Dressing the boundary: on soliton solutions of the nonlinear Schrödinger equation on the half-line
DDressing the boundary: on soliton solutions ofthe nonlinear Schr¨odinger equation on the half-line
Cheng Zhang [email protected]
Department of Mathematics, Shanghai University99 Shangda Road, Baoshan DistrictShanghai, 200444, China
Abstract
Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin,we provide a direct method for computing soliton solutions of the focusing nonlinearSchr¨odinger (NLS) equation on the half-line. The integrable BCs at the origin arerepresented by constraints of the Lax pair, and our method lies on dressing the Laxpair by preserving those constraints in the Darboux-dressing process. The method isapplied to two classes of solutions: solitons vanishing at infinity and self-modulatedsolitons on a constant background. Half-line solitons in both cases are explicitly com-puted. In particular, the boundary-bound solitons, that are static solitons boundedat the origin, are also constructed. We give a natural inverse scattering transforminterpretation of the method as evolution of the scattering data determined by theintegrable BCs in space.
Key words: integrable boundary conditions, the NLS equaiton on the half-line, solitonsolutions, dressing transformations, inverse scattering transform
Research in initial-boundary-value problems for integrable nonlinear PDEs represents oneof the basic problems in integrable systems. One crucial aspect is that the boundaryconditions (BCs) are inherent in the definition of integrability. For instance, in the inversescattering transform (IST) approach to integrating the Korteweg-de Vries (KdV) equation,the vanishing BCs (the KdV field and its derivatives vanish as the space variable tendsto infinity) are explicitly imposed [1]. In the Hamiltonian formulation of integrable fieldtheory, the vanishing BCs are also needed in order to ensure the existence of infinitelymany conserved quantities [2,3]. Indeed, one could argue that integrable PDEs are said tobe integrable only if they obey certain BCs that preserve the integrability of the system.The common choices are vanishing BCs or periodic BCs as for the case of KdV.One of the systematic approaches to boundary-value problems to nonlinear PDEs weredue to Sklyanin [4] in the framework of the Hamiltonian field theory, cf. [5]. In Sklyanin’sapproach, the central object is the so-called classical reflection equation involving theclassical r matrix and boundary K ± matrices. The BCs at the two ends of an intervalare encoded into K ± . Then the notion of integrability in the presence of BCs is clearlydefined as the existence of infinitely many commuting flows. In fact, the theory is relatedto a far-reaching context as it represents the classical aspects of the quantum theory ofintegrability [6, 7]. 1 a r X i v : . [ n li n . S I] S e p he IST is the analytic approach for solving initial-value problems for integrable PDEs,cf. [5, 8]. To address to integrable PDEs on the half-line, or more generally on an inter-val with generic BCs, the IST has been remarkably generalized by Fokas to the unifiedtransform method [9–11]. The key idea of the unified transform method is lying on thesimultaneous treatments of both the initial and boundary data in the direct scatteringprocess. Then the scattering data are put into certain functional constraints usually for-mulated as Riemann-Hilbert problems in the inverse space.Although Fokas’ approach has the great advantage that both the initial and boundarydata are regarded on an equal footing, in general, it is a difficult task to solve the inverse(Riemann-Hilbert) problems to derive explicit solutions of the original PDEs. Moreover,in contrast to Sklyanin’s approach, there is no clear definition of integrable BCs in theunified transform method. Note that a special class of BCs, known as linearizable BCs,do exist in Fokas’ approach [10, 11]. The linearizable BCs reflect certain symmetry of thescattering data that can be used to simplify the Riemann-Hilbert problems. For certainmodels, they coincide with the integrable BCs, cf. [4, 10]. However, linearizable BCs are,a priori, not equivalent to integrable BCs (see Section 4 for a detailed example). As animportant application of the unified transform method, asymptotic solutions of integrablePDEs at large time, mostly accompanied with the linearizable BCs, can be derived [10,11].In this paper, we study the focusing nonlinear Schr¨odinger (NLS) equation iq t + q xx − | q | q = 0 , q := q ( x, t ) , (1.1)restricted to the half-line domain, i.e. x ≥
0, under the Robin BCs at x = 0. It is knownthat the Robin BCs are integrable BCs [4], and our aim is to compute exact solutions ofthe NLS equation on the half-line in imposing the integrable BCs.The NLS equation is an important model in mathematical physics (see for instance[5, 12, 13] for its integrable aspects). It can be solved by the IST under the vanishingBCs [12], or on a constant background that has non-vanishing BCs [14–16]. The first caseadmits the usual (bright) soliton solutions and the second has self-modulated solitons thatare solitonic envelops propagating on a background.The NLS model on the half-line has also been extensively studied in the literature.Sklyanin was the first to derive integrable BCs of the model on an interval. His workwas then followed by important contributions [17, 19–21] in which the main idea was toincorporate the integrable BCs into the powerful IST framework. The unified transformmethod has also been applied to the half-line NLS model [10,22]. In particular, the authorshad the Robin BCs as linearizable BCs, and derived asymptotic soliton solutions at largetime. In spite of a good understanding of the model, exact soliton solutions of NLS onthe half-line were only obtained rather recently in [23] where a nonlinear mirror-imagetechnique was applied. The method consists in extending the half-line space domain tothe whole axis. This reflects the space-inverse symmetry of NLS and allows to obtainsolutions of the model using the usual IST by uniquely looking at the positive semi axis.The technique was successfully applied to the vector NLS model [24], and was used toobtain boundary-bound solitons [25] that are static solitons subject to the BCs. Note thata recent study of the model was reported in [26] following a functional analytical approach.Our approach to solving the NLS equation on the half-line is based on the theory ofintegrable BCs. It is well-known that soliton solutions of integrable PDEs can be obtainedin an algebraic fashion using the Darboux-dressing transformations (DTs), cf. [27–29].The essential idea of our method is to “dress” the integrable boundary structures at theboundary point x = 0 in preserving the integrability. In other words, we are dressing thesystem at the boundary. Two classes of seed solutions are considered: the zero seed solu-ion, which correspond to the usual (bright) solitons, and the non-zero seed solution, whichhas self-modulated solitons on a constant background. In both cases, soliton solutions onthe half-line, as well as boundary-bound solitons, arise directly in the boundary-dressingprocess. One of the particular advantage of our method is that it does not require exten-sion of the space domain to the whole axis. It also admits a natural IST interpretation:the “boundary profile” at x = 0 is encoded into scattering data that evolve linearly onthe semi axis, then the half-line solutions can be obtained in solving the inverse problems.Although the main content of the paper is focusing on a particular model that is the NLSequation on the half-line, the scope of the paper is to address to a rather general questionof how to determine exact solutions of integrable PDEs accompanied with integrable BCs.The paper is organized as follows: we review Sklyanin’s approach to integrable BCsand the DT approach to generating soliton solutions in Section 2 and 3 respectively. Thislays the basis for the boundary-dressing method presented in Section 4. In Section 5 and6, the method is applied to two classes of soliton solutions of NLS. An IST interpretationof boundary-dressing method the is given in Section 7. We start by a brief summary of Sklyanin’s formalism [4] (see also [30] for recent studies)to derive integrable BCs for the NLS equation on an interval. This formalism is based onthe Hamiltonian formulation of integrable PDEs, cf. [5]. r -matrix structure The NLS equation is the result of the compatibility between the linear differential equations
U φ = φ x , V φ = φ t . (2.1)Here, the matrix-valued functions U and V , known as the Lax pair, depend on x, t andalso on a spectral parameter λ . They are in the forms U = − iλσ + Q , V = − iλ σ + 2 λ Q − iQ x σ − iQ σ , (2.2)where σ = (cid:18) − (cid:19) , Q = (cid:18) q − ¯ q (cid:19) . (2.3)We will call U and V the x -part and t -part of the Lax pair respectively.Now, restricting the space domain to an interval [ x − , x + ] with x ± being the two ends. Aparticularly important quantity called monodromy matrix T ( λ ), obtained from the x -partof the Lax pair at a fixed time t , is the following ordered exponential function T ( λ ) = (cid:120) exp (cid:90) x + x − U ( ξ, t , λ ) dξ , (2.4)for it encodes the spectral properties in the process of the direct scattering in the IST. Inthe Hamiltonian formulation of the NLS equation [5], two monodromy matrices satisfy { T ( λ ) ⊗ (cid:48) T ( η ) } = [ r ( λ − η ) , T ( λ ) ⊗ T ( η )] , (2.5) We drop the the x , t and λ dependence for conciseness unless there is ambiguity. here ⊗ is the standard tensor product for matrices, and the operation {· ⊗ (cid:48) ·} standsfor the Poisson bracket in tensor-product form. The quantity r ( λ ) is called the classical r matrix, and it satisfies the classical Yang-Baxter equation[ r ( λ − η ) , r ( λ ) + r ( η )] + [ r ( λ ) , r ( η )] = 0 . (2.6)It is in the form r ( λ ) = 12 λ P , P = , (2.7)where P is the permutation matrix acting on C ⊗ C . The formula (2.5) is a universalintegrable structure characterizing wide classes of classical integrable models [5]. It isentirely determined by the form of U , and is independent of conditions imposed at theboundaries x ± . It can be used to prove the integrability of the NLS equation under thequasi-periodic BCs [5]. However, for generic BCs, the above relations are not enough toguarantee the integrability. Another equation involving the BCs is needed. Remark 2.1
The quasi-periodic BCs for NLS read q ( x − ) = e iϕ q ( x + ) , (2.8) where ≤ ϕ < π . Both the vanishing and non-vanishing BCs can be seen as the specialcases of the quasi-periodic BCs [5]. We proceed to the Sklyanin’s formalism. Recall that x − and x + are the boundaries. Onecan show, if there exist nonsingular spectrum-dependent matrices K ± ( λ ) satisfying[ r ( λ − η ) , K ± ( λ ) ⊗ K ± ( η )] = ( I ⊗ K ± ( η )) r ( λ + η ) ( K ± ( λ ) ⊗ I ) − ( K ± ( λ ) ⊗ I ) r ( λ + η ) ( I ⊗ K ± ( η )) , (2.9)with I being the identity matrix, then the integrable algebra (2.5) is deformed to {T ( λ ) ⊗ (cid:48) T ( η ) } =[ r ( λ − η ) , T ( λ ) ⊗ T ( η )] + ( T ( λ ) ⊗ I ) r ( λ + η ) ( I ⊗ T ( η )) − ( I ⊗ T ( η )) r ( λ + η ) ( T ( λ ) ⊗ I ) , (2.10)with T ( λ ) = T ( λ ) K − ( λ ) T − ( − λ ). In this setting, a generating function τ ( λ ) can beconstructed as τ ( λ ) = tr ( K + ( λ ) T ( λ )) , (2.11)and satisfies { τ ( λ ) , τ ( η ) } = 0 . (2.12) The explicit form of r ( λ ) depends on the Poisson structure of the model. In this paper, we use { q ( x ) , ¯ q ( y ) } = − iδ ( x − y ) . he function τ ( λ ) is interpreted as a generating function of commuting integrals of mo-tion. In order that τ ( λ ) generates infinitely many conserved quantities, it should betime-independent, which is equivalent to the following conditions K + ( λ ) V ( λ, x + ) = V ( − λ, x + ) K + ( λ ) , (2.13) K − ( λ ) V ( − λ, x − ) = V ( λ, x + ) K − ( λ ) . (2.14)Therefore, the complete integrability of the NLS equation in the presence of boundariesis obtained. The integrable BCs are encoded into the boundary matrices K ± that satisfyboth Eq (2.9) and Eqs (2.13, 2.14). Remark 2.2
It is worth noting that as x ± tend to ±∞ , both vanishing BCs and non-vanishing BCs have K ± ∝ some constant diagonal matrices as solutions. They are trivialsolutions of Eqs (2.9) and (2.13, 2.14). Both cases are thus integrable. DT approach is a direct method for generating soliton solutions of integrable PDEs. Theunderlying structures are connected to the IST involving studies of the scattering proper-ties of the Lax system (2.1). Here we are mainly following the notations used in [29].
DTs are gauge transformations preserving the forms of the Lax pair. For integrable PDEs,B¨acklund transformations are results of DTs at the level of solutions, as an “old” solutionis transformed into a “new” solution. For notational purpose, we use U [0], V [0] and φ [0]to denote the undressed Lax system (2.1).DT for NLS can be defined as follows: suppose there is a gauge-like transformation φ [1] = D [1] φ [0] . (3.1)Now φ [1] satisfies the newly transformed system U [1] φ [1] = φ [1] x , V [1] φ [1] = φ [1] t . (3.2)The structure of U [1] , V [1] is required to be identical to that of U [0] , V [0]. They areconnected by D [1] x = D [1] U [0] − U [1] D [1] , D [1] t = D [1] V [0] − V [1] D [1] . (3.3)The matrix D [1] is called dressing matrix. There are several equivalent representations of D [1], and here we adopt the polynomial (in λ ) form D [1] = (cid:0) λ − ¯ λ (cid:1) + (cid:0) ¯ λ − λ (cid:1) P [1] , P [1] = ψ ψ † ψ † ψ . (3.4)The 2-vector ψ = ( µ , ν ) (cid:124) is a special solution of the undressed Lax system (2.1) at λ = λ , ψ † denotes the transpose conjugate of ψ . The dressing matrix D [1] defines aone-fold Darboux transformation, as it adds a pair of zeros to the undressed Lax system . This can be easily seen by taking the determinant of D [1]:det D [1] = ( λ − λ ) ( λ − ¯ λ ) . (3.5) utting D [1] into Eq (3.3) gives the form of Q [1] in terms of Q [0] and P [1]: Q [1] = Q [0] − i ( λ − ¯ λ )[ σ , P [1]] . (3.6)This is the reconstruction formula , and a one-soliton solution q [1] can be easily obtained.The function q [0] is commonly known as seed solution . In the process of DTs, the choiceof q [0] is important, because the Lax system (2.1) cannot be integrated with arbitrary q [0]. In practice, we choose the zero seed solution q [0] = 0 and the non-zero constantbackground seed solution q [0] = ρ e iρ t , ρ >
0. Both cases will be studied later in thepresence of integrable BCs. N -soliton solutions N -fold DTs can be constructed by iteration. Assume that there exist N linearly-independentspecial solutions ψ j = ( µ j , ν j ) (cid:124) of the undressed Lax system (2.1) evaluated at λ j , j =1 . . . N , then the N -fold dressing matrix D [ N ] is in the form D [ N ] = (cid:0) ( λ − ¯ λ N ) + (¯ λ N − λ N ) P [ N ] (cid:1) · · · (cid:0) ( λ − ¯ λ ) + (¯ λ − λ ) P [1] (cid:1) , (3.7)where P [ j ] = ψ j [ j − ψ † j [ j − ψ † j [ j − ψ j [ j − , ψ j [ j −
1] = D [ j − (cid:12)(cid:12) λ = λ j ψ j . (3.8)A series expansion of D [ N ] in λ leads to D [ N ] = λ N + λ N − Σ + λ N − Σ · · · + Σ N , (3.9)with the matrices Σ j , j = 1 . . . N to be determined. Inserting φ [ N ] = D [ N ] φ [0] into U [ N ] φ [ N ] = φ [ N ] x , V [ N ] φ [ N ] = φ [ N ] t , (3.10)and taking account of the forms of Q [0] and Q [ N ], one can obtain the reconstructionformula for N -soliton solutions q [ N ] = q [0] + 2 i Σ (1 , , (3.11)where Σ (1 , is the (1 ,
2) entry of Σ . It can be put into the following compact form q [ N ] = q [0] + 2 i ∆ ∆ , (3.12)where ∆ = (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 µ · · · − λ NN µ N ¯ λ N ¯ ν · · · ¯ λ NN ¯ ν N λ N − ν · · · λ N − N ν N ¯ λ N − ¯ µ · · · ¯ λ N − N ¯ µ N · · · · · · · · · · · · · · · · · · ν · · · ν N ¯ µ · · · ¯ µ N λ N − µ · · · λ N − N µ N − ¯ λ N − ¯ ν · · · − ¯ λ N − N ¯ ν N · · · · · · · · · · · · · · · · · · µ · · · µ N − ¯ 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) , (3.13)∆ = (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 − N ν N ¯ λ N − ¯ µ · · · ¯ λ N − N ¯ µ N · · · · · · · · · · · · · · · · · · ν · · · ν N ¯ µ · · · ¯ µ N λ N − µ · · · λ N − N µ N − ¯ λ N − ¯ ν · · · − ¯ λ N − N ¯ ν N · · · · · · · · · · · · · · · · · · µ · · · µ N − ¯ 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) . (3.14)This formula will be used later in the computation of half-line solutions of the NLS equa-tion. A brief proof is given in Appendix A. Dressing integrable BCs
Now we proceed to the main results of the paper. The Sklyanin’s formalism, described inSection 2, characterizes the integrability of the NLS equation on an interval x − ≤ x ≤ x + .An half-line problem can be easily realized as a special case of the interval problem bysetting x − = 0 and x + → ∞ .Our aim is to compute exact solutions of the NLS equation on the half-line withoutspecial treatments such as extending the space domain into the whole line, or performingasymptotic analysis at large t . One knows that the integrable BCs are completely deter-mined by the t -part of the Lax pair through Eqs (2.13, 2.14) (with the boundary matricesalso satisfying Eq (2.9)), and that the DT approach to generating soliton solutions can,in principle, be defined in any space-time domain. The problem remains to find suitableDT for the NLS model on the half-line, with the t -part of the Lax pair satisfying theconstraints (2.13, 2.14). According to the Sklyanin’s formalism, the boundary matrices K ± ( λ ) can be treated sep-arately . In the rest of the paper, we assume that K + is proportional to some constantdiagonal matrices, which are trivial solutions of Eqs (2.14) and (2.9). This assumption canbe justified since we only consider two classes of solutions: solutions under the vanishingBCs as x → ∞ lim x →∞ q = 0 , lim x →∞ q x = 0 , (4.1)which have K + ∝ I , and solutions under the “constant” background condition (non-vanishing BCs) lim x →∞ q = ρ e iρ t , lim x →∞ q x = 0 , (4.2)with ρ being a positive real constant acting as the background, which have K + ∝ σ .For conciseness, we use K instead of K − , then one has to solve the boundary constraint K ( λ ) V ( − λ ) (cid:12)(cid:12) x =0 = V ( λ ) (cid:12)(cid:12) x =0 K ( λ ) , (4.3)with K ( λ ) also satisfying (2.9). This constraint corresponds to the linearizable BCs in theFokas’ unified transform method [10, 22]. A class of solutions is the Robin BCs: q x (cid:12)(cid:12) x =0 = α q (cid:12)(cid:12) x =0 , α ∈ R , (4.4)which has the boundary matrix K ( λ ) in the form K ( λ ) = (cid:18) f α ( λ ) 00 1 (cid:19) , f α ( λ ) = i α + 2 λi α − λ . (4.5)One can easily check that K − ( λ ) = K ( − λ ). The real parameter α controls the boundarybehavior. The limiting cases of α give rise either to Dirichlet BCs q (cid:12)(cid:12) x =0 = 0 , α → ∞ , K = I , (4.6)or to the Neumann BCs q x (cid:12)(cid:12) x =0 = 0 , α = 0 , K = − σ . (4.7)We exclude the special cases λ = ± iα due to the non-degeneracy of K ( λ ). One cancheck that the boundary matrix K ( λ ) defined in (4.5) satisfies the classical reflectionequation (2.9), which concludes that the Robin BCs (4.4) are integrable BCs, cf. [4,18,19]. emark 4.1 It is interesting to see that the Robin BCs are not the only solutions of theboundary constraint (4.3) . For instance, one can have q (cid:12)(cid:12) x =0 = ρ e iθ , ¯ q x (cid:12)(cid:12) x =0 = e − iθ q x (cid:12)(cid:12) x =0 , (4.8) as the BCs, with ρ (cid:54) = 0 , θ ∈ R . The corresponding boundary matrix K ( λ ) is in the form K ( λ ) = (cid:32) ie iθ λρρ − λ ie − iθ λρρ − λ (cid:33) . (4.9) This solution fits perfectly into the definition of linearizable BCs in the unified transformmethod. They can be in theory used in the simultaneous treatments of both the x -part and t -part of the Lax pair, in which the boundary matrix K ( λ ) will imply certain reductions ofthe scattering coefficients of the initial-boundary data. However, the matrix K ( λ ) in thiscase does not satisfy the classical reflection equation (2.9) . Therefore, the BCs (4.8) arenot integrable. The boundary constraint (4.3) with the boundary matrix K ( λ ) defined in (4.5) can beinterpreted as a DT for certain Lax pair with its space domain extended to the whole axis.Imposing the following relation (cid:101) Q ( x, t ) = Q ( − x, t ) . (4.10)One can verify that (cid:101) V (cid:12)(cid:12) x =0 = σ V ( − λ ) (cid:12)(cid:12) x =0 σ , (4.11)with (cid:101) V defined in terms of (cid:101) Q ( x, t ). The relation (4.10) is actually a B¨acklund transforma-tion of NLS due to its space-inverse symmetry. Taking account of (4.3), the above formulacan be expressed as a DT for V at x = 0: D ( λ ) V ( λ ) (cid:12)(cid:12) x =0 − (cid:101) V ( λ ) (cid:12)(cid:12) x =0 D ( λ ) = 0 , D ( λ ) = σ K ( − λ ) . (4.12)The boundary constraint (4.3) is just reformulated as a DT (4.12), with Q transformed to (cid:101) Q as the action of the DT. In order that the DT (4.12) is also defined for the x -part ofthe Lax pair U , one needs to introduce a functional Π in the formΠ( x, t, λ ) = H ( x ) Q ( x, t ) + H ( − x ) D ( − λ ) (cid:101) Q ( x, t ) D ( λ ) , (4.13)where H is the Heaviside function. Now the functional Π, combining both Q and (cid:101) Q , is afield defined on the whole real axis. It satisfies (cid:101) Π( x, t, λ ) := Π( − x, t, − λ ) = D ( λ ) Π( x, t, λ ) D ( − λ ) . (4.14)Using the extended field Π instead of Q in the Lax pair, i.e. U = − iλ σ + Π , V = − iλ σ + 2 λ Π − i Π x σ − i Π σ , (4.15)one can verify the following relations D ( λ ) U ( λ ) (cid:12)(cid:12) x =0 − (cid:101) U ( λ ) (cid:12)(cid:12) x =0 D ( λ ) = 0 , D ( λ ) V ( λ ) (cid:12)(cid:12) x =0 − (cid:101) V ( λ ) (cid:12)(cid:12) x =0 D ( λ ) = 0 . (4.16)hese relations incorporate the boundary constraint (4.3) into the Lax pair U, V with anextended field Π. The equivalence between the extended Lax pair (4.15) and the integrableBCs is thus established. The apparent advantage of the introduction of the extended fieldΠ is that it allows us to perform the usual IST. This corresponds to the mirror-imagemethod for solving the NLS equations [23]. However, there are some analytic difficultiesin dealing with the functional field Π as it contains the Heaviside function which requiresspecial care at x = 0. Moreover, extending the space domain to the whole real axis doesnot seem to be the most natural approach to a half-line problem. The construction of exact solutions of the NLS equation on the half-line is based on thefollowing steps. At the seed solution level, taking account of the boundary constraint (4.3),given ψ ( λ ) as a special solution of the undressed Lax system U [0] and V [0], if there existanother paired special solution ψ ( λ ) obeying ψ (0 , t, λ ) = K ( λ ) ψ (0 , t, λ ) , λ = − λ , ¯ λ (cid:54) = − λ , (4.17)then the seed solution q [0] satisfies the BCs imposed by (4.3). This can be easily understoodby looking at the t -part equation of the Lax system at x = 0 V [0](0 , t, λ j ) ψ j (0 , t, λ j ) = ψ j t (0 , t, λ j ) , j = 1 , . (4.18)By inserting the relation (4.17) into the previous equations one gets K ( λ j ) V [0]( − λ j ) K ( − λ j ) ψ j = V [0]( λ j ) ψ j = ψ j t ( λ j ) , j = 1 , , (4.19)at x = 0, which implies V [0] is subject to the boundary constraint (4.3). Note that weexclude for the moment the case where λ is a pure imaginary (because of the condition¯ λ (cid:54) = − λ ). In fact, the existence of (4.17) is a strong condition which allows to constructsoliton solutions satisfying the BCs imposed by (4.3). Proposition 4.1 (Dressing the boundary)
Consider the half-line NLS model with RobinBCs (4.4) . Assume that there exist paired special solutions { ψ , (cid:98) ψ } of the undressed Laxsystem (2.1) associated with the parameters { λ , (cid:98) λ } such that (cid:98) ψ (0 , t, (cid:98) λ ) = K ( (cid:98) λ ) ψ (0 , t, λ ) , (cid:98) λ = − λ , ¯ λ (cid:54) = − λ , (4.20) where K ( λ ) is given in (4.5) , then a two-fold DT using such pair leads to a V [2] satisfying K ( λ ) V [2]( − λ ) | x =0 = V [2]( λ ) K ( λ ) | x =0 . (4.21) The so-constructed q [2] satisfies the Robin BCs. We use (cid:98) q [1] to denote such q [2] . Proof: In order that (cid:98) q [1] is an exact solution of the NLS equation on the half-line, weneed to prove the relation (4.21). Let D [2]( λ ) be the dressing matrix constructed using { ψ , (cid:98) ψ } . One knows that V [2]( λ ) and V [0]( λ ) are connected by V [2]( λ ) = D [2] t ( λ ) D [2] − ( λ ) + D [2]( λ ) V [0]( λ ) D [2] − ( λ ) . (4.22)One can easily verify that if D [2]( λ ) satisfies D [2]( λ ) K ( λ ) | x =0 = K ( λ ) D [2]( − λ ) | x =0 , (4.23)hen V [2]( λ ) satisfies (4.21). Multiplying both sides of Eq (4.23) by an irrelevant factor( i α − λ ). Since D [2] can be expressed as a matrix polynomial of degree 2 in λ and that( iα − λ ) K ( λ ) = i αI + 2 λσ , the l.h.s. and r.h.s. of (4.23) are thus matrix polynomialsof degree 3. We use L ( λ ) and R ( λ ) to denote them L ( λ ) = D [2]( λ ) K ( λ ) = λ L + λ L + λ L + L , (4.24) R ( λ ) = K ( λ ) D [2]( − λ ) = λ R + λ R + λ R + R . (4.25)Clearly, L = R , L = R , and L , L , R , R can be determined by the zeros of L ( λ ) , R ( λ )and the associated kernel vectors, cf. [31, Chapiter (3.10)]. Following the zeros of D [2]( λ ), i.e. D [2]( λ ) ψ = 0 and D [2]( (cid:98) λ ) (cid:98) ψ = 0, and the relation between ψ and (cid:98) ψ (4.20), onehas R ( λ ) | λ = − λ ψ = 0 , L ( λ ) | λ = − λ ψ = 0 , R ( λ ) | λ = λ (cid:98) ψ = 0 , L ( λ ) | λ = λ (cid:98) ψ = 0 , (4.26)evaluated at x = 0. Moreover, let ϕ = σ ¯ ψ , σ = (cid:18) − ii (cid:19) , one can verify (see alsoAppendix A) R ( λ ) | λ = − ¯ λ ϕ = 0 , L ( λ ) | λ = − ¯ λ ϕ = 0 , R ( λ ) | λ =¯ λ (cid:98) ϕ = 0 , L ( λ ) | λ =¯ λ (cid:98) ϕ = 0 , (4.27)at x = 0, where (cid:98) ϕ is defined in the same way as ϕ . The above formulae show that L ( λ )and R ( λ ) share the same zeros and the associated kernel vectors, which, in turn, impliesthat L ( λ ) = R ( λ ). This completes the proof.The above construction is the realization of “dressing the boundary”. In fact, theexistence of the paired special solutions { ψ ( λ ) , (cid:98) ψ ( (cid:98) λ ) } implies that the t -part of un-dressed Lax pair V [0] satisfies the boundary constraint (4.3). Thus the seed solution q [0]is “presumed” to be subject to the integrable BCs (4.4). Dressing V [0] using the pair { ψ ( λ ) , (cid:98) ψ ( (cid:98) λ ) } preserves the boundary constraint (4.3), hence the BCs (4.4). The so-constructed solution (cid:98) q [1] represents a one-soliton solution on the half-line, although twospecial solutions are involved. This can be naturally understood as follows: the paired spe-cial solutions { ψ ( λ ) , (cid:98) ψ ( (cid:98) λ j ) } create asymptotically at large t two independent solitonswith opposite vilocities, as time evolves from −∞ to ∞ , only one soliton remains in x ≥ N paired special solutions. Proposition 4.2 ( N -soliton solutions) Consider the half-line NLS model with RobinBCs (4.4) . Assume that there exist N paired special solutions { ψ j ( λ j ) , (cid:98) ψ j ( (cid:98) λ j ) } , j =1 , . . . , N , of the undressed Lax system (2.1) such that (cid:98) ψ j (0 , t, (cid:98) λ j ) = K ( (cid:98) λ j ) ψ j (0 , t, λ j ) , (cid:98) λ j = − λ j , ¯ λ j (cid:54) = − λ j , (cid:98) λ k (cid:54) = λ j , (4.28) where K ( λ ) is given (4.5) . Then, the so-constructed q [2 N ] corresponds to an N -solitonsolution of NLS on the half-line. We use (cid:98) q [ N ] to denote such q [2 N ] . he requirement (cid:98) λ k (cid:54) = λ j ensures that all the special solutions are independent. By con-struction, the integrable structures such as the boundary constraint (4.3) and the BCs (4.4)are preserved at each step of the dressing. It remains to find such pairs { ψ j ( λ j ) , (cid:98) ψ j ( (cid:98) λ j ) } .Also note that the case where ψ j has pure imaginary spectral parameter is not includedinto the above formalism. These problems are considered in the next two Sections whenwe deal with the concrete examples. We consider the zero seed solution q [0] = 0, which has the vanishing BCs as x → ∞ . Thespecial case where ψ j has pure imaginary parameter is also treated. The latter correspondsto static solitons bounded to the boundary. It is straightforward to apply Prop. 4.2. The zero seed solution implies that the specialsolution ψ j ( λ j ), j = 1 , . . . , N , is in the form ψ j ( λ j ) = (cid:18) µ i ν i (cid:19) = e ( − iλ j x − iλ j t ) σ (cid:18) u j v j (cid:19) , (5.1)where λ j is a complex parameter. Here ( u j , v j ) (cid:124) is a constant vector. Obviously, (cid:98) ψ j ( (cid:98) λ j ) = K ( − λ j ) ψ j ( − λ j ) , (5.2)is the paired special solution of ψ j ( λ j ). Now having the data { ψ j ( λ j ) , (cid:98) ψ j ( (cid:98) λ j ) } , we areready to compute the N -soliton solutions of the NLS equation on the half-line. Twosolitons interacting with the boundary are illustrated in Fig. 1 and 2.Figure 1: Two-soliton solution satisfying the Dirichlet BCs (left) and Neumann BCs (right) Static soliton solutions arise as the special solutions ψ j ( λ j ) having pure imaginary param-eters λ j . Since ¯ λ j = − λ j , each ψ j needs to be paired with itself to eventually make theboundary constraint preserved (4.3) in the boundary-dressing process. - - Figure 2: Two-soliton solution satisfying the Robin BCs ( α = 3) Proposition 5.1 (Boundary-bound solitons)
Assume that, associated with N distinctpure imaginary parameters λ j = iκ j , κ j ∈ R , there exist N special solutions ψ j ( λ j ) , j = 1 , . . . , N , of the undressed Lax system (2.1) with zero seed solution, in the forms ψ j ( λ j ) = ψ j ( iκ j ) = (cid:18) µ i ν i (cid:19) = e ( κ j x +2 iκ j t ) σ (cid:18) u j (cid:19) , x ≥ , (5.3) such that κ j satisfies f α ( iκ j ) = α + 2 κ j α − κ j < , (5.4) with α being a real parameter and u j being defined as u j = (cid:113) − f α ( iκ j ) ( − N , (5.5) then the so-construction q [ N ] corresponds to a static N -soliton solution satisfying ( q [ N ] x − αq [ N ]) | x =0 = 0 . (5.6)Proof: The proof is split into two cases: when N is odd and when N is even. Moreover,for simplicity, we only consider the cases where N = 1 and N = 2 since the odd N andthe even N cases can be understood as the generalizations.N=1: since the zero seed solutions is imposed, one has V [0] = − iλ σ . The one-stepDT involves a dressing matrix D [1] constructed from a special solution ψ ( λ ). Using theidentity ( iaI ± λσ ) = ( ia ∓ λ ) K ( ± λ ), one can show if D [1] obeys( iaI − λσ ) D [1]( λ ) | x =0 = D [1]( − λ )( iaI + 2 λσ ) | x =0 , (5.7)then the dressed V [1] = D [1] t D [1] − + D [1] V [0] D [1] − satisfies the boundary constraints(4.3). It remains to prove the relation (5.7). Knowing that D [1]( λ ) is a matrix polynomialof degree 1 in λ , the l.h.s. and r.h.s. of (5.7) are thus polynomials of degree 2. Denotethem by L ( λ ) =( iaI − λσ ) D [1]( λ ) = λ L + λ L + L , (5.8) R ( λ ) = D [1]( − λ )( iaI + 2 λσ ) = λ R + λ R + R . (5.9)learly, L = R , L = R . This explains the “presumed” form (5.7): D [1]( λ ) is of odddegree in λ , thus (5.7) ensures that the leading and zero-degree terms of the both sidesare equal. The equality (5.7) holds, if K ( λ ) ψ | x =0 = σ ¯ ψ | x =0 . (5.10)In fact, let ϕ = σ ¯ ψ , D [1]( − λ ) ϕ = 0. One can show that L ( λ ) | λ = λ ψ = 0 , R ( λ ) | λ = λ ψ = 0 , L ( λ ) | λ = − λ ϕ = 0 , R ( λ ) | λ = − λ ϕ = 0 , (5.11)meaning that L ( λ ) and R ( λ ) share the same zeros and the associated kernel vectors, thus L ( λ ) = R ( λ ). Moreover, the constraint (5.10) imposes f α ( iκ ) u = − i ¯ v , v = i ¯ u , (5.12)where u , v are elements of the constant vector appearing in the general expression of ψ j (6.7). The above constraints lead to f α ( iκ ) | u | = −| v | . (5.13)Because of the requirement that f α ( iκ ) <
0, and without loss of generality, letting | v | =1 and u be real, then u = 1 / (cid:112) − f α ( iκ ). One recovers the statements (5.5) for N = 1.Note that the assumption (5.7) is needed for any odd N , this impose similar constraintson u j , v j as shown in (5.13).N=2: similarly, one needs the following identity( iaI − λσ ) D [2]( λ ) | x =0 = D [2]( − λ )( iaI − λσ ) | x =0 . (5.14)Then the dressed V [2] = D [2] t D [2] − + D [2] V [0] D [2] − satisfies the boundary constraints(4.3). As previously explained, this identity ensures the equality of the leading and zero-degree terms of the both sides of (5.14). The relation holds if K ( − λ j ) ψ j | x =0 = σ ¯ ψ j | x =0 , j = 1 , . (5.15)In compononts, it reads f α ( − iκ j ) | u j | = 1 f α ( iκ j ) | u j | = −| v j | , j = 1 , . (5.16)Again let | v j | = 1 and u j be real, one obtains (5.5) for N = 2. This condition is true forany even N . Remark 5.1
The requirement f α ( iκ j ) < excludes the Dirichlet BCs, i.e. α → ∞ , forthe boundary-bound solitons. Following the above proposition, one can easily compute static solitons bounded tothe boundary under the Robin BCs. When there are multi-static solitons bounded tothe bounadry, the interference phenomena take place (see Fig. 3). One can dress theboundary using both the moving and static soliton data for the boundary constraint (4.3)is preserved at each step of the dressing process (see Fig. 4). Note that similar resultswere obtained in [25] using the mirror-image technique.igure 3: Boundary-bound solitons under the Robin BCs ( α = 3): the magnitude (norm)is constant along the boundary for the one-soliton case (left); an interference phenomenontakes place for a doubly-boundary-bound soliton (right) - - - Figure 4: Interaction between a (moving) soliton and a doubly-boundary-bound solitonunder the Robin BCs ( α = 3) In this section, we are dealing with the non-zero seed solution q [0] = ρ e iρ t , ρ > , (6.1)where ρ represents the constant background. The model is more involved than the zeroseed solution case because a two-valued function related to the spectral parameter of thespecial solution appears, cf. [5, 14–16]. It admits self-modulated soliton solutions. We briefly recall the DT approach to the model, cf. [32]. It follows from a gauge transfor-mation φ → D ρ φ to the undressed Lax system (2.1) with D ρ = (cid:18) e iρ t (cid:19) , (6.2)that the Lax pair is transformed to two constant matrices U = − iλ σ + (cid:18) ρ − ρ (cid:19) , V = − i U + 2 λ U + i ( λ I + 2 ρ ) , (6.3)he eigenvalues of U are ± iξ with ξ satisfying ξ = λ − ρ , (6.4)and those of V are i ( ρ ± λ ξ ). Here ξ can be seen as a the two-valued function of λ .The matrices U , V can be simultaneously diagonalized following M − U M = iξσ , M − V M = i ( ρ I + 2 λ ξσ ) , (6.5)where M := M ( λ, ξ ) = (cid:18) i ( λ − ξ ) /ρ i ( λ − ξ ) /ρ (cid:19) . (6.6)Combining the above analysis, a special solution of the undressed Lax system (2.1) withthe constant background seed solution (6.1) is in the form ψ j ( λ j , ξ j ) = (cid:18) µ i ν i (cid:19) = D − ρ M ( λ j , ξ j ) e i ( ρ tI + ξ j ( x +2 λ j t ) σ ) (cid:18) u j v j (cid:19) , (6.7)where ξ j and λ j are related by (6.4), and u j , v j are constants. Although ξ j depends on λ j ,we put the explicit dependence of ξ j because the sign is important in later determinationof soliton solutions on the half-line. Clearly, inserting µ i , ν i , j = 1 , . . . , N , into (3.12) givesrise to N -soliton solutions of the model. One can easily check that the seed solution (6.1) is subject to the Neumann BCs and σ V [0]( − λ ) | x =0 = V [0]( λ ) | x =0 σ , (6.8)with V [0] being the t -part of the undressed Lax pair. Now we are looking to dress the Laxpair by preserving the boundary constraint (6.8).Consider the case λ j (cid:54) = − ¯ λ j . Given ψ j ( λ j , ξ j ) in the form (6.7), let the paired specialsolution be in the form (cid:98) ψ j ( (cid:98) λ j , (cid:98) ξ j ) = (cid:18)(cid:98) µ i (cid:98) ν i (cid:19) = D − ρ M ( − λ j , − ξ j ) e i ( ρ tI − ξ j ( x − λ j t ) σ ) (cid:18) − u j v j (cid:19) . (6.9)Using the identity σ M ( λ j , ξ j ) = − M ( − λ j , − ξ j ) σ , one can verify that (cid:98) ψ j ( (cid:98) λ j , (cid:98) ξ j ) | x =0 = σ ψ j ( λ j , ξ j ) | x =0 . (6.10)One can see that the paired special solution requires not only (cid:98) λ j = − λ j but also (cid:98) ξ j = − ξ j .It is straightforward to apply Prop. 4.2 to obtain N -soliton solutions on the half-line underthe Neumann BCs.For static solitons ( λ j = − ¯ λ j ), one can used the similar ideas as presented in Prop. 5.1.By letting u j = v j = 1, one can show the boundary constraint (6.8) is preserved at eachstep of the dressing. Examples of half-line self-modulated solitons are illustrated in Fig. 5.Moreover, one can combine the (moving) soliton and boundary-bound solitons together(see Fig. 6).igure 5: A self-modulated soliton on a constant background interacting with the boundary(left) and a doubly-boundary-bound soliton (right) - Figure 6: Interaction between a (moving) soliton and a boundary-bound soliton under theNeumann BCs on a constant background
The IST is an analytic method for solving initial-value problems for integrable PDEs withthe space-time domain restricted to x ∈ R , t ≥ t . It is essentially made of three steps:first, the direct scattering process where the initial conditions at t = t are transformedinto scattering data using the x -part of the Lax pair; second, the time-evolution process where the scattering data evolve linearly in time according to the t -part of the Lax pair;and last, the inverse problem where the scattering data are put into the reconstructionformulae to recover solutions of the integrable PDEs.In contrast to the usual IST, the unified transform approach to half-line problems isto restrict the space-time domain to x ≥ t ≥ t . Then, in the direct scattering processone encodes both the initial and boundary conditions into the scattering data. However,it is a hard problem to solve the inverse problem even as for a rather simple situation suchas the NLS equation on the half-line where exact solutions do exist.In order to overcome this problem and to fit our approach to exact solutions on thehalf-line into the IST scheme, one needs to extend the space-time domain into x ≥ t ∈ R . This can be compared with the usual IST, and in turn, a boundary-value problemis defined where the “initial boundary profile” is imposed by the BCs at x = 0. It turnsout that the boundary-value problem can be solved using a space-evolution process wherescattering data are determined by the t -part of the Lax pair at x = 0 and evolve linearlyin space for x ≥ t -partof the Lax pair V at x = 0. For simplify, we only consider the zero seed solution case.This imposes the vanishing asymptotic conditions under which the NLS field q and its x derivatives vanish as t → ±∞ . Then one can have the Jost solutionslim t →±∞ φ ± (0 , t ; λ ) = e − iλ tσ . (7.1)Due to the λ -dependence of the spectral parameter, the analytical domain of Jostsolutions can be naturally split into four quadrants, which leads to a “time” monodromymatrix S ( k ) in the form S ( λ ) = (cid:18) a (24) ( λ ) ¯ b (13) ( λ ) b (24) (¯ λ ) ¯ a (13) (¯ λ ) (cid:19) . (7.2)Here the subscript of a (24) ( λ ) means that the scattering function a (24) ( λ ) can be analyti-cally continued to the union of the quadrants (2) and (4) (see Fig. 7 for the distributionof the quadrants). This applies also to other scattering functions.(1)(2) (4)(3) ¯ λ j ¯ (cid:98) λ j λ j (cid:98) λ j Figure 7: Zeros of the scattering function a (24) ( λ ) in the presence of Robin BCsApparently, the direct scattering of V at x = 0 differs from the usual IST only by theuse of potential. Here the potential is Q T as V can be written as V = − iλ σ + Q T , Q T = 2 λ Q − iQ x σ − iQ σ . (7.3)This switches the roles of initial and boundary conditions: instead of characterizing initialprofile, the BCs are first considered and encoded into S ( k ). Then following the asymptoticconditions imposed to the NLS field q as t → ±∞ , one can easily show that the scatteringdata evolve linearly in x as S ( λ ) obeys ∂S ( k ) ∂x = − iλ [ σ , S ( k )] . (7.4)The Jost solutions can be put into certain ( x, t )-dependent Riemann-Hilbert problem,which eventually lead to soliton solutions of NLS with zeros of a (24) ( λ ) appearing in theunion of the quadrants (2) and (4).Having the above space-evolution process in mind, we are ready to implement theRobin BCs into the system. Since V is required to obey the boundary constraint (4.3), anadditional constraint on S appears S ( − λ ) = K ( − λ ) S ( λ ) K ( λ ) , (7.5)ith the boundary matrix K ( λ ) defined in (4.5). Consequently, the Robin BCs impliesthat if λ j is a zero of a (24) ( λ ), so does − λ j because a (24) ( − λ ) = a (24) ( λ ) . (7.6)The associated norming constants are related by b (24) ( − λ j ) = f a ( − λ ) b (24) ( λ j ) . (7.7)Therefore, the paired zeros of a (24) ( λ ) (see Fig. 7) and the relation between the pairednorming constants give the underlying reason of the paired special solutions { ψ j ( λ j ) , (cid:98) ψ j ( (cid:98) λ j ) } in boundary-dressing process in Prop. 4.2. Note that the relation (7.5) is in contrast tothe mirror-image technique where the pairing of zeros of the scattering function is relatedby spectra { λ j , − ¯ λ j } , cf. [23, Equation (2.33)]. By carefully reviewing the integrable BCs for the NLS equation, we provide a directmethod for computing exact solutions of the focusing NLS equation on the half-line underthe Robin BCs. The method is lying on dressing the integrable boundary constraints byappropriate pairing of special solutions in the Darboux-dressing process. Two classes ofseed solutions are considered, which lead to usual (bright) solitons on the half-line andself-modulated solitons on the half-line respectively. In particular, the boundary-boundsolitons are computed in both cases. The method has the advantage that it is simpleand direct. It admits a natural IST interpretation as evolution in space of the integrableboundary data.It is believed that the boundary-dressing approach can be applied to a wide rangeof problems where the integrable boundary structures exist. As to the NLS case, onecan, for instance, compute half-line dark solitons which correspond to exact solutions ofthe defocusing NLS equation on the half-line . Applications of the boundary-dressingmethod to computing exact half-line solutions of the sine-Gordon equation was recentlyobtained [33]. Other extensions of the method can be related to dressing the boundary ona star-graph [34] where similar boundary constraints appear, or to tackle integrable PDEson an interval where the algebraic-geometric integration technique may be involved [35,36]. Acknowledgments
The author is supported by NSFC (No.11601312) and Shanghai Young Eastern Scholarprogram (2016-2019).
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Appendix A: Derivation of D [ N ] Recall the series expansion of D [ N ] D [ N ] = λ N + λ N − Σ + λ N − Σ · · · + Σ N . (A.1)The complete determination of Σ j relies on the identification of the kernel vectors of D [ N ]( λ ) at its zeros λ j , ¯ λ j . This can be done with the help of the following lemma. Lemma A.1
Let ψ j = ( µ j , ν j ) (cid:124) , j = 1 . . . N , be special solutions of the Lax system (2.1) evaluated at λ j . Then a set of N vector functions ϕ j , defined by ϕ j = cσ ¯ ψ j = (cid:18) − ¯ ν j ¯ µ j (cid:19) , c = − i , σ = (cid:18) − ii (cid:19) . (A.2) satisfy U (¯ λ j ) ϕ j = ϕ jx , V (¯ λ j ) ϕ j = ϕ jt . (A.3)The proportionality constant c in (A.2) is irrelevant and can be replaced by any non-zeronumber. It is easy to check ψ j and ϕ j satisfy the orthogonality condition ϕ † j ψ j = 0 , j = 1 , · · · , N . (A.4)The dressing matrix D [ N ] has kernel vectors ϕ j , j = 1 . . . N at λ = ¯ λ j , thus ψ j and ϕ j give us a complete characterisation of Σ j . Lemma A.2