Dressing a new integrable boundary of the nonlinear Schrödinger equation
DDressing a new integrable boundary of the nonlinearSchrödinger equation
K.T. Gruner ∗ [email protected] Universität zu Köln, Mathematisches Institut,Weyertal 86-90, 50931 Köln, Germany
August 07, 2020
Abstract
We further develop the method of dressing the boundary for the focusing non-linear Schrödinger equation (NLS) on the half-line to include the new boundarycondition presented by Zambon. Additionally, the foundation to compare the so-lutions to the ones produced by the mirror-image technique is laid by explicitlycomputing the change of scattering data under the Darboux transformation. Inparticular, the developed method is applied to insert pure soliton solutions.
Keywords:
NLS equation, integrable boundary conditions, half-line, initial-boundaryvalue problems, soliton solutions, dressing transformation, inverse scattering method.
The nonlinear Schrödinger equation is one of the well-known examples, where the modelof a physical phenomenon incorporates both nonlinearity and dispersion in such a waythat a soliton—(i) a wave of permanent form (ii) which is localized (iii) and can interactstrongly with other solitons and retain its identity—emerges. For integrable modelsas the NLS equation is, these special solutions have been worked out extensively andprimarily by the inverse scattering method. In this context it should be noted thatmany physical phenomena naturally arise as initial-boundary value problems, due to thelocalized character of the problem. Nonetheless, characterizing soliton solutions for theseproblems is substantially less developed than for the corresponding initial value problems,which is therefore an objective of this work.Similar to the model of the initial value problem, the integrability is a crucial propertyto be able to derive soliton solutions. Besides the usually addressed boundary conditions, ∗ The author is partially supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebraand Dynamics’, funded by the DFG. a r X i v : . [ m a t h - ph ] A ug he Dirichlet, Neumann and Robin boundary condition [2], in that category for the NLSequation, a new integrable boundary has been derived by dressing the Dirichlet boundarywith a defect [6]. To then find soliton solutions in these models different method havebeen successfully applied.One of these methods, which is called “dressing the boundary”, utilizes the Darbouxtransformation in a way which preserves the integrable structures of the system at theboundary. Again, the usual boundary conditions in the case of the NLS equation provideda suitable framework to apply this method [7], after it had already been used to produceresults in the case of the sine-Gordon equation on the half-line together with integrableboundary conditions [8]. Most importantly, for these boundary conditions, it has beenestablished that solitons which travel with a specific velocity need to have a counterpart,we call mirror soliton, with equal amplitude and opposite velocity. This pair of solitonsstand for the reflection happening at the boundary, where the soliton interchanges itsrole with the corresponding mirror soliton.Every soliton has a specific set of so called scattering data in the context of theinverse scattering method, from which it can be described uniquely. In [2], apart fromusing a different method, the mirror-image technique, to construct soliton solutions in theaforementioned usually addressed models with integrable boundary conditions, relationsregarding the scattering data of the soliton and of the corresponding mirror soliton havebeen established. Having these relations facilitates the analysis and understanding of thesoliton behavior.The paper is organized as follows: In Section 2, we review the inverse scatteringmethod for the NLS equation in order to derive the expression of a one-soliton solutionand its dependency on the scattering data. Then, we adapt the method of dressing theboundary to the new boundary conditions in Section 2.2. Using these results, we takepure soliton solutions in Section 3.1, which can be constructed in this theory, and pointout common relations between the scattering data of this solution. Finally, we visualizesaid solutions in Section 3.2. In the following, we give a brief summary of the inverse scattering transform of thefocusing NLS equation. As in [2] and [3], it will serve as a guideline in order to implementadditional results. Therefore, following the analysis given in [1], we introduce the NLSequation iu t + u xx + 2 | u | u = 0 ,u (0 , x ) = u ( x ) (2.1)for u ( t, x ) : R × R (cid:55)→ C and the initial condition u ( x ) . The equation can be expressedin an equivalent compatibility condition of the following linear spectral problems ψ x = U ψ,ψ t = V ψ, (2.2)2here ψ ( t, x, λ ) and the matrix operators U = − iλσ + Q, V = − iλ σ + (cid:101) Q (2.3)are × matrices. The potentials Q and (cid:101) Q of U and V are defined by Q ( t, x ) = (cid:18) u − u ∗ (cid:19) , (cid:101) Q ( t, x, λ ) = (cid:18) i | u | λu + iu x − λu ∗ + iu ∗ x − i | u | (cid:19) and σ = (cid:18) − (cid:19) . In this context, the matrices U and V form a so-called Lax pair, depending not only on t and x , but also on a spectral parameter λ . Hereafter, the asterix denotes the complexconjugate, C + = { λ ∈ C : (cid:61) ( λ ) > } as well as C − = { λ ∈ C : (cid:61) ( λ ) < } and ψ (cid:124) isthe transpose of ψ . For a solution ψ ( t, x, λ ) of the Lax system (2.2) the compatibilitycondition ψ tx = ψ xt for all λ ∈ C is equivalent to u ( t, x ) satisfying the NLS equation(2.1). Moreover, we will refer to U and V as the x and t part of the Lax pair, respectively. In that regard, given a sufficiently fast decaying function u ( t, x ) → and derivative u x ( t, x ) → as | x | → ∞ , it is reasonable to assume that there exist × -matrix-valued solutions, we call modified Jost solutions under time evolution, (cid:98) ψ ( t, x, λ ) = ψ ( t, x, λ ) e iθ ( t,x,λ ) σ , where θ ( t, x, λ ) = λx + 2 λ t , of the modified Lax system (cid:98) ψ x + iλ [ σ , (cid:98) ψ ] = Q (cid:98) ψ, (cid:98) ψ t + 2 iλ [ σ , (cid:98) ψ ] = (cid:101) Q (cid:98) ψ with constant limits as x → ±∞ and for all λ ∈ R , (cid:98) ψ ± ( t, x, λ ) → , as x → ±∞ . They are solutions to the following Volterra integral equations: (cid:98) ψ − ( t, x, λ ) = + (cid:90) x −∞ e − iθ (0 ,x − y,λ ) σ Q ( t, y ) (cid:98) ψ − ( t, y, λ ) e iθ (0 ,x − y,λ ) σ d y, (cid:98) ψ + ( t, x, λ ) = − (cid:90) ∞ x e − iθ (0 ,x − y,λ ) σ Q ( t, y ) (cid:98) ψ + ( t, y, λ ) e iθ (0 ,x − y,λ ) σ d y. (2.4) Lemma 1.
Let u ( t, · ) ∈ H , ( R ) = { f ∈ L ( R ) : xf, f x ∈ L ( R ) } . Then, for every λ ∈ R , there exist unique solutions (cid:98) ψ ± ( t, · , λ ) ∈ L ∞ ( R ) satisfying the integral equations (2.4) . Thereby, the second column vector of (cid:98) ψ − ( t, x, λ ) and the first column vector of (cid:98) ψ + ( t, x, λ ) can be continued analytically in λ ∈ C − and continuously in λ ∈ C − ∪ R ,while the first column vector of (cid:98) ψ − ( t, x, λ ) and the second column vector of (cid:98) ψ + ( t, x, λ ) can be continued analytically in λ ∈ C + and continuously in λ ∈ C + ∪ R . Analogously, the columns of ψ ± ( t, x, λ ) can be continued analytically and continu-ously into the complex λ -plane, ψ (2) − and ψ (1)+ can be continued analytically in λ ∈ C − and continuously in λ ∈ C − ∪ R , while ψ (1) − and ψ (2)+ can be continued analytically in λ ∈ C + and continuously in λ ∈ C + ∪ R . 3he limits of the Jost solutions and the zero trace of the matrix U gives det ψ ± = 1 for all x ∈ R . Further, ψ ± are both fundamental matrix solutions to the Lax system(2.2), so there exists an x - and t -independent matrix A ( λ ) such that ψ − ( t, x, λ ) = ψ + ( t, x, λ ) A ( λ ) , λ ∈ R . The scattering matrix A is determined by this system and therefore we can also write A ( λ ) = ( ψ + ( t, x, λ )) − ψ − ( t, x, λ ) , whereas its entries can be written in terms of Wron-skians. In particular, a ( λ ) = det[ ψ (1) − | ψ (2)+ ] and a ( λ ) = − det[ ψ (2) − | ψ (1)+ ] , implying thatthey can respectively be continued in λ ∈ C + and λ ∈ C − . The eigenfunction inherit thesymmetry relation of the Lax pair ψ ± ( t, x, λ ) = − σ (cid:0) ψ ± ( t, x, λ ∗ ) (cid:1) ∗ σ, (2.5)which directly gives a ( λ ) = a ∗ ( λ ∗ ) and a ( λ ) = − a ∗ ( λ ) . The asymptotic behaviorof the modified Jost functions and scattering matrix as λ → ∞ is (cid:98) ψ − = + 12 iλ σ Q + 12 iλ σ (cid:90) x −∞ | u ( t, y ) | d y + O (1 /λ ) , (cid:98) ψ + = + 12 iλ σ Q − iλ σ (cid:90) ∞ x | u ( t, y ) | d y + O (1 /λ ) , and A ( λ ) = + O (1 /k ) .Let u ( t, · ) ∈ H , ( R ) be generic. That is, a ( λ ) is nonzero in C + except at a finitenumber of points λ , . . . , λ N ∈ C + , where it has simple zeros a ( λ j ) = 0 , a (cid:48) ( λ j ) (cid:54) = 0 , j = 1 , . . . , N . This set of generic functions u ( t, · ) is an open dense subset of H , ( R ) usually denoted by G . By the symmetry mentioned above, a ( λ j ) = 0 if and only if a ( λ ∗ j ) = 0 for all j = 1 , . . . , N . At these zeros of a and a , we obtain for theWronskians the following relation for j = 1 , . . . , N , ψ (1) − ( t, x, λ j ) = b j ψ (2)+ ( t, x, λ j ) , ψ (2) − ( t, x, ¯ λ j ) = ¯ b j ψ (1)+ ( t, x, ¯ λ j ) , (2.6)where we defined ¯ λ j = λ ∗ j . Whereas for j = 1 , . . . , N , the relations then provide residuerelations used in the inverse scattering method Res λ = λ j (cid:16) (cid:98) ψ (1) − a (cid:17) = C j e iθ ( t,x,λ j ) (cid:98) ψ (2)+ ( t, x, λ j ) , Res λ =¯ λ j (cid:16) (cid:98) ψ (2) − a (cid:17) = ¯ C j e − iθ ( t,x, ¯ λ j ) (cid:98) ψ (1)+ ( t, x, ¯ λ j ) , where the weights are C j = b j (cid:0) d a ( λ j )d λ (cid:1) − and ¯ C j = ¯ b j (cid:0) d a (cid:48) (¯ λ j )d λ (cid:1) − , and they satisfy thesymmetry relations ¯ b j = − b ∗ j and ¯ C j = − C ∗ j .The inverse problem can be formulated using the jump matrix J ( t, x, λ ) = (cid:18) | ρ ( λ ) | e − iθ ( t,x,λ ) ρ ∗ ( λ ) e iθ ( t,x,λ ) ρ ( λ ) 0 (cid:19) , ρ ( λ ) = a ( λ ) /a ( λ ) for λ ∈ R . Defining sectionallymeromorphic functions M − = ( (cid:98) ψ (1)+ , (cid:98) ψ (2) − /a ) , M + = ( (cid:98) ψ (1) − /a , (cid:98) ψ (2)+ ) , we can give the method of recovering the solution u ( t, x ) from the scattering data. Riemann–Hilbert problem 1.
For given scattering data ( ρ, { λ j , C j } Nj =1 ) as wellas t, x ∈ R , find a × -matrix-valued function C \ R (cid:51) λ (cid:55)→ M ( t, x, λ ) satisfying1. M ( t, x, · ) is meromorphic in C \ R .2. M ( t, x, λ ) = 1 + O (1 /λ ) as | λ | → ∞ .3. Non-tangential boundary values M ± ( t, x, λ ) exist, satisfying the jump condition M + ( t, x, λ ) = M − ( t, x, λ )(1 + J ( t, x, λ )) for λ ∈ R .4. M ( t, x, λ ) has simple poles at λ , . . . , λ N , ¯ λ , . . . , ¯ λ N with Res λ = λ j M ( t, x, λ ) = lim λ → λ j M ( t, x, λ ) (cid:18) C j e iθ ( t,x,λ j ) (cid:19) , Res λ =¯ λ j M ( t, x, λ ) = lim λ → ¯ λ j M ( t, x, λ ) (cid:18) C j e − iθ ( t,x, ¯ λ j ) (cid:19) . After regularization, the Riemann–Hilbert problem 1 can be solved via Cauchy projectors,and the asymptotic behavior of M ± ( t, x, λ ) as λ → ∞ yields the reconstruction formula u ( t, x ) = − i N (cid:88) j =1 C ∗ j e − iθ ( t,x,λ ∗ j ) [ (cid:98) ψ ∗ + ] ( t, x, λ j ) − π (cid:90) ∞−∞ e − iθ ( t,x,λ ) ρ ∗ ( λ )[ (cid:98) ψ ∗ + ] ( t, x, λ ) d λ. In the reflectionless case, we have ρ ( λ ) = 0 for λ ∈ R and the one-soliton solution with λ = ξ + iη can be calculated as u ( t, x ) = − iη C ∗ | C | e − i (2 ξx +4( ξ − η ) t ) sech (cid:16) η ( x + 4 ξt ) − log | C | η (cid:17) . We change the notation so that u ( t, x ) = u s ( t, x ; ξ, η, x , φ ) has the following expression u s ( t, x ; ξ, η, x , φ ) = 2 ηe − i (2 ξx +4( ξ − η ) t +( φ + π/ sech(2 η ( x + 4 ξt − x )) , (2.7)where φ = arg( C ) and x = η log | C | η . 5 .2 Dressing the boundary As mentioned in the Introduction, a new integrable boundary condition for the NLSequation on the half-line has been obtained in [6] by dressing a Dirichlet boundary witha “jump-defect”. In this section we want to introduce this model on the half-line andcompute soliton solutions via the dressing method. Therefore, consider the NLS equation(2.1) for ( t, x ) ∈ R + × R + and complement it with boundary condition at x = 0 , which,in our notation, are of the following form u x = iu t − u Ω2 + u | u | − uα , (2.8)where Ω = (cid:112) β − | u | , α and β real parameters. Then, the NLS equation has again acorresponding Lax system and the boundary condition can be written in the form of aboundary constraint ( K ) t ( t, x, λ ) (cid:12)(cid:12) x =0 = ( V ( t, x, − λ ) K ( t, x, λ ) − K ( t, x, λ ) V ( t, x, λ )) (cid:12)(cid:12) x =0 , (2.9)where the boundary matrix K ( t, x, λ ) is given by K = 1(2 λ − i | β | ) − α (cid:18) λ + 4 iλ Ω[0] − ( α + β ) 4 iλu [0]4 iλu [0] ∗ λ − iλ Ω[0] − ( α + β ) (cid:19) . (2.10)The boundary matrix is scaled by ((2 λ − i | β | ) − α ) − , so that ( K ( t, x, λ )) − = K ( t, x, − λ ) . (2.11)Note that the scaling could also be chosen as ((2 λ + i | β | ) − α ) − leaving the constructedsolution unchanged. The property (2.11) is needed in the case of the half-line to properlycalculate the zeros and associated kernel vectors of the special solutions. Moreover,contrary to the boundary constraint on two half-lines, the boundary constraint on onehalf-line has a limitation to the t part of the Lax pair, as already mentioned in [4].Nevertheless, it is possible to compute soliton solutions in this model. Therefore, weintroduce the relation of the boundary matrix K ( t, x, λ ) to defect matrices, which arelinear in λ , see [4]. Proposition 1.
The boundary matrix K ( t, x, λ ) can be viewed, up to a function of λ ,as product of two defect matrices λG ,α ( t, x, λ ) = 2 λ + (cid:18) α ± i (cid:112) β − | u | iuiu ∗ α ∓ i (cid:112) β − | u | (cid:19) , where α, β ∈ R \ { } and ˜ u is subject to the Dirichlet boundary condition. In fact, ((2 λ − iβ ) − α ) K ( t, x, λ ) = 4 λ G ,α ( t, x, λ ) G , − α ( t, x, λ ) .
6n particular, it is of importance that the product K ( t, x, λ ) of the two defect ma-trices G ,α ( t, x, λ ) and G , − α ( t, x, λ ) is commutative. Thereby, it is comprehensiblethat a kernel vector for each of the matrices G ,α ( t, x, λ ) and G , − α ( t, x, λ ) at partic-ular, different λ , λ introduce the same kernel vectors for K ( t, x, λ ) at these valuesof λ . In this approach we will leave out boundary-bound soliton solutions. This isdue to the fact that the number of zeros and associated kernel vectors given throughthe special solutions is halved when working with boundary-bound soliton solutions onone half-line. Referring to the analysis in [4], we also introduce the function space X = { f ∈ H , t ( R + ) , ∂ x f ∈ H , t ( R + ) } , where H , t ( R + ) = { f ∈ L ( R + ) : tf ∈ L ( R + ) } and H , t ( R + ) = { f ∈ L ( R + ) : ∂ t f, tf ∈ L ( R + ) } . As in the reference, this functionspace is essential when it comes to identifying the exact signs of the entries in the diag-onal of the constructed boundary matrix corresponding to the dressed solution. Proposition 2.
Consider a solution u [0]( t, x ) to the NLS equation on the half-line subjectto the new boundary conditions (2.9) with parameters α, β ∈ R \ { } and at x = 0 inthe function space u [0]( · , ∈ X . Take two solutions { ψ , (cid:98) ψ } of the undressed Laxsystem corresponding to u [0] for λ = λ = − α + iβ and λ = (cid:98) λ = − λ . Further, take N solutions ψ j of the undressed Lax system corresponding to u [0] for distinct λ = λ j ∈ C \ (cid:0) R ∪ i R ∪ { λ , λ ∗ , − λ , − λ ∗ } (cid:1) , j = 1 , . . . , N . Constructing K ( t, x, λ ) as in (2.10) with u [0] , α and β , we assume that there exist paired solutions (cid:98) ψ j of the undressed Laxsystem corresponding to u [0] for λ = (cid:98) λ j = − λ j , j = 1 , . . . , N , satisfying (cid:98) ψ j (cid:12)(cid:12) x =0 = ( K ( t, x, λ j ) ψ j ) (cid:12)(cid:12) x =0 , (cid:98) λ k (cid:54) = λ j . (2.12) Then, a N -fold Darboux transformation D [2 N ] using { ψ , (cid:98) ψ , . . . , ψ N , (cid:98) ψ N } and theirrespective spectral parameter lead to the solution u [2 N ] to the NLS equation on the half-line. In particular, the boundary condition is preserved and we denote such a solution u [2 N ] by (cid:98) u [ N ] .Proof. With { ψ , (cid:98) ψ , . . . , ψ N , (cid:98) ψ N } , we have N linear independent solutions to the un-dressed Lax system (2.2), since λ , . . . , λ N , (cid:98) λ , . . . , (cid:98) λ N are distinct due to (cid:98) λ k (cid:54) = λ j . There-fore, the Darboux transformation is uniquely determined and the dressed solution (cid:98) u [ N ] satisfies the NLS equation.In order to proof that there is a matrix K N ( t, x, λ ) satisfying ( K N ) t ( t, x, λ ) (cid:12)(cid:12) x =0 = ( V [2 N ]( t, x, − λ ) K N ( t, x, λ ) − K N ( t, x, λ ) V [2 N ]( t, x, λ )) (cid:12)(cid:12) x =0 , it is of advantage to consider the equivalent equality ( D [2 N ]( t, x, − λ ) K ( t, x, λ )) (cid:12)(cid:12) x =0 = ( K N ( t, x, λ ) D [2 N ]( t, x, λ )) (cid:12)(cid:12) x =0 , (2.13)where we need to remark that K ( t, x, λ ) is multiplied by ((2 λ − i | β | ) − α ) / to simplifyfurther notation. In view of this equality, it becomes plausible to assume that the matrix,we wish to find, is of second order in λ , i.e. K N ( t, x, λ ) = λ + λK (1) ( t, x ) + K (0) ( t, x ) .Our goal will be to construct this matrix K N ( t, x, λ ) as a Darboux transformation with7pectral parameters λ and − λ and corresponding kernel vectors which we need todetermine in the following paragraph. We will restrict the argumentation to one of thespectral parameters λ and note that it can be reproduced analogously for the other one − λ .Since, up to a function of λ , K ( t, x, λ ) = G ,α ( t, x, λ ) G , − α ( t, x, λ ) , we can deduceas in [4] that there exist two vectors υ and (cid:98) υ at two spectral parameters respectively λ and (cid:98) λ = − λ for which G ,α ( t, x, λ ) υ = 0 , G , − α ( t, x, (cid:98) λ ) (cid:98) υ = 0 . Therefore, K ( t, x, λ ) can be seen as two-fold dressing matrix with the inherited kernelvectors of G ,α and G , − α at respectively λ and (cid:98) λ , so that K ( t, x, λ ) υ = 0 , K ( t, x, (cid:98) λ ) (cid:98) υ = 0 . These kernel vectors υ and (cid:98) υ are either linear dependent or linear independent of { ψ , (cid:98) ψ , ϕ , (cid:98) ϕ } . Further, these vectors will serve as a means to construct the kernelvectors for the dressing matrix K N ( t, x, λ ) . Thereby, we distinguish the two cases:1. The kernel vector υ of K ( t, x, λ ) can be expressed as a linear combination of { ψ , ϕ } at λ = λ and x = 0 . Then, w.l.o.g. υ = ψ , again to simplify notation.Since ψ is linearly independent of ψ , . . . , ψ N , it is possible to define a new vector υ (cid:48) = D [2 N ]( t, x, λ ) υ , which will serve as one of the kernel vectors for the dressing matrix K N ( t, x, λ ) . Itis important to note that constructing K N ( t, x, λ ) in this manner will result in thefollowing relations for the vector ψ and the orthogonal vector ϕ at x = 0 : D [2 N ]( t, x, − λ ) K ( t, x, λ ) ψ = K N ( t, x, λ ) D [2 N ]( t, x, λ ) ψ = 0 ,D [2 N ]( t, x, − λ ∗ ) K ( t, x, λ ∗ ) ϕ = K N ( t, x, λ ∗ ) D [2 N ]( t, x, λ ∗ ) ϕ = 0 . (2.14)2. The kernel vector υ of K ( t, x, λ ) can not be expressed as a linear combinationof { ψ , ϕ } at λ = λ and x = 0 . In this case, making out the kernel vectordirectly turns out to be not as easy. Since we assumed that the kernel vector υ and { ψ , ϕ } are linearly independent, we know that at x = 0 we can define a vector (cid:101) ψ = K ( t, x, λ ) ψ (cid:54) = 0 , which, in particular, solves the t -part of the undressedLax system at x = 0 and λ = − λ , due to K ( t, x, λ ) satisfying the boundaryconstraint (2.9). Similarly, the relations (A.2) for the dressing matrix D [2 N ]( t, x, λ ) imply that ψ (cid:48) = D [2 N ]( t, x, λ ) ψ and (cid:101) ψ (cid:48) = D [2 N ]( t, x, − λ ) (cid:101) ψ are also solutionsthe t -part of the dressed Lax system at x = 0 , λ = λ and λ = − λ , respectively.Now, connecting these three transformation, we require that there exists a matrix D [2 N ]( t, x, − λ ) K ( t, x, λ )( D [2 N ]( t, x, λ )) − , we also call K N ( t, x, λ ) , which then,at x = 0 , satisfies ( D [2 N ]( t, x, − λ ) K ( t, x, λ ) ψ ) = ( K N ( t, x, λ ) D [2 N ]( t, x, λ ) ψ ) (cid:54) = 0 , ( D [2 N ]( t, x, − λ ∗ ) K ( t, x, λ ∗ ) ϕ ) = ( K N ( t, x, λ ∗ ) D [2 N ]( t, x, λ ∗ ) ϕ ) (cid:54) = 0 . (2.15)8urther, evaluating the determinant of K N ( t, x, λ ) at the spectral parameter λ or λ ∗ and x = 0 , we obtain det( K N ( t, x, λ )) = det( K N ( t, x, λ ∗ )) = 0 . Thisimplies that there exist two kernel vectors of K N ( t, x, λ ) corresponding to one ofthe spectral parameters each. Hence, in this case we have found the kernel vectorwith which we want to construct the dressing matrix K N ( t, x, λ ) , whereas it thensatisfies (2.15).Now, given we have constructed a -fold Darboux transformation using λ and − λ andthe appropriately chosen kernel vectors through the already mentioned procedure, wewant to proof that this dressing matrix K N ( t, x, λ ) indeed satisfies the equality (2.13).First, we will write the equality as matrix polynomials of degree N + 2 in λ and denotethem as L ( λ ) and R ( λ ) . Hence, L ( λ ) = D [2 N ]( t, , − λ ) K ( t, , λ ) = λ N +2 L N +2 + λ N +1 L N +1 + · · · + λL + L ,R ( λ ) = K N ( t, , λ ) D [2 N ]( t, , λ ) = λ N +2 R N +2 + λ N +1 R N +1 + · · · + λR + R . The structure of the matrices yields L N +2 = = R N +2 , which also stems from the factthat we multiplied K ( t, x, λ ) by ((2 λ − i | β | ) − α ) / . Regarding the property (2.11),the special solutions provide N zeros and associated kernel vectors R ( λ ) (cid:12)(cid:12) λ = λ j ψ j = 0 , L ( λ ) (cid:12)(cid:12) λ = λ j ψ j = 0 ,R ( λ ) (cid:12)(cid:12) λ = (cid:98) λ j (cid:98) ψ j = 0 , L ( λ ) (cid:12)(cid:12) λ = (cid:98) λ j (cid:98) ψ j = 0 , at x = 0 for j = 1 , . . . , N . For R ( λ ) the equalities are clear from the definition of theDarboux transformation and with the assumption (2.12), the equalities for L ( λ ) followanalogously. With the orthogonal vectors ϕ j = σ ψ ∗ j and (cid:98) ϕ j = σ (cid:98) ψ ∗ j , we obtain R ( λ ) (cid:12)(cid:12) λ = λ ∗ j ϕ j = 0 , L ( λ ) (cid:12)(cid:12) λ = λ ∗ j ϕ j = 0 ,R ( λ ) (cid:12)(cid:12) λ = (cid:98) λ ∗ j (cid:98) ϕ j = 0 , L ( λ ) (cid:12)(cid:12) λ = (cid:98) λ ∗ j (cid:98) ϕ j = 0 , whereby these equalities hold at x = 0 for j = 1 , . . . , N . This is however not enough toensure equality in (2.13), since the determinant is of power N + 2 in λ and we only have N zeros. Further, the choice of K N implies R ( λ ) (cid:12)(cid:12) λ = λ ψ = L ( λ ) (cid:12)(cid:12) λ = λ ψ ,R ( λ ) (cid:12)(cid:12) λ = (cid:98) λ (cid:98) ψ = L ( λ ) (cid:12)(cid:12) λ = (cid:98) λ (cid:98) ψ , and with the orthogonal vectors ϕ = σ ψ ∗ and (cid:98) ϕ = σ (cid:98) ψ ∗ , we obtain R ( λ ) (cid:12)(cid:12) λ = λ ∗ ϕ = L ( λ ) (cid:12)(cid:12) λ = λ ∗ ϕ ,R ( λ ) (cid:12)(cid:12) λ = (cid:98) λ ∗ (cid:98) ϕ = L ( λ ) (cid:12)(cid:12) λ = (cid:98) λ ∗ (cid:98) ϕ , x = 0 . At this point, it is important that all vectors are linearly independent. In viewof the additional vectors from the construction of K N ( t, x, λ ) , at λ = λ , we see thatin both cases either a linear combination of ψ and ϕ or ψ itself is given. Therefore,this provides a linear independent vector and the same is true for the second spectralparameter. As mentioned in the second case, these vectors are not necessarily kernelvectors for L ( λ ) and R ( λ ) . However, they are indeed kernel vectors of the difference C ( λ ) = L ( λ ) − R ( λ ) = λ N +1 C N +1 + · · · + C , which can therefore be calculatedexplicitly with the given amount of zeros and kernel vectors ( C N +1 , · · · , C ) λ N +10 ψ · · · ( (cid:98) λ ∗ N ) N +1 (cid:98) ϕ N ... ... ... ψ · · · (cid:98) ϕ N = 0 . Consequently, the matrix coefficients of C ( λ ) are zero and so is C ( λ ) , implying L ( λ ) = R ( λ ) . Furthermore, this equality gives us that in both cases, the kernel vectors are indeedas described in the first case equal to D [2 N ]( t, x, λ ) υ and D [2 N ]( t, x, (cid:98) λ ) (cid:98) υ with υ and (cid:98) υ kernel vectors of K ( t, x, λ ) respectively at λ = λ and λ = − λ . Even though, thelinear dependence of the kernel vector to the pair { ψ , ϕ } is not implied in both cases.Given K N ( t, x, λ ) of the form λ + λK (1) ( t, x ) + K (0) ( t, x ) , we want to determinethe matrix coefficients in terms of the solution at x = 0 to confirm that the boundaryconditions are preserved. Thereby, the symmetry of V ∗ ( t, x, λ ∗ ) = σV ( t, x, λ ) σ − , where σ = (cid:18) − (cid:19) , is inherited by K N ( t, x, λ ) , so that we can identify K N ( t, , λ ) = λ + λ (cid:32) K (1)11 ( t ) K (1)12 ( t ) − (cid:0) K (1)12 ( t ) (cid:1) ∗ (cid:0) K (1)11 ( t ) (cid:1) ∗ (cid:33) + (cid:32) K (0)11 ( t ) K (0)12 ( t ) − (cid:0) K (0)12 ( t ) (cid:1) ∗ (cid:0) K (0)11 ( t ) (cid:1) ∗ (cid:33) . The equality L N +1 = R N +1 gives at x = 0 on the off-diagonal of K (1) ( t, that K (1)12 ( t ) = iu [2 N ]( t, and K (1)21 ( t ) = − (cid:0) K (1)12 ( t ) (cid:1) ∗ = iu ∗ [2 N ]( t, . For the entries on thediagonal of K (1) ( t, x ) , we obtain from the same equality K (1)11 ( t ) = i Ω[0] − ) , (cid:0) K (1)11 ( t ) (cid:1) ∗ = − i Ω[0] − ∗ ) , (2.16)where Σ is defined as the matrix coefficient of λ N − of the matrix D [2 N ]( t, x, λ ) , see(A.4). To determine the remaining entries of the matrix coefficients, we need to extractinformation from the determinant of K N ( t, x, λ ) , which can be calculated as det( K N ( t, , λ )) = det( D [2 N ]( t, , − λ )) det( K ( t, , λ )) det(( D [2 N ]( t, , λ )) − ) , and using that det( D [2 N ]( t, , − λ )) = det( D [2 N ]( t, , λ )) , due to the fact that for eachspectral parameter λ j we also use − λ j for j = 1 , . . . , N , we obtain = det( K ( t, , λ )) = λ − α − β λ + ( α + β ) . K N ( t, , λ ) in polynomial form asabove, we can match the coefficients such that tr( K (1) ( t, , tr( K (0) ( t, K (1) ( t, − α − β , (cid:60) ( K (1)11 ( t ) (cid:0) K (0)11 ( t ) (cid:1) ∗ ) − (cid:0) K (0)12 ( t ) (cid:1) ∗ (cid:61) ( u [2 N ]( t, , det( K (0) ( t, α + β ) . (2.17)Combining the first line in (2.17) with the expressions we have for K (1)11 ( t ) and its complexconjugate, see (2.16), we can deduce that (cid:60) ( K (1)11 ( t )) = 0 . Further, evaluating the nextequality of (2.13) at x = 0 , which is L N = R N , implies L N = Σ − i Σ (cid:18) Ω[0] u [0] u [0] ∗ − Ω[0] (cid:19) − α + β = Σ + K (1) ( t, + K (0) ( t,
0) = R N , where again Σ is the matrix coefficient of λ N − of the matrix D [2 N ]( t, x, λ ) . Matchingthe (12) -entry of this equality, we derive ( u [2 N ] − u [0]) Ω[0]2 − iu [0](Σ ) = − i K (1)11 ( t )( u [2 N ] − u [0]) + iu [2 N ](Σ ∗ ) + K (0)12 ( t ) , and using the expressions in (2.16) we have for (Σ ) and (Σ ∗ ) , we obtain after can-cellation K (0)12 ( t ) − iu [2 N ] (cid:60) ( K (1)11 ( t )) . However, we already calculated that (cid:60) ( K (1)11 ( t )) needs to be zero in order for the de-terminants to be equal. Hence, also K (0)12 ( t ) and thereby the off-diagonal of K (0) ( t, vanishes. It follows by the third equation of (2.17) that (cid:61) ( K (0)11 ( t )) = 0 and then, by thefourth equation we have K (0) ( t,
0) = ± α + β . To verify that it is indeed minus as for K ( t, , λ ) , we confirm with the equality of L = R at x = 0 , which is − α + β N = K (0) ( t, N , where Σ N is the zero-th order matrix coefficient of the dressing matrix D [2 N ]( t, x, λ ) .For this to be satisfied for all t ∈ R + , we need to have K (0) = − α + β . Thereby, weobtain tr( K (0) ( t, − α + β . Thus, the second equation of (2.17) implies that K (1)11 ( t ) = ± i (cid:112) β − | u [2 N ] | , (cid:0) K (1)11 ( t ) (cid:1) ∗ = ∓ i (cid:112) β − | u [2 N ] | . K (1) ( t, to be able toconstitute that K N ( t, , λ ) preserves the boundary constraint, i.e. we need to show thatthe signs coincide with the signs in the same entry of K ( t, , λ ) in front of Ω[0] .Therefore, a similar analysis as in [4] is needed, where we use the fact that under theDarboux transformation functions u [0]( · , in the function space X are mapped ontofunctions, here u [2 N ]( · , , which lie in the function space X . Further, we have that K ( t, , λ ) has a positive sign in the (11) -entry in front of Ω[0] . As before, we have thekernel vectors υ and (cid:98) υ of K ( t, x, λ ) at x = 0 and respectively λ = λ and λ = (cid:98) λ .Then, for K ( t, , λ ) multiplied by ((2 λ − i | β | ) − α ) / , we have as t goes to infinity that lim t →∞ K ( t, , λ ) = diag( λ + i | β | λ − ( α + β )4 , λ − i | β | λ − ( α + β )4 )= (cid:40) diag(( λ − λ )( λ − (cid:98) λ ∗ ) , ( λ − λ ∗ )( λ − (cid:98) λ )) , if β > , diag(( λ − λ ∗ )( λ − (cid:98) λ ) , ( λ − λ )( λ − (cid:98) λ ∗ )) , if β < . In turn, this implies that the limits of the kernel vectors of K ( t, , λ ) are υ ∼ (cid:40) e , if β > ,e , if β < , (cid:98) υ ∼ (cid:40) e , if β > ,e , if β < , as t goes to infinity, where e and e are unit vectors. Since the dressing matrix D [2 N ]( t, x, λ ) is also diagonal as t goes to infinity, see [4], the kernel vectors υ (cid:48) , (cid:98) υ (cid:48) of K N ( t, , λ ) inherit the long time behavior of their corresponding vector. Therefore,the signs can be determined to be positive in the (11) -entry and negative in the (22) -entryof K (1) . In conclusion, if we assume u [0]( · , ∈ X , we can find a Darboux transformation K N ( t, x, λ ) for which V [2 N ] satisfies (2.9) regarding x = 0 , where K N ( t, x, λ ) is similarto K ( t, x, λ ) with an updated function (cid:98) u [ N ] . Remark 1.
Similar to the analysis of the long time behavior of the kernel vectors,one could look at the long time behavior of the dressing matrix D [2 N ]( t, x, λ ) to de-duce the same result through the equality of K N ( t, x, λ ) with the product of matrices D [2 N ]( t, x, − λ ) K ( t, x, λ )( D [2 N ]( t, x, λ )) − . Nevertheless, this is closely related to oneanother, since the limit behavior of the kernel vectors of D [2 N ]( t, x, λ ) determines thedistribution of factors λ − λ j , λ − (cid:98) λ j , λ − λ ∗ j and λ − (cid:98) λ ∗ j for j = 1 , . . . , N in the diagonalentries as t goes to infinity. Therewith, we have shown that the method of dressing the boundary can be appliedto the new boundary conditions constituted in [6]. Unlike in [7], where the boundary ma-trices G ( λ ) and G N ( λ ) = G ( λ ) are provided at the beginning and the proof is to checkthat they satisfy the equality (2.13) together with the dressing matrix D [2 N ]( t, x, λ ) ,we only provide K ( t, x, λ ) and the proof is to construct a suitable boundary matrix K N ( t, x, λ ) satisfying the equality (2.13). Afterwards, we need to verify that the con-structed matrix K N ( t, x, λ ) is in terms of the solution space indeed the boundary matrixwe need for the boundary constraint with respect to the dressed solution (cid:98) u [ N ] . The12eason why we need a different approach is due to the boundary matrix. In the case ofthe Robin boundary condition, the structure of the boundary matrix G ( λ ) is such thatwhen comparing L ( λ ) with R ( λ ) it is already clear with regard to λ that the (2 N + 1) -thand zero-th order matrix coefficients are equal. Hence, the zeros and associated kernelvectors of the dressing matrix D [2 N ]( t, x, λ ) are sufficient to derive the equality of L ( λ ) and R ( λ ) . However defining K N ( t, x, λ ) similarly to K ( t, x, λ ) as in (2.10) with u [0] and Ω[0] updated to u [2 N ] and Ω[2 N ] , we only have the equality of the (2 N + 1) -th ordermatrix coefficient and consequently the zeros and associated kernel vectors of the dress-ing matrix D [2 N ]( t, x, λ ) are insufficient to derive the equality. Even though the newboundary condition does fit in the proof initially suggested in [7], the Robin boundarycondition very well fits into this one as we will put forward in Appendix B. In this section, we want to derive relations for multi-soliton solutions between the weights C j and (cid:98) C j , which correspond to a pair of zeros λ j and (cid:98) λ j of a ( λ ) , for j = 1 , . . . , N .Consider the zero seed solution u [0] = 0 and C \ ( R ∪ i R ) (cid:51) λ = − α + iβ . Also, K ( t, x, λ ) = 1(2 λ − i | β | ) − α (cid:18) λ + 4 iλ | β | − ( α + β ) 00 4 λ − iλ | β | − ( α + β ) (cid:19) . Then, in order to apply Proposition 2, we take distinct λ = λ j ∈ C \ (cid:0) R ∪ i R ∪{ λ , λ ∗ , − λ , − λ ∗ } (cid:1) , j = 1 , . . . , N , and at the corresponding spectral parameter a so-lution ψ j , j = 0 , . . . , N , to the Lax system regarding u [0] . The solutions ψ j to the zeroseed solution are readily produced ψ j = (cid:18) µ j ν j (cid:19) = e − i ( λ j x +2 λ j t ) σ (cid:18) u j v j (cid:19) , where ( u j , v j ) (cid:124) ∈ C for j = 0 , . . . , N . Particularly, the choice for ψ will be ( u , v ) =(1 , or ( u , v ) = (0 , respectively for β > or β < inspired by the first case inthe proof of Proposition 2. Now, given the relation (2.12), we also take solutions to thesame Lax system at λ = (cid:98) λ j = − λ j defined by (cid:98) ψ j = e − i ( (cid:98) λ j x +2 (cid:98) λ j t ) σ (ˆ u j , ˆ v j ) (cid:124) , j = 1 , . . . , N .Whereas, the relation is equivalent to ˆ u j ˆ v j = (2 λ j + i | β | ) − α (2 λ j − i | β | ) − α u j v j , j = 1 , . . . , N. Note that if λ j ∈ C + , then (cid:98) λ j ∈ C − which, in turn, implies that (cid:98) ψ j has opposite limitbehavior as ψ j for x → ±∞ . Such that in order to apply Theorem 1 to the Darbouxtransformation corresponding to (cid:98) λ j and (cid:98) ψ j , we instead use the counterpart (cid:98) λ ∗ j and (cid:98) ϕ j .Since with (cid:98) λ ∗ j ∈ C + , the vector (cid:98) ϕ j = e − i ( (cid:98) λ ∗ j x +2( (cid:98) λ ∗ j ) t ) σ ( − ˆ v ∗ j , ˆ u ∗ j ) (cid:124) , admits the same limit13ehavior as ψ j for x → ±∞ . Similar to Remark 3 following Theorem 1, we can deducefor a two-fold Darboux transformation consisting of { λ , ψ , (cid:98) λ ∗ , (cid:98) ϕ } that the weights inthe scattering data can be calculated as C (2)1 = − v u ( λ − λ ∗ )( λ − (cid:98) λ ) λ − (cid:98) λ ∗ , C (2)2 = − ˆ u ∗ − ˆ v ∗ ( (cid:98) λ ∗ − λ ∗ )( (cid:98) λ ∗ − (cid:98) λ ) (cid:98) λ ∗ − λ . This results in C (2)1 ( C (2)2 ) ∗ = − λ · (2 λ + i | β | ) − α (2 λ − i | β | ) − α · (cid:61) ( λ ) (cid:60) ( λ ) , where it is obvious that the factor (2 λ + i | β | ) − α (2 λ − i | β | ) − α is the only difference to the analogousresult in the case of the Robin boundary condition, where we have iα − λ iα +2 λ . Nevertheless,by defining λ = ξ + iη and (cid:98) λ ∗ = − ξ + iη as well as the corresponding weights C = 2 η e η x + iφ = C (2)1 and (cid:98) C = 2 η e η (cid:98) x + i (cid:98) φ = C (2)2 , we obtain a relation betweenthe initial positions and phases of the two inserted solitons x + (cid:98) x = 12 η log (cid:16) η ξ (cid:17) + 14 η log (cid:16) (4 ξ − α − (2 η + | β | ) ) + (4 ξ (2 η + | β | )) (4 ξ − α − (2 η − | β | ) ) + (4 ξ (2 η − | β | )) (cid:17) ,φ − (cid:98) φ = 2 arg( λ ) + arg (cid:16) ξ − α − (2 η + | β | ) + i ξ (2 η + | β | )4 ξ − α − (2 η − | β | ) + i ξ (2 η − | β | ) (cid:17) + π. Remark 2.
In general, we can construct a N -Darboux transformation using the in-formation given by { λ , ψ , . . . , λ N , ψ N , (cid:98) λ ∗ , (cid:98) ϕ , . . . , (cid:98) λ ∗ N , (cid:98) ϕ N } , where λ j = ξ j + iη j andconsequently (cid:98) λ ∗ = − ξ j + iη j for j = 1 , . . . , N with corresponding solutions to the un-dressed Lax system as above. Then for j = 1 , . . . , N , the relation for a pair of initialpositions x j and (cid:98) x j = x N + j as well as phases φ j and (cid:98) φ j = φ N + j amounts to x j + (cid:98) x j = 12 η j log (cid:16) η j ξ j (cid:17) + 14 η j log (cid:16) (4 ξ j − α − (2 η j + | β | ) ) + (4 ξ j (2 η j + | β | )) (4 ξ j − α − (2 η j − | β | ) ) + (4 ξ j (2 η j − | β | )) (cid:17) − η j N (cid:88) (cid:48) k =1 log [( ξ j − ξ k ) + ( η j − η k ) ][( ξ j + ξ k ) + ( η j − η k ) ][( ξ j + ξ k ) + ( η j + η k ) ][( ξ j − ξ k ) + ( η j + η k ) ] ,φ j − (cid:98) φ j = 2 arg( λ j ) + arg (cid:16) ξ j − α − (2 η j + | β | ) + i ξ j (2 η j + | β | )4 ξ j − α − (2 η j − | β | ) + i ξ j (2 η j − | β | ) (cid:17) + π − N (cid:88) (cid:48) k =1 arg (cid:16) [( ξ j − ξ k ) + i ( η j − η k )][( ξ j + ξ k ) + i ( η j − η k )][( ξ j + ξ k ) + i ( η j + η k )][( ξ j − ξ k ) + i ( η j + η k )] (cid:17) , whereas the product of a pair of weights C j and ˆ C j = C N + j is C j ˆ C ∗ j = − λ j (2 λ j + i | β | ) − α (2 λ j − i | β | ) − α (2 η j ) (2 ξ j ) (cid:104) N (cid:89) (cid:48) k =1 ( λ j − λ ∗ k )( λ j + λ k )( λ j − λ k )( λ j + λ ∗ k ) (cid:105) , where the prime indicates that the term with k = j is omitted from the sum and product. K N ( t, x, λ ) corresponding to the dressed solution (cid:98) u [ N ] has kernel vec-tors ψ (cid:48) = D [2 N ]( t, x, λ ) ψ and the orthogonal vector as before ϕ (cid:48) = D [2 N ]( t, x, λ ∗ ) ϕ respectively at the spectral parameter λ = λ and λ ∗ as described in the first case. Ad-ditionally, note that the second case, in which the solution ψ and the kernel vector υ are linearly independent, can also occur. An example is given by the non-zero seed so-lution u [0] = ρe iρ t with constant background ρ > in the case of the Robin boundarycondition, where the solutions to the Lax system can not be connected to the kernelvectors. The Darboux transformation presented in Appendix A gives the algebraic means toderive N -soliton solutions simply by calculating the (12) -entry of the projector matrices ( P [ j ]) for j = 1 , . . . , N recursively and then sum them up or by the direct calculationof the quotient of two N × N matrices, which represents the (12) -entry of the sumof projector matrices, i.e. (Σ ) , as presented in [7]. Motivated by Section 3.1, thepure soliton solutions in the case of the new boundary condition, which we can obtain,are constructed by choosing pairs of spectral parameter λ j and (cid:98) λ ∗ j , j = 1 , . . . , N , andassociated constants u j , v j , − ˆ v ∗ j and ˆ u ∗ j as explained therein.For N = 1 , consider the spectral parameter λ = ξ + iη , where it is comprehensiblethat, with regard to (2.7), ξ and η respectively describe the velocity and the amplitudeof the physical one-soliton. Further, the quotient of the constants u and v is highlyrelated to the initial position x and phase φ of the soliton. Consequently, the mirrorsoliton corresponding to (cid:98) λ ∗ = − ξ + iη has opposite velocity to and the same amplitudeas the physical soliton. Particularly, we have visualized said behavior in Figures 1 and2. Whereas, the Dirichlet boundary condition u ( t,
0) = 0 occur as a special case of thenew boundary condition (2.8), when for example | α | → ∞ , | β | → ∞ or β → . Indeed,structurally these cases correspond to the boundary matrix K ( t, x, λ ) = . Thereby, weplotted in Figure 1 on the left the reflection of a one-soliton solution | (cid:98) u [1]( t, x ) | subject tothe Dirichlet boundary condition as well as on the right a contour plot, which includes themirror soliton (dashed). In Figure 2, we chose particular parameter α = 1 and β = 2 toplot an example of a one-soliton solution | (cid:98) u [1]( t, x ) | in the case of soliton reflection withrespect to the new boundary condition in three dimensions on the left and as a contourplot together with the mirror soliton on the right. It is observable that in these casesthe physical soliton and the mirror soliton change roles after the usual soliton interactionwith the physical soliton visible before and the mirror soliton visible after the interactionwith the boundary. Additionally, in the case of the Dirichlet boundary condition theinteraction of the pair of solitons is such that the whole solution is zero at the boundary x = 0 .Subsequently, we used the mentioned algorithm to include higher order soliton so-lutions in the results. First of all, inspired by the breather in the case of the Dirichletboundary condition, see Figure 6 in [2], we plotted a similar breather solution as solitonreflection in the case of the new boundary condition with parameter α = 1 and β = 2 on15 ig. 1. Dirichlet boundary condition at x = 0 : one-soliton reflection with ξ = 1 , η = 1 , x = 5 and φ = 0 . Left: 3D plot of | (cid:98) u [1]( t, x ) | . Right: contour plot showingthe mirror soliton (dashed) to the left of x = 0 . Fig. 2.
New boundary condition ( α = 1 , β = 2 ) at x = 0 : one-soliton reflection with ξ = 1 , η = 1 , x = 5 and φ = 0 . Left: 3D plot. Right: contour plot. Fig. 3.
New boundary condition ( α = 1 , β = 2 ) at x = 0 : two-soliton reflection with ξ = ξ = 1 / , η = 1 / , η = 3 / , x = x = 5 and φ = φ = 0 . Left: 3D plot.Right: contour plot. ig. 4. New boundary condition ( α = 1 , β = 2 ) at x = 0 : three-soliton reflectionwith ξ = 3 / , ξ = 1 / , ξ = 5 / , η = 1 , η = 3 / , η = 1 / , x = 5 , x = 8 , x = 11 and φ = φ = φ = 0 . Left: 3D plot. Right: contour plot. the left and the contour together with the mirror soliton on the right of Figure 3. As onewould suspect, the main difference can be observed at the boundary x = 0 . Ultimately,we went one step further and even plotted the reflection of a three-soliton solution in thecase of the new boundary condition, again with the same parameters, on the left and itscontour including the mirror soliton on the right of Figure 4. The choice of parameters,which is needed in order to comply with the conditions of Proposition 2, is described inSection 3.1. Conclusion
In this work, we further developed the method of dressing the boundary to be applicableto the NLS equation on the half-line with the new boundary condition. The boundarycondition corresponds to a time dependent gauge transformation (2.10) and this timedependence together with the polynomial degree with respect to the spectral parameterof the transformation thin out the solution space for the new boundary condition. As wehave seen in [4], for the time dependence we need the solution to go to zero as the timegoes to infinity. Moreover, the polynomial degree disables the consideration of boundary-bound solitons. Nonetheless, we were able to show that it is possible to construct reflectedpure soliton solutions of arbitrary (even) order in this model and to visualize the result.17 ppendices
A Darboux transformation
The Darboux transformation can be viewed as gauge transformation acting on forms ofthe Lax pair U , V . For that, the undressed Lax system (2.2) will be written as U [0] , V [0] and ψ [0] and the transformed, structural identical system as U [ N ] , V [ N ] and ψ [ N ] with N ∈ N . The transformed vector ψ [ N ] = D [ N ] ψ [0] satisfies the transformed system ψ [ N ] x = U [ N ] ψ [ N ] , ψ [ N ] t = V [ N ] ψ [ N ] , (A.1)whereas they are connected by D [ N ] x = U [ N ] D [ N ] − D [ N ] U [0] , D [ N ] t = V [ N ] D [ N ] − D [ N ] V [0] . (A.2)For given N distinct column vector solutions ψ j = ( µ j , ν j ) (cid:124) of the undressed Lax system(2.2) evaluated at λ = λ j , j = 1 . . . N , we construct an iteration of the one-fold dressingmatrix D [1] in the following sense D [ N ] = (( λ − λ ∗ N ) + ( λ ∗ N − λ N ) P [ N ]) · · · (( λ − λ ∗ ) + ( λ ∗ − λ ) P [1]) , where P [ j ] are projector matrices defined by 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 . (A.3)Then to reconstruct the solution u [ N ] , we need to insert ψ [ N ] = D [ N ] ψ [0] into thetransformed Lax system (A.1) and extract the information of the coefficient of λ N − ofthe first line. Therefore, we need the coefficient of λ N − of D [ N ] which we denote by Σ ,i.e. Σ = N (cid:88) j =1 − λ ∗ j + ( λ ∗ j − λ j ) P [ j ] . (A.4)Consequently, the reconstruction formula can be computed as Q [ N ] = Q [0] − i N (cid:88) j =1 ( λ j − λ ∗ j )[ σ , P [ j ]] . This calculation can be used to recursively construct N -soliton solutions. Especially, weuse it to compute the solutions in Section 3.2. Moreover, the change of the scatteringdata under Darboux transformations has been investigated, among others, for the NLSequation, see [5]. With that book serving as a basis, we give a brief overlook for therelevant theorem in our notation. 18 .1 Change of scattering data under Darboux transformations With scattering data ( ρ, { λ j , C j } Nj =1 ) , λ j ∈ C + for all j = 1 , . . . , N , we want to givethe relevant information needed to retrace the change of scattering data under Darbouxtransformations. It is of renewed importance that the solution and its derivative withrespect to x connected to the scattering data is a sufficiently fast decaying function for | x | → ∞ . Then, given a spectral parameter λ ∈ C + \{ λ , . . . , λ N } and a column solutionof the undressed Lax system ψ = u ψ (1) − ( t, x, λ ) + v ψ (2)+ ( t, x, λ )= u (cid:98) ψ (1) − ( t, x, λ ) e − iθ ( t,x,λ ) + v (cid:98) ψ (2)+ ( t, x, λ ) e iθ ( t,x,λ ) . Defining the ratio of the second and the first component to be q = [ (cid:98) ψ − ] ( t, x, λ ) + v u [ (cid:98) ψ + ] ( t, x, λ ) e iθ ( t,x,λ ) [ (cid:98) ψ − ] ( t, x, λ ) + v u [ (cid:98) ψ + ] ( t, x, λ ) e iθ ( t,x,λ ) , we obtain, in turn, an expression for the ratio of v and u , i.e. − v u = [ (cid:98) ψ − ] ( t, x, λ ) − q [ (cid:98) ψ − ] ( t, x, λ )[ (cid:98) ψ + ] ( t, x, λ ) − q [ (cid:98) ψ + ] ( t, x, λ ) e − iθ ( t,x,λ ) . (A.5)Also, the one-fold Darboux transformation corresponding to λ and ψ takes the form D [1] = λ + 11 + | q | (cid:18) − λ − λ ∗ | q | ( λ ∗ − λ ) q ∗ ( λ ∗ − λ ) q − λ ∗ − λ | q | (cid:19) . The properties of the Jost functions imply lim x →−∞ q = ∞ , lim x → + ∞ q = 0 . Thereby, adding a pole to the scattering data under Darboux transformations can beexplained by the following
Theorem 1.
Let the scattering data a ( λ ) , λ ∈ C + ∪ R , a ( λ ) , λ ∈ R and b j for j = 1 , . . . , N be given. Applying the Darboux transformation with λ ∈ C + \ { λ , . . . , λ N } and ψ = u ψ (1) − ( t, x, λ ) + v ψ (2)+ ( t, x, λ ) , where u , v ∈ C \ { } , we add an eigenvalueto the scattering data leaving the original eigenvalues unchanged. Further, a (cid:48) ( λ ) = λ − λ λ − λ ∗ a ( λ ) ,a (cid:48) ( λ ) = a ( λ ) ,b (cid:48) j = b j ,b (cid:48) = − v u , λ ∈ C + ∪ R ,λ ∈ R ,j = 1 , . . . , N, ρ (cid:48) ( λ ) = λ − λ ∗ λ − λ ρ ( λ ) ,C (cid:48) j = λ j − λ ∗ λ j − λ C j ,C (cid:48) = − v u λ − λ ∗ a ( λ ) . λ ∈ R ,j = 1 , . . . , N, roof. The scattering data rely heavily on the Jost functions. That is why, the first stepis to find the behavior of the Jost functions in the transformed system. Therefore, weneed to see what the limit values of the Darboux transformation are lim x →−∞ D [1] = diag( λ − λ ∗ , λ − λ ) , lim x → + ∞ D [1] = diag( λ − λ , λ − λ ∗ ) . Then, we can deduce that the transformed Jost functions can be expressed through ( ψ (1) − ) (cid:48) ( t, x, λ ) = D [1] λ − λ ∗ ψ (1) − ( t, x, λ ) , ( ψ (2)+ ) (cid:48) ( t, x, λ ) = D [1] λ − λ ∗ ψ (2)+ ( t, x, λ ) , which also is passed onto ( (cid:98) ψ (1) − ) (cid:48) and ( (cid:98) ψ (2)+ ) (cid:48) . As already mentioned in Section 2, a ( λ ) =det[ ψ (1) − | ψ (2)+ ] . It follows that for λ ∈ C + ∪ R , the limit values of [ (cid:98) ψ − ] and [ (cid:98) ψ + ] are a ( λ ) as x goes to + ∞ and −∞ , respectively. So that we have a (cid:48) ( λ ) = lim x →∞ ([ (cid:98) ψ − ] ) (cid:48) = λ − λ λ − λ ∗ a ( λ ) . Analogously, we find for a ( λ ) that a ( λ ) = [ ψ + ] [ ψ − ] − [ ψ + ] [ ψ − ] = ([ (cid:98) ψ + ] [ (cid:98) ψ − ] − [ (cid:98) ψ + ] [ (cid:98) ψ − ] ) e iθ ( t,x,λ ) , and therefore the limit values of [ (cid:98) ψ − ] and − [ (cid:98) ψ + ] behave as a ( λ ) e − iθ ( t,x,λ ) as x goesto + ∞ and −∞ , respectively. Consequently, a (cid:48) ( λ ) = lim x →∞ ([ (cid:98) ψ − ] ) (cid:48) = a ( λ ) . Also resulting in ρ (cid:48) ( λ ) = λ − λ ∗ λ − λ ρ ( λ ) . Since the Jost functions we relate in order to obtain b j are changed identically with D [1] / ( λ − λ ∗ ) , b j remain unchanged, i.e. b (cid:48) j = b j for j = 1 , . . . , N . Then, by the definition of C j we can calculate C (cid:48) j = b (cid:48) j (cid:16) d a (cid:48) ( λ j )d λ (cid:17) − = λ j − λ ∗ λ j − λ C j , j = 1 , . . . , N. At the new eigenvalue λ = λ , we have that the transformed Jost function are alsoidentically changed by D [1]( t, x, λ )( λ − λ ∗ ) = 11 + | q | (cid:18) | q | − q ∗ − q (cid:19) . Hence, as we calculated already in (A.5), we obtain b (cid:48) = ([ ψ − ] ) (cid:48) ( t, x, λ )([ ψ + ] ) (cid:48) ( t, x, λ ) = [ ψ − ] ( t, x, λ ) − q [ ψ − ] ( t, x, λ )[ ψ + ] ( t, x, λ ) − q [ ψ + ] ( t, x, λ ) = − v u . Subsequently, the weight for the added eigenvalue is readily obtained by C (cid:48) = b (cid:48) (cid:16) d a (cid:48) ( λ )d λ (cid:17) − = − v u λ − λ ∗ a ( λ ) . emark 3. A particular example is dressing a pure soliton solution from the zero seedsolution for which a ( λ ) = 1 , a ( λ ) = 0 , whereby ρ ( λ ) = 0 . Then, inserting poles λ , . . . , λ N ∈ C + with corresponding u j , v j ∈ C \ { } , j = 1 , . . . , N , results in the (rele-vant) scattering data a ( N )11 ( λ ) = N (cid:89) j =1 λ − λ j λ − λ ∗ j , a ( N )12 ( λ ) = 0 , C ( N ) j = − v j u j N (cid:89) k =1 ( λ j − λ ∗ k ) (cid:16) N (cid:89) (cid:48) k =1 ( λ j − λ k ) (cid:17) − , where the prime indicates that the term with k = j is omitted from the product. B Robin boundary condition
As mentioned in Section 2, the proof, which we tailored to fit the new boundary condition,is also applicable to the Robin boundary condition. In fact, we will proceed and writeit down explicitly not only for the convenience of the interested reader but also since itwas a guiding step between the proof for the defect conditions connecting two half-lines,see [4], and Proposition 2.
B.1 Dressing the Robin boundary condition
As before, we look at the NLS equation (2.1) for ( t, x ) ∈ R + × R + and complement itwith Robin boundary condition at x = 0 , which, in our notation, is u x = αu, α ∈ R . (B.1)Then, the NLS equation has again a corresponding Lax system and the boundary con-dition can be written in the form of a boundary constraint V ( t, x, − λ ) G ( λ ) − G ( λ ) V ( t, x, λ )) (cid:12)(cid:12) x =0 , (B.2)where the boundary matrix G ( λ ) is given by G ( λ ) = 1 iα + 2 λ (cid:18) iα − λ iα + 2 λ (cid:19) (B.3)and is, in particular, independent of t and x . Similarly to the boundary matrix (2.10), G ( λ ) is scaled by ( iα + 2 λ ) − , so that ( G ( λ )) − = G ( − λ ) , which is crucial for theproof of Proposition 3. Remark 4.
The kernel vectors for G ( λ ) can be easily obtained at λ = iα and λ = − iα ,we have respectively e and e . .2 Dressing the boundary In this approach we will also leave out boundary-bound soliton solutions. Due to thefact that for boundary-bound soliton solutions the proof needs to be slightly changed.The proof will be not as detailed as for Proposition 2. Nonetheless, we will point out thedifferences rather than the similarities.
Proposition 3.
Consider a solution u [0]( t, x ) to the NLS equation on the half-line subjectto the Robin boundary conditions (B.2) with parameter α ∈ R . Take one solution ψ of theundressed Lax system corresponding to u [0] for λ = λ = − iα . Further, take N solutions ψ j of the undressed Lax system corresponding to u [0] for distinct λ = λ j ∈ C \ (cid:0) R ∪ i R (cid:1) , j = 1 , . . . , N . Constructing G ( λ ) as in (B.3) with α , we assume that there exist pairedsolutions (cid:98) ψ j of the undressed Lax system corresponding to u [0] for λ = (cid:98) λ j = − λ j , j =1 , . . . , N , satisfying (cid:98) ψ j (cid:12)(cid:12) x =0 = ( G ( λ j ) ψ j ) (cid:12)(cid:12) x =0 , (cid:98) λ k (cid:54) = λ j . (B.4) Then, a N -fold Darboux transformation D [2 N ] using { ψ , (cid:98) ψ , . . . , ψ N , (cid:98) ψ N } and theirrespective spectral parameter lead to the solution u [2 N ] to the NLS equation on the half-line. In particular, the boundary condition is preserved and we denote such a solution u [2 N ] by (cid:98) u [ N ] .Proof. Defining the matrix G N ( t, λ ) = 2 λσ + G (0) ( t ) similar to K N ( t, x, λ ) throughthe transformed kernel vectors D [2 N ]( t, x, λ ) e and D [2 N ]( t, x, λ ∗ ) e at x = 0 andrespectively λ = λ and λ = λ ∗ , we can derive that the equality ( D [2 N ]( t, x, − λ ) G ( λ )) (cid:12)(cid:12) x =0 = ( G N ( t, λ ) D [2 N ]( t, x, λ )) (cid:12)(cid:12) x =0 (B.5)holds, whereas G ( λ ) is multiplied by λ + iα .To reconstruct the expression of G N ( t, λ ) , we analyze the equality (B.5). In particularfor the equality of the matrix coefficients regarding λ of order N , we obtain for the off-diagonal entries of G (0) ( t ) that G (0)12 ( t ) = 0 and G (0)21 ( t ) = 0 . Then, for the diagonalentries, we need to evaluate the determinant of G N ( t, λ ) in two ways. Firstly, as aproduct of matrices det( G N ( t, λ )) = det( D [2 N ]( t, , − λ )) det( G ( λ )) det(( D [2 N ]( t, , λ )) − ) = − λ − α . Secondly, through the definition G N ( t, λ ) = 2 λσ + G (0) ( t ) and the partial result for theoff-diagonal entries, we obtain consequently G (0)11 ( t ) − G (0)22 ( t ) = 0 , det( G (0) ( t )) = − α . (B.6)Hence, we need to have G (0) ( t ) = ± iα . However, by the equality (B.5) of the matrixcoefficients regarding λ of zero-th order, we can verify, that the structure of G ( λ ) ispreserved, since we need to have G (0) ( t ) = − iα in order for − iα Σ N = G (0) ( t )Σ N (B.7)to hold. 22his proposition serves as a means to insert solitons in the case of the zero seedsolution and also the non-zero seed solution with constant background as presented in[7]. Similarly, to the argumentation therein, we would also need to proof distinctly thedressing of boundary-bound solitons with pure imaginary spectral parameters λ j ∈ i R , j = 1 , . . . , N . However, since the reasoning is similar and it is presumably not applicablein the case of the new boundary condition, we omit that consideration here.Nonetheless, we hereby gave the proof to dress solitons corresponding to λ j ∈ C \ (cid:0) R ∪ i R (cid:1) , j = 1 , . . . , N , in the dressing the boundary method we adapted to the newboundary conditions. It turns out that the equality (B.7), from which we then followthe definite form of G N ( λ ) , functions as an intermediate step between the ideas in theproof for the defect conditions connecting two half-lines [4] and for the new boundaryconditions, see Proposition 2. B.3 Relations between scattering data
Consider the zero seed solution u [0] = 0 and C \ R (cid:51) λ = − iα . Following the stepsin Section 3.1, we take ψ j = e − i ( λ j x +2 λ j t ) σ ( u j , v j ) (cid:124) for λ = λ j and with respect tothe relation (cid:98) ψ j (cid:12)(cid:12) x =0 = ( G ( λ j ) ψ j ) (cid:12)(cid:12) x =0 , − λ k (cid:54) = λ j for all j, k ∈ { , . . . , N } , we also take (cid:98) ψ j = e − i ( (cid:98) λ j x +2 (cid:98) λ j t ) σ (ˆ u j , ˆ v j ) (cid:124) for λ = (cid:98) λ j = − λ j , whereas ˆ u j ˆ v j = iα − λ j iα + 2 λ j u j v j , j = 1 , . . . , N. Analogously, if we apply a two-fold Darboux transformation consisting of { λ , ψ , (cid:98) λ ∗ , (cid:98) ϕ } ,we obtain the scattering data, which particularly results in the relation C (2)1 ( C (2)2 ) ∗ = − λ · iα − λ iα + 2 λ · (cid:61) ( λ ) (cid:60) ( λ ) , (B.8)which is up to notation the same as in [2]. To align the notation, one would need tocomplex conjugate (B.8) and then it would be equal to the equation (2.36) in their paperwith k = − λ ∗ . This is due to the differently defined potential (cid:101) Q of the matrix V , whichas a consequence gives the existence of Jost solutions with different asymptotic behaviorand continuations into different parts of the complex plane. Analogously to Section 3.1,we obtain the following relations between the initial positions and phases of N insertedsolitons. Remark 5.
In general, we can construct a N -Darboux transformation using the in-formation given by { λ , ψ , . . . , λ N , ψ N , (cid:98) λ ∗ , (cid:98) ϕ , . . . , (cid:98) λ ∗ N , (cid:98) ϕ N } , where λ j = ξ j + iη j andconsequently (cid:98) λ ∗ = − ξ j + iη j for j = 1 , . . . , N with corresponding solutions to the un-dressed Lax system as above. Then for j = 1 , . . . , N , the relation for a pair of initial ositions x j and (cid:98) x j = x N + j as well as phases φ j and (cid:98) φ j = φ N + j amounts to x j + ˆ x j = 12 η j log (cid:16) η j ξ j (cid:17) + 14 η j log (cid:16) (2 ξ j ) + ( α − η j ) (2 ξ j ) + ( α + 2 η j ) (cid:17) − η j N (cid:88) (cid:48) k =1 log [( ξ j − ξ k ) + ( η j − η k ) ][( ξ j + ξ k ) + ( η j − η k ) ][( ξ j + ξ k ) + ( η j + η k ) ][( ξ j − ξ k ) + ( η j + η k ) ] ,ϕ j − ˆ ϕ j = 2 arg( λ j ) + arg (cid:16) ξ j + i (2 η j − α )2 ξ j + i (2 η j + α ) (cid:17) − N (cid:88) (cid:48) k =1 arg (cid:16) [( ξ j − ξ k ) + i ( η j − η k )][( ξ j + ξ k ) + i ( η j − η k )][( ξ j + ξ k ) + i ( η j + η k )][( ξ j − ξ k ) + i ( η j + η k )] (cid:17) , whereas the product of a pair of weights C j , (cid:98) C j = C N + j is C j (cid:98) C ∗ j = − λ j iα − λ j iα + 2 λ j (2 η j ) (2 ξ j ) (cid:104) N (cid:89) (cid:48) k =1 ( λ j − λ ∗ k )( λ j + λ k )( λ j − λ k )( λ j + λ ∗ k ) (cid:105) . References [1] M. J. Ablowitz, B. Prinari, and A. D. Trubatch.
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