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An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line

We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4x4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation.


I. INTRODUCTION
The initial value problems for many integrable nonlinear evolution partial differential equations (PDEs) with the Lax pairs can be solved in terms of the inverse scattering transform (IST) [1][2][3]. After that, there exist some important extensions of the IST such as the Deift-Zhou nonlinear steepest descent method [4] and the Fokas unified method [5][6][7]. Particularly, the Fokas unified method can be used to study the initialboundary value problems for both linear and nonlinear integrable evolution PDEs with 2 × 2 Lax pairs on the half-line and the finite interval, such as the nonlinear Schrödinger equation [6,[8][9][10][11], the sine-Gordon equation [12,13], the KdV equation [14], the mKdV equation [15,16], the derivative nonlinear Schrödinger equation [17], Ernst equations [18], and etc. (see Refs. [19][20][21] and references therein). Recently, Lenells extended the Fokas method to study the initial-boundary value (IBV) problems for integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line [22]. After that, the idea was extended to study IBV problems of some integrable nonlinear evolution equations with 3 × 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation [23], the Sasa-Satsuma equation [24], the coupled nonlinear Schrödinger equations [25][26][27][28], and the Ostrovsky-Vakhnenko equation [29]. To the best of our knowledge, so far there is no work on the IBV problems of integrable equations with 4 × 4 Lax pairs on the half-line.
The rest of this paper is organized as follows. In Sec. 2, we study the spectral analysis of the associated 4 × 4 Lax pair of Eq. (1). Sec. 3 presents the corresponding 4 × 4 matrix RH problem in terms of the jump matrices found in Sec. 2. The global relation is used to establish the map between the Dirichlet and Neumann boundary values in Sec. 4.

II. THE SPECTRAL ANALYSIS OF THE LAX PAIR
In this subsection, we will simultaneously consider the spectral analysis of the Lax pair (4) to present sectionally its analytic eigenfunctions in order to formulate a 4 × 4 matrix RH problem defined in the complex k-plane. The spin-1 GP equations (1) admits the 4 × 4 Lax pair [30] ψ x + ikσ 4 ψ = U (x, t)ψ, ψ t + 2ik 2 σ 4 ψ = V (x, t, k)ψ, where ψ = ψ(x, t, k) is a 4×4 matrix-valued or 4 × 1 column vector-valued spectral function, k ∈ C is an isospectral parameter, σ 4 = diag(1, 1, −1, −1). and the 4 × 4 matrix-valued functions U (x, t) and V (x, t, k) are defined by A new eigenfunction µ = µ(x, t, k) is defined by the transform such that the Lax pair (4) is changed into an equivalent form whereσ 4 µ = [σ 4 , µ],σ 4 denote the commutator with respect to σ 4 and the operator acting on a 4 × 4 matrix such that e xσ4 X = e xσ4 Xe −xσ4 . The Lax pair (7) leads to a full derivative form where the closed one-form W (x, t, k) is Fig. 1(a)), . Thus for the point (ξ, τ ) on the each contour, we have It follows from the one-form (8) that we can use the Volterra integral equations to define its three eigenfunctions {µ j } 3 1 on the above-mentioned three contours {γ j } 3 where I = diag(1, 1, 1, 1), the integral is over a piecewise smooth curve from (x j , t j ) to (x, t), W j (x, t, k) is given by Eq. (9) with µ(x, t, k) replaced by µ j (x, t, k). Since the one-form W j is closed, thus µ j is independent of the path of integration. If we choose the paths of integration to be parallel to the x and t axes, then the integral Eq. (11) becomes (j = 12, 3) Eq. (12) implies that the first, second, third, and fourth columns of the matrices µ j (x, t, k)'s contain these

exponentials
To analyse the bounded domains of the eigenfunctions {µ j } 3 1 in the complex k-plane, we need to use the curve K = {k ∈ C|Re f (k) · Re g(k) = 0, f (k) = ik, g(k) = ik 2 }, to separate the complex k-plane into four regions (see Fig. 2): Thus it follows from Eqs. (10), (13) and (14) that the domains, where the different columns of eigenfunctions {µ j } 3 1 are bounded and analytic in the complex k-plane, are presented as follows: where Re f (k) = −Im k < 0, g + (k) =: Re g(k) = −2 Re k Im k > 0, and g − (k) =: Re g(k) = −2 Re k Im k < 0.

(d) The minors of eigenfunctions
The cofactor matrix X A (or the transpose of the adjugate) of a 4 × 4 matrix X is given by where m ij (X) denote the (ij)th minor of X and (X A ) T X = adj(X)X = det X. It follows from Eq. (7) that be shown that the matrix-valued functions µ A j 's satisfy the Lax pair whose solutions can be expressed as by using the Volterra integral equations, where U T and V T denote the transposes of U and V , respectively.
It is easy to check that the regions of boundedness of µ A j : is bounded for k ∈ (C + , C + , C − , C − ), which are symmetric ones of µ j about the Re k-axis (cf. Eq. (15)).  Let us introduce the 4 × 4 matrix-valued functions S(k), s(k), and S(k) by µ j , j = 1, 2, 3 Evaluating system (23) at (x, t) = (0, 0) and (x, t) = (0, T ), respectively, we have These relations among µ j are displayed in Fig. 3. Thus these three functions S(k), s(k), and S(k) are dependent such that we only consider two of them, e.g., S(k) and s(k).
According to the definition (12) of µ j , Eq. (24) implies that where µ j (0, t, k), j = 1, 2 and µ 3 (x, 0, k), 0 < x < ∞, 0 < t < T satisfy the Volterra integral equations Thus, it follows from Eqs. (25) and (26) that s(k) and S(k) are determined by U (x, 0, k) and V (0, t, k), i.e., by the initial data q j (x, t = 0) and the Dirichlet-Neumann boundary data q j (x = 0, t) and q jx (x = 0, t), j = 1, 0, −1, respectively. In fact, µ 3 (x, 0, k) and µ 1,2 (0, t, k) satisfy the x-part and t-part of the Lax pair (7) at t = 0 and x = 0, respectively, that is, Moreover, the functions {S(k), s(k)} and {S A (k), s A (k)} have the following boundedness: It follows from the third one in Eq. (24) that we have the so-called global relation where µ 3 (0, t, k), 0 < t < T satisfies the Volterra integral equation In each domain D n , n = 1, 2, 3, 4 of the complex k-plane, the solution M n (x, t, k) of Eq. (7) is via the Volterra integral equations, where W n (x, t, k) is given by Eq. (9) with µ(x, t, k) replaced with M n (x, t, k), and the definition of the contours (γ n ) lj 's is given by The definition (32) of (γ n ) lj implies that the matrices γ n (n = 1, 2, 3, 4) are of the forms According to the similar proof for the 3 × 3 Lax pair in [22] and the above-mentioned properties of µ(x, t, k), we have the bounedness and analyticity of M n : Proposition 2.1. The matrix-valued functions M n (x, t, k), n = 1, 2, 3, 4 are weill defined by Eq. (31) for k ∈D n and (x, t) ∈Ω. For any fixed point (x, t), M n 's are the bounded and analytic function of k ∈ D n away from a possible discrete set of singularity {k j } at which the Fredholm determinants vanish. M n (x, t, k) also admits the bounded and continuous extensions toD n and The new spectral functions S n (k) (n = 1, 2, 3, 4) are introduced by Let M (x, t, k) stand for the sectionally analytic function on the Riemann k-spere which is equivalent to with the jump matrices J mn (x, t, k) defined by Proposition 2.2. The matrix-valued functions S n (x, t, k) (n = 1, 2, 3, 4) defined by can be determined by the entries of s(k) = (s ij ) 4×4 , S(k) = (S ij ) 4×4 (cf. Eq. (24)) as follows: where n i1j1,i2j2 (X) denotes the determinant of the sub-matrix generated by taking the cross elements of i 1,2 th rows and j 1,2 th columns of the 4 × 4 matrix X and ) denotes the determinant of the matrix generated by choosing the first and second columns of S(k) and the third and fourth columns of s(k), and Proof. Let γ x0 3 with x 0 > 0 denote the contour (x 0 , 0) → (x, t) in the (x, t)-plane and µ 3 (x, t, k; x 0 ) be determined by Eq. (11) with j = 3 and the contour γ 3 replaced by γ x0 3 . M n (x, t, k; x 0 ) is defined by Eq. (31) with the contour γ 3 replaced by γ x0 3 . We introduce the functions R n (k; x 0 ), S n (k; x 0 ), and T n (k; x 0 ) in the form It follows from Eq. (40) that we have the relations and which can in general deduce the functions {R n (k; x 0 ), S n (k; x 0 ), T n (k; x 0 )} for the given spectral functions {s(k), S(k)}.
Moreover, we can also determine some entries of {R n (k; by using Eqs. (31) and (40). System (42) contains 32 scalar equations for 32 unknowns. Thus it follows from system (42) that we have S n (k; x 0 ). Then taking the limit x 0 → ∞ of S n (k; x 0 ) yields Eq. (39).
We use {k j } N 1 to denote the above-mentioned possible zeros and suppose that they satisfy the following assumption.
Assumption 2.3. We suppose that • n 33,44 (s)(k) admits n 1 possible simple zeros in D 1 denoted by {k j } n1 1 ; • n 11,22 (s)(k) admits N − n 3 possible simple zeros in D 4 denoted by {k j } N n3+1 ; and that none of these simple zeros coincide. Moreover, none of these functions are assumed to have zeros on the boundaries of the D n 's (n = 1, 2, 3, 4).
1 be the eigenfunctions given by Eq. (31) and suppose that the set {k j } N 1 of singularities are as the above-mentioned Assumption 2.3. Then we have the following residue conditions: where the overdot denotes the derivative with resect to the parameter k and θ = θ(k) = −i(kx + 2k 2 t).
Proof. It follows from Eqs. (38) and (39) that we find the four columns of M 1 (x, t, k) as For the case that k j ∈ D 1 is a simple zero of n 33 whose residues at k = k j , k j ∈ D 1 yield Eq. (44). Similarly, we can show Eq. (45) for k j ∈ D 2 , Eq. (46) for k j ∈ D 3 , and Eq. (47) for k j ∈ D 4 by studying Eqs. (38) and (39) for n = 2, 3, 4.

III. THE 4 × 4 MATRIX RIEMANN-HILBERT PROBLEM
By using the district contours γ j (j = 1, 2, 3, 4), the integral solutions of the revised Lax pair (7), and S n due to {S(k), s(k)}, we have defined the sectionally analytic function M n (x, t, k), n = 1, 2, 3, 4, which solves a 4 × 4 matrix Riemann-Hilbert (RH) problem. This RH problem can be formulated on basis of the initial conditions of the Schwartz class q j (x, t = 0) and Dirichlet-Neumann boundary data q j (x = 0, t) and q jx (x = 0, t), j = 1, 0, −1. Thus the solution of Eq. (1) for all values of x, t can be refound by solving the RH problem.
We use the initial and boundary data to define the jump matrices J mn (x, t, k), n, m = 1, ..
• Across the contoursD n ∪D m (n, m = 1, 2, 3, 4), M (x, t, k) satisfies the jump condition (36). Proof. System (50) can be deduced from the large k asymptotics of the eigenfunctions. We can follow the similar one in Refs. [6,11] to show the rest proof of the Theorem.

IV. NONLINEARIZABLE BOUNDARY CONDITIONS
The main difficulty of the initial-boundary value problems is to find the boundary values for a well-posed problem. All boundary conditions are required for the definition of S(k), and hence for the formulate the 4 × 4 matrix RH problem. Our main conclusion exhibits the unknown boundary condition on basis of the prescribed boundary condition and the initial condition in terms of the solution of a system of nonlinear integral equations.
which can be written as It follows from Eq. (7) that we have the asymptotics of eigenfunctions where we have introduced the following functions j,14 = where the functions {µ jl (x, t)} 3 1 , i = 1, 2 are independent of k. We define the matrix-valued function Ψ(t, k) = (Ψ ij (t, k)) 4×4 as By using the asymptotic of Eq. (53) and the boundary data at x = 0, we find Thus we have the the Dirichlet-Neumann boundary data at x = 0: For the vanishing initial values, it follows from Eqs. (52a) and (52b) that we have the following asymptotic of c 24 (t, k) and c 1j (t, k), j = 3, 4. Proof.

(c) The map between Dirichlet and Neumann problems
In the following we mainly show that the spectral functions S(k) and S L (k) can be expressed in terms of the prescribed Dirichlet and Neumann boundary data and the initial data using the solution of a system of integral equations.
Define the new notations as The sign ∂D j , j = 1, ..., 4 stands for the boundary of the jth quadrant D j , oriented so that D j lies to the left of ∂D j . ∂D 0 3 denotes the boundary contour which has not contain the zeros of Σ − (k) and ∂D 0 3 = −∂D 0 1 .
1−1 (t ′ )dt ′ + lower order terms, can be generated at each step from the known Neumann boundary data u [n] 1j , j = 1, 0, −1 such that we know that the Dirichlet boundary data u [n] 0j , j = 1, 0, −1 can then be given by Eqs. (105a)-(105c). Remark 4.3. We can also give the corresponding Gelfand-Levitan-Marchenko representations for the Dirichlet and Neumann boundary value problems and the IVB of spin-1 GP equation on the finite interval, which will be studied in another paper. The analogous analysis of the Fokas unified method can also be extended to study the IBV problems for other integrable nonlinear evolution PDEs with 4 × 4 Lax pairs on the half-line or the finite interval.