with respect to x and using f ,x = − iξ f + ˜ uf , f ,x = − ¯˜ uf + iξ f , u x − ˜ u x = − i Re ξ ( u − ˜ u ) − ( u + ˜ u ) q ξ ) − | u − ˜ u | . (3.2)Setting ξ = a + ib , equation (3.2) becomes (2.9a). Similarly, by differentiating (3.1) with respectto t and by using the fact that ( f , f ) T satisfies (2.1b) with ˜ u and λ = ξ , we obtain (2.9b).In light of the above arguments, we can use the DT to construct the solution of the NLSequation in the presence of the boundary condition (1.2). The strategy consists of the followingtwo steps: (1) find a solution ˜ u of the NLS equation satisfying the Dirichlet boundary condition˜ u | x =0 = 0 and find a solution ( f , f ) T of the corresponding Lax pair system; (2) by using theDT (3.1) with ξ = a + ib , construct a new solution u of the NLS equation. Such a solution u satisfies the time-dependent boundary condition (1.2), due to the fact that the boundary (1.2)can be derived by dressing the Dirichlet boundary with the integrable defect condition (thefrozen BT). In particular, if we find a N -soliton solution of ˜ u in the first step, then we can, afterusing the second step, generate the ( N + 1)-soliton solution u satisfying the boundary condition(1.2).We next implement the above two steps in details. It is evident that the trivial solution˜ u = 0 satisfies the Dirichlet boundary condition. The solution ( f ( ξ ) , f ( ξ )) T of the associatedLax pair system can be taken in the form of( f ( ξ ) , f ( ξ )) T = (cid:16) A e − iξ x − iξ t , B e iξ x +2 iξ t (cid:17) T , (3.3)where ξ = a + ib , and A and B are arbitrary constants. By substituting (3.3) into (3.1), weobtain the following single soliton satisfying the boundary condition (1.2), u = 4 b f ¯ f | f | + | f | = 2 b exp( − iax − i ( a − b ) t + iθ )cosh( bx + 4 abt + θ ) , (3.4)where exp( iθ ) = A ¯ B | A || B | and exp( θ ) = (cid:12)(cid:12)(cid:12) A B (cid:12)(cid:12)(cid:12) .It has been shown that non-trivial soliton solutions ˜ u satisfying the Dirichlet boundarycondition can be constructed by using the DT [16]. More precisely, the N -soliton solution, byapplying the DT to dress the Dirichlet boundary, are given by˜ u [ n ] = 2 i ∆ [ N ]∆ [ N ] , (3.5)1where∆ [ N ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ N − ν λ N − ν · · · λ N − N ν N ¯ λ N − ¯ µ ¯ λ N − ¯ µ · · · ¯ λ N − N ¯ µ N λ N − ν λ N − ν · · · λ N − N ν N ¯ λ N − ¯ µ ¯ λ N − ¯ µ · · · ¯ λ N − N ¯ µ N · · · · · · · · · · · · · · · · · · · · · · · · ν ν · · · ν N ¯ µ ¯ µ · · · ¯ µ N λ N − µ λ N − µ · · · λ N − N µ N − ¯ λ N − ¯ ν − ¯ λ N − ¯ ν · · · − ¯ λ N − N ¯ ν N λ N − µ λ N − µ · · · λ N − N µ N − ¯ λ N − ¯ ν − ¯ λ N − ¯ ν · · · − ¯ λ N − N ¯ ν N · · · · · · · · · · · · · · · · · · · · · · · · µ µ · · · µ N − ¯ ν − ¯ ν · · · − ¯ ν N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.6)and ∆ [ N ] are defined by the same determinant as ∆ [ N ] but with the first row replaced by (cid:0) − λ N µ , − λ N µ , · · · , − λ N N µ N , ¯ λ N ¯ ν , ¯ λ N ¯ ν , · · · , ¯ λ N N ¯ ν N (cid:1) , and λ j = − λ j − , ¯ λ j − = − λ j − , λ k − = − λ j − , j = 1 , · · · , N,µ j − = c j e − iλ j − x − iλ j − t , ν j − = d j e iλ j − x +2 iλ j − t , j = 1 , · · · , N,µ j = c j e iλ j − x − iλ j − t , ν j = d j e − iλ j − x +2 iλ j − t , j = 1 , · · · , N, (3.7)with c j and d j being arbitrary constants. The solution of associated Lax pair system are givenby ˜ φ [ N ] = ( ˜ φ [ N ] , ˜ φ [ N ]) T = D [ N ] ˜ φ [0] , (3.8)where ˜ φ [0] = (cid:16) Ae − iλx − iλ t , Be iλx +2 iλ t (cid:17) T ,D [ N ] = N − Y k =0 (cid:0) ( λ − ¯ λ N − k ) I + (¯ λ N − k − λ N − k ) P [2 N − k ] (cid:1) ,P [ j ] = 1 | µ j [ j − | + | ν j [ j − | | µ j [ j − | µ j [ j − ν j [ j − µ j [ j − ν j [ j − | ν j [ j − | ! , j = 1 , · · · , N, µ j [ j − ν j [ j − ! = j − Y k =1 (cid:0) ( λ − ¯ λ j − k ) I + (¯ λ j − k − λ j − k ) P [ j − k ] (cid:1)(cid:12)(cid:12) λ = λ j µ j ν j ! , j = 1 , · · · , N. (3.9)Inserting(3.5) and (3.8) into (3.1), we obtain the following ( N + 1)-soliton solution u = 2 i ∆ [ N ]∆ [ N ] − i ( ξ − ¯ ξ ) ˜ φ [ N ] ¯˜ φ [ N ] | ˜ φ [ N ] | + | ˜ φ [ N ] | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ = ξ , (3.10)where ξ = a + ib , ξ = λ j , j = 1 , · · · , N . Such a solution satisfies the NLS equation togetherwith the boundary condition (1.2).2 We have established the integrability of the NLS equation in the presence of time-dependentboundary condition (1.2) both via the Sklyanin’s formalism and via the tool of B¨acklund trans-formations together with a reduction technique. We also presented a direct method to constructsoliton solutions of such a boundary problem. Our method for the construction of solutions isbased on the Darboux transformation method in conjunction with a boundary dressing tech-nique. It is worth mentioning that an analogous time-dependent boundary for the integrablediscrete NLS equation was studied very recently in [11], where the nonlinear mirror imagemethod [17, 18] was applied to construct the solutions. We believe that the nonlinear mirrorimage method can also be applied to solve the boundary problem (1.2) for the NLS equation.The key point is to derive an additional symmetry (induced by the boundary) on the scatteringdata. A possible way to derive such a symmetry is by using the fact that the boundary (1.2)can be interpreted as arising from the B¨acklund transformation connected two fields with anodd reduction (see section 2.2). The study on this topic is not trivial, it will be performed inthe future.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant No.11771186).
References [1] C. Zambon, The classical nonlinear Schr¨odinger model with a new integrable boundary, J. HighEnerg. Phys. 2014 (2014) 36.[2] P. Bowcock, E. Corrigan and C. Zambon, Classically integrable field theories with defects, Int. J.Mod. Phys. A 19S2 (2004) 82.[3] E. Corrigan and C. Zambon, Jump-defects in the nonlinear Schr¨odinger model and other non-relativistic field theories, Nonlinearity 19 (2006) 1447.[4] I. Habibullin and A. Kundu, Quantum and classical integrable sine-Gordon model with defect, Nucl.Phys. B 795 (2008) 549.[5] V. Caudrelier, On a systematic approach to defects in classical integrable field theories, Int. J. Geom.Meth. Mod. Phys. vol. 5, No. 7 (2008) 1085.3