Abstract We construct multi-soliton solutions of the n-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called dressing the boundary, which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided. Keywords：polarizer effect;solitons on the half-line;vector nonlinear Schrödinger equation;integrable boundary conditions;boundary-bound states
The concept of integrable boundary conditions (BCs), mainly developed by Sklyanin [1], represents one of the most successful approaches to initial-boundary-value problems for two-dimensional integrable nonlinear partial differential equations (PDEs). The idea lies in translating the integrability of soliton equations with boundaries into certain algebraic constraints known as reflection equations, cf [1–3]. As consequences, classes of soliton models, restricted on a finite interval, are integrable subject to integrable BCs [1].
In this paper, we consider the focusing vector nonlinear Schrödinger (VNLS) equation, also known as the Manakov model [4], restricted to the half-line space domain. The equation reads$\begin{eqnarray}{\rm{i}}{{\boldsymbol{r}}}_{t}+{{\boldsymbol{r}}}_{{xx}}+2{\boldsymbol{r}}{}^{\dagger }\,{\boldsymbol{H}}\,{\boldsymbol{r}}\,{\boldsymbol{r}}={\bf{0}},\quad {\boldsymbol{r}}:= {\boldsymbol{r}}(x,t),\end{eqnarray}$where ${\boldsymbol{r}}={({r}_{1},\cdots ,{r}_{n})}^{{\rm{T}}}$, 0 denotes the zero n-vector, and r† denotes the conjugate transpose of r. Each component rj is a complex field, and H is an n × n positive definite Hermitian matrix modeling interactions among the components. There is a natural ${ \mathcal U }(n)$-invariance of the model under the transformation r ↦ Tr, where ${\boldsymbol{T}}\in { \mathcal U }(n)$. Let T diagonalize H, then the VNLS equation (1), up to certain scaling, can be reduced to its standard form$\begin{eqnarray}{\rm{i}}{{\boldsymbol{r}}}_{t}+{{\boldsymbol{r}}}_{{xx}}+2{{\boldsymbol{r}}}^{\dagger }{\boldsymbol{rr}}={\bf{0}}.\end{eqnarray}$
The VNLS equation is a vector generalization of the (scalar) NLS equation by allowing internal degrees of freedom. Physically, it is a relevant model to describe optical solitons and collective states in low-temperature physics, cf [5, 6]; mathematically, the nontrivial interactions of vector solitons are related to the notion of Yang–Baxter maps, cf [7–10].
Integrable BCs for the VNLS equation, as well as soliton solutions to the VNLS equation on the half-line, were derived in [11] by means of a nonlinear mirror-image technique [12] (see also [13]) that extends the half-line space domain to the whole axis. However, there was severe difficulty constructing N-soliton solutions on the half-line as the soliton data can only be computed recursively. In practice, the computations are becoming increasingly complicated for N ≥ 2 (see for instance [11], conclusions).
We provide an efficient approach to deriving N-soliton solutions of the VNLS equation on the half-line. The construction is based on the so-called dressing the boundary, introduced recently by one of the authors [14, 15]. The essential ideas are: given integrable BCs of the VNLS equation (or any integrable PDEs), by generating soliton solutions using the Darboux-dressing transformations (DTs), we look for those DTs that preserve the integrable BCs. This gives rise to exact solutions of the underlying integrable model on the half-line, and admits a natural inverse scattering transform (IST) interpretation. The true powers of our construction consist of: (i) the N-soliton solutions can be obtained in compact forms (this was highly complicated following the mirror-image method [11, 12], see the discussion provided in section 6); (ii) it does not require any extension of the space domain. This reveals that dressing the boundary represents a natural approach to solve the VNLS equation (or classes of PDEs) on the half-line equipped with integrable BCs.
Note that Fokas’ unified transform method [16, 17] represents a systematic approach to treating initial-boundary-value problems for integrable PDEs. This method can be regarded as a generalization of the IST, cf [18–21], and was already applied to the NLS [22] and VNLS [23] equations. However, it is a difficult task to obtain exact solutions within the Fokas’ method, although asymptotic solutions at large times could be derived.
The outline and main results of the paper are as follows. First, DTs for generating soliton solutions of the VNLS equation are reviewed in section 2. Then, we recall in section 3 results on integrable BCs for the VNLS equation on the half-line [11]. There are two classes of integrable BCs: the homogeneous Robin BCs and mixed Neumann/Dirichlet BCs. In section 4, we apply the approach of dressing the boundary to the VNLS equation on the half-line, which gives rise to explicit N-soliton solutions on the half-line. Our results provide a clear answer to the question of obtaining general N-soliton solutions in the presence of a boundary [11]. Moreover, we can construct stationary vector solitons subject to the Robin BCs at the boundary. These correspond to boundary-bound solitons. In section 5, we provide explicit examples of vector solitons interacting with the boundary. In particular, by combining the effects of the mixed Neumann/Dirichlet BCs and the ${ \mathcal U }(n)$-invariance of the VNLS equation, the boundary can act as a polarizer that tunes components of solitons after interacting with the boundary. We provide connections of our construction to the IST in section 6. Possible connections of the half-line VNLS equation under integrable BCs to the Gross–Pitaevskii (GP) equation are also discussed.
2. DTs and soliton solutions
The n-component VNLS equation (2) is equivalent to the compatibility of the linear differential system$\begin{eqnarray}{{\boldsymbol{\Phi }}}_{x}={\boldsymbol{U}}\,{\boldsymbol{\Phi }},\quad {{\boldsymbol{\Phi }}}_{x}={\boldsymbol{V}}\,{\boldsymbol{\Phi }}.\end{eqnarray}$Here, U, V, known as the Lax pair, are (n + 1 × n + 1) matrix-valued functions$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{U}} & = & -{\rm{i}}\lambda {\boldsymbol{\Sigma }}+{\boldsymbol{Q}},\quad {\boldsymbol{V}}=-2{\rm{i}}{\lambda }^{2}{\boldsymbol{\Sigma }}+{{\boldsymbol{Q}}}_{{\boldsymbol{T}}},\\ {{\boldsymbol{Q}}}_{{\boldsymbol{T}}} & = & 2\lambda {\boldsymbol{Q}}-{\rm{i}}{{\boldsymbol{Q}}}_{x}\,{\boldsymbol{\Sigma }}-{\rm{i}}{{\boldsymbol{Q}}}^{2}\,{\boldsymbol{\Sigma }},\end{array}\end{eqnarray}$where λ is the spectral parameter, and Σ and Q are block matrices$\begin{eqnarray}{\boldsymbol{\Sigma }}=\left(\begin{array}{cc}{{\boldsymbol{I}}}_{n} & {\bf{0}}\\ {{\bf{0}}}^{{\rm{T}}} & -1\end{array}\right),\quad {\boldsymbol{Q}}=\left(\begin{array}{cc}{{\boldsymbol{O}}}_{n} & {\boldsymbol{r}}\\ -{{\boldsymbol{r}}}^{\dagger } & 0\end{array}\right),\end{eqnarray}$with In, On being the identity and zero square matrices of size n, respectively. There is a natural gauge group acting on the Lax pair (3)$\begin{eqnarray}\begin{array}{rcl}\widetilde{{\boldsymbol{U}}} & = & {\boldsymbol{G}}\,{\boldsymbol{U}}\,{{\boldsymbol{G}}}^{-1}+{{\boldsymbol{G}}}_{x}\,{{\boldsymbol{G}}}^{-1},\\ \widetilde{{\boldsymbol{V}}} & = & {\boldsymbol{G}}\,{\boldsymbol{V}}\,{{\boldsymbol{G}}}^{-1}+{{\boldsymbol{G}}}_{t}\,{{\boldsymbol{G}}}^{-1},\end{array}\end{eqnarray}$and DTs can be represented by G that preserves the forms of U, V by extracting the singular structures, cf [24–27]. A one-step DT for the VNLS equation amounts to the map ${\boldsymbol{\Phi }}\mapsto \widetilde{{\boldsymbol{\Phi }}}={\boldsymbol{D}}[1]{\boldsymbol{\Phi }}$, where D[1] is called a dressing factor of degree 1$\begin{eqnarray}{\boldsymbol{D}}[1](\lambda )={{\boldsymbol{I}}}_{n+1}+\left(\displaystyle \frac{{\lambda }_{1}^{* }-{\lambda }_{1}}{\lambda -{\lambda }_{1}^{* }}\right){{\boldsymbol{\Pi }}}_{1},\quad {{\boldsymbol{\Pi }}}_{1}=\displaystyle \frac{{{\boldsymbol{\Psi }}}_{1}{{\boldsymbol{\Psi }}}_{1}^{\dagger }}{{{\boldsymbol{\Psi }}}_{1}^{\dagger }{{\boldsymbol{\Psi }}}_{1}}.\end{eqnarray}$Here, $\Psi$1 is a particular solution of the undressed Lax pair (3) associated with λ1. Having a set of N particular solutions {$\Psi$j, λj}, j = 1,…,N, one can iterate the DTs and construct the dressing factor D[N] of degree N. For simplicity $\Psi$j's are assumed to be vectors (of rank 1). In the IST formalism, D[N] plays the role of the scattering matrix: one adds a pair of complex zero/pole $\{{\lambda }_{j},{\lambda }_{j}^{* }\}$ to the scattering system at each step of the DTs . There are two important properties of DTs: (1) the Bianchi permutativity, meaning that the order of adding $\Psi$j is irrelevant; and (2) the action of D[N](λ) can be expressed in compact forms (usually in terms of determinant structures).
Since we are focusing on soliton solutions, the zero seed solution r = 0 is imposed to the undressed Lax pair. Without loss of generality, let $\Psi$j's be in the forms$\begin{eqnarray}{{\boldsymbol{\Psi }}}_{j}={{\rm{e}}}^{-{\rm{i}}({\lambda }_{j}x+2{\lambda }_{j}^{2}t){\boldsymbol{\Sigma }}}\left(\begin{array}{c}{{\boldsymbol{b}}}_{j}\\ 1\end{array}\right),\end{eqnarray}$where bj's are constant complex n-vectors called norming vectors. Now, encoding the soliton data into {bj, λj}, j = 1,…,N, with distinct λj, the N-soliton solutions to the VNLS equation (2) are, cf [21, 26]$\begin{eqnarray}{r}_{{\ell }}=\displaystyle \frac{2{\rm{i}}}{\det {\boldsymbol{M}}}\left|\begin{array}{cccc}0 & 1 & \cdots & 1\\ {\beta }_{1;{\ell }} & & & \\ \vdots & & {\boldsymbol{M}} & \\ {\beta }_{N;{\ell }} & & & \end{array}\right|,\quad {\ell }=1,\ldots ,n,\end{eqnarray}$where rℓ is the ℓth component of r, ${{\boldsymbol{\beta }}}_{j}:= {{\boldsymbol{\beta }}}_{j}(x,t)={e}^{-2{\rm{i}}({\lambda }_{j}x+2{\lambda }_{j}^{2}t)}{{\boldsymbol{b}}}_{j}$ with βj;ℓ being its ℓth component, and the N × N matrix M has the components ${M}_{j{\ell }}=\tfrac{{{\boldsymbol{\beta }}}_{{\ell }}^{\dagger }{{\boldsymbol{\beta }}}_{j}+1}{{\lambda }_{j}^{* }-{\lambda }_{{\ell }}}$. As an illustration, the one-soliton data {b1, λ1}, with ${\lambda }_{1}=\tfrac{1}{2}({\mu }_{1}+{\rm{i}}{\nu }_{1})$, ν1 > 0, lead to the one vector soliton solution$\begin{eqnarray}{\boldsymbol{r}}(x,t)={\boldsymbol{p}}\,\displaystyle \frac{{\nu }_{1}{{\rm{e}}}^{-{\rm{i}}({\mu }_{1}x+({\mu }_{1}^{2}-{\nu }_{1}^{2})t-\pi )}}{\cosh ({\nu }_{1}(x+2{\mu }_{1}t+{\rm{\Delta }}x))}\equiv {\boldsymbol{p}}\,r(x,t).\end{eqnarray}$Here, ${\rm{\Delta }}x=\tfrac{\mathrm{log}| {{\boldsymbol{b}}}_{1}| }{{\nu }_{1}}$, ${\boldsymbol{p}}=\tfrac{{{\boldsymbol{b}}}_{1}}{| {{\boldsymbol{b}}}_{1}| }$, the solution r(x, t) is composed by the usual scalar NLS soliton solution r(x, t), where the amplitude ν1 and the velocity 2μ1 are controlled by the imaginary and real parts of λ1, respectively, and a unit polarization vector p.
3. Integrable BCs for VNLS
Now we restrict the space domain of the VNLS equation to the positive semi-axis. Integrable BCs for the half-line VNLS equation were investigated in [11] (see also [28] in which only the vector Robin BCs were derived). The integrability in the presence of a boundary was translated into a constraint on the t-part of the Lax pair$\begin{eqnarray}{\boldsymbol{K}}(\lambda ){\boldsymbol{V}}(-\lambda ){| }_{x=0}={\boldsymbol{V}}(\lambda ){| }_{x=0}\,{\boldsymbol{K}}(\lambda ).\end{eqnarray}$Here, the boundary matrixK(λ) is assumed to be nondegenerate. As solutions of the boundary constraint (11), two classes of BCs were obtained [11]: (i) the homogeneous vector Robin BCs:$\begin{eqnarray}({{\boldsymbol{r}}}_{x}-2\alpha {\boldsymbol{r}}){| }_{x=0}={\bf{0}},\quad \alpha \in {\mathbb{R}},\end{eqnarray}$having the boundary matrix$\begin{eqnarray}{\boldsymbol{K}}(\lambda )=\left(\begin{array}{cc}{f}_{a}(\lambda ){{\boldsymbol{I}}}_{n} & {\bf{0}}\\ {{\bf{0}}}^{{\rm{T}}} & 1\end{array}\right),\quad {f}_{\alpha }(\lambda )=\displaystyle \frac{{\rm{i}}\alpha +\lambda }{{\rm{i}}\alpha -\lambda }.\end{eqnarray}$The real parameter α controls the boundary behavior: the Neumann (rx∣x=0 = 0) and Dirichlet BCs (r∣x=0 = 0) appear as special cases of (12) as α = 0 and ∣α∣ → ∞, respectively; (ii) the mixed Neumann/Dirichlet (mND) BCs:$\begin{eqnarray}{r}_{{{\ell }}_{x}}{| }_{x=0}=0,\quad {r}_{j}{| }_{x=0}=0,\quad \mathrm{for}\ j\ne {\ell },\end{eqnarray}$where rℓ is the ℓth component of r. Accordingly, one has the boundary matrix$\begin{eqnarray}{\boldsymbol{K}}=\mathrm{diag}({\delta }_{1},...,{\delta }_{n},1),\quad {\delta }_{{\ell }}=1,\quad {\delta }_{j}=-1,\quad \mathrm{for}\ j\ne {\ell },\end{eqnarray}$where the + / − sign of δℓ corresponds to Neumann/Dirichlet BCs.
The boundary constraint (11) was derived in [11] by considering the space-reverse symmetry of the VNLS equation as a Bäcklund transformation. The same constraint was also introduced in Fokas’ unified transform, known as linearizable BCs. Note that the boundary matrix ${\boldsymbol{K}}$ is related to a far-reaching context as it represents solutions of the classical and quantum reflection equations [1–3].
The integrable BCs are compatible with the ${ \mathcal U }(n)$-invariance of the VNLS equation. The transformation ${\boldsymbol{r}}\mapsto \widetilde{{\boldsymbol{r}}}={\boldsymbol{T}}\,{\boldsymbol{r}}$, ${\boldsymbol{T}}\in { \mathcal U }(n)$, is trivial to the Robin BCs, because a collective change in the components of ${\boldsymbol{r}}$ takes place at the boundary under ${\boldsymbol{T}}$. However, ${\boldsymbol{T}}$ induces a nontrivial effect under the mND BCs: since the components of ${\boldsymbol{r}}$ can interact differently with the boundary in two ways that are Neumann and Dirichlet BCs, the action of ${\boldsymbol{T}}$ can mix the two interactions and make transmissions among the different components appear. These transmission phenomena have the interpretation that the boundary acts as a ‘polarizer’ tuning the polarizations of the incoming solitons and, after interacting with the boundary, changes in the polarizations among the solitons take place [11].
4. Dressing the boundary
The integrable BCs for the VNLS equation on the half-line are completely determined by the t-part of the Lax pair through the boundary constraint (11). By dressing the boundary [14], we mean that in the process of DTs to generate exact solutions, the boundary constraint is preserved at each step of the DTs. By construction, this leads to exact solutions of the VNLS equation subject to the integrable BCs. In practice this requires one to find appropriate particular solutions in DTs.
[Dressing the boundary] Let ${\boldsymbol{U}},{\boldsymbol{V}}$ be the undressed Lax pair. Assume that ${\boldsymbol{V}}$ satisfies the boundary constraint (11), and that the Lax pair admits a pair of particular solutions ${{\boldsymbol{\Psi }}}_{j},{\widetilde{{\boldsymbol{\Psi }}}}_{j}$, associated with ${\lambda }_{j},{\widetilde{\lambda }}_{j}$, respectively (assume ${\lambda }_{j}$ is not pure imaginary), such that$\begin{eqnarray}{\widetilde{{\boldsymbol{\Psi }}}}_{j}{| }_{x=0}={\boldsymbol{K}}(-{\lambda }_{j}){{\boldsymbol{\Psi }}}_{j}{| }_{x=0},\quad {\widetilde{\lambda }}_{j}=-{\lambda }_{j},\end{eqnarray}$where ${\boldsymbol{K}}(\lambda )$ is the boundary matrix; then the boundary constraint (11) is preserved after dressing ${\boldsymbol{V}}$ using ${{\boldsymbol{\Psi }}}_{j},{\widetilde{{\boldsymbol{\Psi }}}}_{j}$.
The proof is closely related to the structure of dressing factors. Similar statements can be found in [14] for the scalar case. To obtain exact solutions on the half-line, it remains to find the paired particular solutions ${{\boldsymbol{\Psi }}}_{j},{\widetilde{{\boldsymbol{\Psi }}}}_{j}$ satisfying (16).
[N-soliton solutions on the half-line] Let $\{{{\boldsymbol{b}}}_{j},{\lambda }_{j}\}$ and $\{{\widetilde{{\boldsymbol{b}}}}_{j},{\widetilde{\lambda }}_{j}\}$, j = 1,…,N, be two sets of N-soliton data. Assume that ${\widetilde{\lambda }}_{j}=-{\lambda }_{j}$ (${\lambda }_{j}$ is not pure imaginary) and ${\widetilde{{\boldsymbol{b}}}}_{j}={\boldsymbol{B}}(-{\lambda }_{j}){{\boldsymbol{b}}}_{j}$ with ${\boldsymbol{B}}(\lambda )={f}_{\alpha }(\lambda ){{\boldsymbol{I}}}_{n}$ (${f}_{\alpha }(\lambda )$ defined in (13)), then the so-constructed solutions restricted to $x\geqslant 0$ correspond to N-soliton solutions on the half-line subject to the Robin BCs (12); if ${\boldsymbol{B}}=-\mathrm{diag}({\delta }_{1},...,{\delta }_{n})$, ${\delta }_{{\ell }}=1$, ${\delta }_{j}=-1$, for $j\ne {\ell }$, then the solutions restricted to $x\geqslant 0$ satisfy the mND BCs (14).
The proof is a direct consequence of lemma 1 by taking into account the forms of the particular solutions (8). Dressing the Lax pair using the N-paired soliton data {bj, λj} and $\{{\widetilde{{\bf{b}}}}_{j},{\widetilde{\lambda }}_{j}\}$ gives rise to 2N-soliton solutions on the whole-line, and the requirements that ${\widetilde{\lambda }}_{j}=-{\lambda }_{j}$ create solitons with opposite velocities. By restricting the space domain to the positive semi-axis, the BCs appear as interactions of solitons with opposite velocities at x = 0, then one obtains N-soliton solutions on the half-line. Although this whole-line picture helps to interpret interactions of solitons as BCs, the derivation of soliton solutions on the half-line can be restricted to x ≥ 0. This is in contrast to the nonlinear mirror-image technique [11, 12], where an extended potential to the whole-line is required.
Note that in the above construction, pure imaginary λj's, corresponding to stationary solitons are excluded. By dressing the boundary, we can also construct stationary solitons satisfying the Robin BCs. These are boundary-bound solitons on the half-line.
[Boundary-bound solitons] Let ${\boldsymbol{\beta }}$ be any real $n$-vector such that $| {\boldsymbol{\beta }}| =1$, and $\{{\widetilde{{\boldsymbol{b}}}}_{j},{\lambda }_{j}\},j=1,\ldots ,N$ be a set of $N$-soliton data. Assume that ${\lambda }_{j}$’s are pure imaginary numbers and distinct, and for given α, satisfy ${f}_{\alpha }({\lambda }_{j})\lt 0$ (${f}_{\alpha }(\lambda )$ defined in (13)). Moreover, assume the following forms of the norming constants ${\widetilde{{\boldsymbol{b}}}}_{j}={\boldsymbol{\beta }}{\left(\sqrt{-{f}_{\alpha }({\lambda }_{j})}\right)}^{{\left(-1\right)}^{N}}$, then the so-constructed solutions restricted to $x\geqslant 0$ correspond to $N$-stationary solitons on the half-line subject to the Robin BCs (12).
The requirement that β is a real vector with ∣β∣ = 1 can be easily obtained by taking any real vector $\widetilde{{\boldsymbol{\beta }}}$ divided by its normal ${\boldsymbol{\beta }}=\tfrac{\widetilde{{\boldsymbol{\beta }}}}{| \widetilde{{\boldsymbol{\beta }}}| }$. Again, the restriction on the soliton data follows the idea of dressing the boundary: the boundary constraint (11) is preserved at each step of the DTs. In computing the boundary-bound solitons, the expressions for the norming constants are different for the odd and even soliton numbers. One also excludes the situation where the stationary solitons are subject to the Dirichlet BCs by assuming fα(λj) < 0. Note that for the scalar NLS case, the boundary-bound states were investigated in [14, 29]. One can put the stationary and moving solitons together by combining the associated soliton data. Due to the Bianchi permutativity of DTs, the order of adding the soliton data is irrelevant.
5. Examples of VNLS soliton interacting with a boundary
It is straightforward to apply Prop. 1 and 2 to obtain soliton solutions of the VNLS equation on the half-line. Fix n = 2; three examples under the Robin BCs are shown in figures 1–3. The left and right figures represent, respectively, the norms of the 1st and 2nd components of the solutions.
Figure 1.
New window|Download| PPT slide Figure 1.Two vector solitons vanish at the boundary subject to the Dirichlet BCs: fα(λj)∣α→∞ = 1. The soliton data are: $\{{\lambda }_{j}=\tfrac{1}{2}({\mu }_{j}+{\rm{i}}{\nu }_{j}),{{\boldsymbol{b}}}_{j}\}$ and $\{{\widetilde{\lambda }}_{j}=-{\lambda }_{j},{\widetilde{{\boldsymbol{b}}}}_{j}={{\boldsymbol{b}}}_{j}\}$, j=1,2, with μ1 = 0.5 ν1 = 2, ${{\boldsymbol{b}}}_{1}^{{\rm{T}}}=(2,1)$, μ2 = 4, ν2 = 1.5, ${{\boldsymbol{b}}}_{2}^{{\rm{T}}}=(4\times {10}^{8},1\times {10}^{8})$.
Figure 2.
New window|Download| PPT slide Figure 2.One soliton interacts with a boundary-bound soliton subject to the Robin BCs (α = 2). The moving soliton data are: $\{{\lambda }_{1}=\tfrac{1}{2}({\mu }_{1}+{\rm{i}}{\nu }_{1}),{{\boldsymbol{b}}}_{1}\}$ and $\{{\widetilde{\lambda }}_{1}=-{\lambda }_{1},{\widetilde{{\boldsymbol{b}}}}_{1}={f}_{2}(-{\lambda }_{1}){{\boldsymbol{b}}}_{1}\}$ with μ1 = 1.4, ν1 = 4, ${{\boldsymbol{b}}}_{1}^{{\rm{T}}}=(3,2);$ the static soliton data are: ${{\boldsymbol{\beta }}}^{{\rm{T}}}=\left(\tfrac{3}{\sqrt{10}},\tfrac{1}{\sqrt{10}}\right)$, ${\lambda }_{2}={\rm{i}}\tfrac{2}{7}$, ${{\boldsymbol{b}}}_{2}=\tfrac{{\boldsymbol{\beta }}\sqrt{3}}{\sqrt{11}}$.
Figure 3.
New window|Download| PPT slide Figure 3.Three stationary solitons interfere with themselves at the boundary subject to the Robin BCs (α = 2). Here, ${{\boldsymbol{\beta }}}^{{\rm{T}}}=\left(\tfrac{4}{\sqrt{65}},\tfrac{7}{\sqrt{65}}\right);$ ${\lambda }_{1}={\rm{i}}\tfrac{6}{2}$, ${{\boldsymbol{b}}}_{1}=\tfrac{{\boldsymbol{\beta }}}{\sqrt{5}};$ ${\lambda }_{2}={\rm{i}}\tfrac{5}{2}$, ${{\boldsymbol{b}}}_{2}=\tfrac{{\boldsymbol{\beta }}}{\sqrt{9}};$ ${\lambda }_{3}={\rm{i}}\tfrac{7}{2}$, ${{\boldsymbol{b}}}_{3}=\tfrac{{\boldsymbol{\beta }}\sqrt{3}}{\sqrt{11}}$.
As for the mND BCs, fix n = 2, and let the transformation matrix ${\boldsymbol{T}}\in { \mathcal S }U(2)$ (following remark 2) parameterized by three parameters ω, θ, ξ be in the form$\begin{eqnarray}{\boldsymbol{T}}=\left(\begin{array}{cc}{{\rm{e}}}^{{\rm{i}}\omega }\cos \theta & -{{\rm{e}}}^{{\rm{i}}\xi }\sin \theta \\ {{\rm{e}}}^{-{\rm{i}}\xi }\sin \theta & {{\rm{e}}}^{-{\rm{i}}\omega }\cos \theta \end{array}\right).\end{eqnarray}$Clearly, ${\boldsymbol{r}}\mapsto {\boldsymbol{Tr}}$ induces a mixture of components of r at the boundary. In the computations of the half-line soliton solution, this amounts to B ↦ TBT−1 for B defined in Prop. 1. Examples of two solitons interacting with an mND boundary are shown below. Having B = diag(1, − 1) gives rise to solitons with the 1st component subject to Neumann BCs and 2nd to Dirichlet BCs (see figure 4); under the action of T, for certain choices of the parameters, one can make one component of the outgoing solitons vanishingly small1(1The complete analysis, requiring some asymptotic estimations of the solutions as t → ± ∞ , is omitted here. Detailed analysis can be found, for instance, in [10].) (see figure 5). In other words, the boundary polarizer switches off the 1st component after solitons interact with the boundary.
Figure 4.
New window|Download| PPT slide Figure 4.Two solitons interact with the mixed Neumann (1st component) and Dirichlet (2nd component) BCs with B = − diag(1, − 1). The soliton data are: $\{{\lambda }_{j}=\tfrac{1}{2}({\mu }_{j}+{\rm{i}}{\nu }_{j}),{{\boldsymbol{b}}}_{j}\}$ and $\{{\widetilde{\lambda }}_{j}=-{\lambda }_{j},{\widetilde{{\boldsymbol{b}}}}_{j}={{\boldsymbol{b}}}_{j}\}$, j = 1, 2 with μ1 = 1, ν1 = 2, ${{\boldsymbol{b}}}_{1}^{{\rm{T}}}=(4,4)$, μ2 = 2, ν2 = 1.5, ${{\boldsymbol{b}}}_{2}^{{\rm{T}}}=(4\times {10}^{4},1\times {10}^{4})$.
Figure 5.
New window|Download| PPT slide Figure 5.Polarizer effect: the boundary tunes the polarizations, and the 1st component becomes vanishingly small after interacting with the boundary. Here, the parameters in T (17) are fixed as ω = 0, θ = 0, ξ = 50.66.
6. Discussion
The approach of dressing the boundary is successfully applied to the VNLS equation on the half-line. Although our construction is rather algebraic, it admits a perfect IST interpretation. Since the integrable BCs are only governed by the t-part of the Lax pair through the boundary constraint (11), the soliton data on the half-line can be derived by performing the IST using the t-part of the Lax pair on the boundary. Precisely, this amounts to the direct scattering analysis of$\begin{eqnarray}{{\boldsymbol{\Phi }}}_{t}={\boldsymbol{V}}\,{\boldsymbol{\Phi }}=\left(-2{\rm{i}}{\lambda }^{2}{\boldsymbol{\Sigma }}+{{\boldsymbol{Q}}}_{{\boldsymbol{T}}}\right){\boldsymbol{\Phi }},\end{eqnarray}$where QT is defined in (4). Let Φ± be the fundamental solutions (3) at x = 0$\begin{eqnarray}\begin{array}{l}{{\boldsymbol{\Phi }}}_{\pm }(0,t;\lambda )={{\rm{e}}}^{-2{\rm{i}}{\lambda }^{2}t{\boldsymbol{\Sigma }}}+{\displaystyle \int }_{\pm \infty }^{t}\\ \quad \times \,{{\rm{e}}}^{-2{\rm{i}}{\lambda }^{2}(t-\tau ){\boldsymbol{\Sigma }}}({{\boldsymbol{Q}}}_{{\boldsymbol{T}}}\,{{\boldsymbol{\Phi }}}_{\pm })(0,\tau ;\lambda ){\rm{d}}\tau ,\quad \lambda \in {\mathbb{R}},\end{array}\end{eqnarray}$they are connected by the scattering system Φ+(λ) = S(λ)Φ−(λ), where S(k) is the scattering matrix. The domain of analyticity of the components of S(k) can be split into the four quadrants of the complex plane. The boundary constraint (11) yields$\begin{eqnarray}{\boldsymbol{K}}(\lambda ){\boldsymbol{S}}(-\lambda )={\boldsymbol{S}}(\lambda ){\boldsymbol{K}}(\lambda ),\end{eqnarray}$which governs the soliton data at the boundary x = 0. It is easy to see that paired singularities {λj, − λj} appear (see figure 6), and the relations between the paired norming constants can also be accordingly extracted. These are the requirements listed in Prop. 1. Then, making the soliton data evolve in x for x > 0 gives rise to N-soliton solutions on the half-line.
Note that the relation (20) is in sharp contrast to the formulae (3.43) and (3.46) in [11], which are the governing relations of the soliton data following the nonlinear mirror-image approach. In [11], the singularities are paired as $\{{\lambda }_{j},-{\lambda }_{j}^{* }\}$, and the paired norming constants, that are nonlinearly coupled, can be only be computed recursively. This makes the computation of N-soliton solutions on the half-line highly complicated. However, the relation (20) only involves linear relations between the paired norming constants (${\widetilde{{\boldsymbol{b}}}}_{j}={\boldsymbol{B}}(-{\lambda }_{j}){{\boldsymbol{b}}}_{j}$ in Prop. 1). Therefore, the half-line N-soliton solutions can be easily derived.
As pointed out by Fokas, the scalar NLS equation on the half-line under the Robin BCs can even model solutions to the GP equation on the whole-line with a Dirac-function potential at the origin, cf [30], introduction. Similarly, the VNLS equation on the half-line under the integrable BCs can also describe certain special cases of the vector GP equation. Fix the number of components n = 2, and one has the vector GP equation$\begin{eqnarray}{\rm{i}}{r}_{1,t}+{r}_{1,{xx}}+2(| {r}_{1}{| }^{2}+| {r}_{2}{| }^{2}){r}_{1}+u(x){r}_{1}=0,\end{eqnarray}$$\begin{eqnarray}{\rm{i}}{r}_{2,t}+{r}_{2,{xx}}+2(| {r}_{1}{| }^{2}+| {r}_{2}{| }^{2}){r}_{2}+v(x){r}_{2}=0,\end{eqnarray}$where u(x), v(x) are the potentials. Let u(x) = v(x) = − 4αδ0(x), and let r1, r2 be even in x. Then, after integration, the Dirac function δ0(x) introduces a jump in the derivatives$\begin{eqnarray}\begin{array}{l}{r}_{j,x}({0}_{+},t)-{r}_{j,x}({0}_{-},t)=-4\alpha {r}_{j}(0,t),\\ \quad \quad \times \,j=1,2,\end{array}\end{eqnarray}$which corresponds to the half-line VNLS equation under the vector Robin BCs (12). Again, let u(x) = 0, v(x) = δ0(x), and let r1 be even and r2 be odd in x. This corresponds to the half-line VNLS equation under the mND BCs (14), with r1 and r2 satisfying, respectively, the Neumann and Dirichlet BCs.
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,, Da-jun ZhangDepartment of Mathematics, 
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