The Fokas Method for the Broer-Kaup System on the Half-Line

We analyze the Broer-Kaup system posed on the half-line by using the unified transform method, also known as the Fokas method. We derive the formal representation of the solution for the Broer-Kaup system in terms of the solution of the matrix Riemann-Hilbert problem formulated in the complex plane of the spectral parameter. The jump matrix is uniquely defined by the spectral functions that satisfy a certain relation, called the global relation involving the initial and boundary values. Furthermore, the spectral functions constructed from the initial values and the boundary values are investigated, plus their associated Riemann-Hilbert problems as the inverse problems.


Introduction
The Broer-Kaup system (BK) arises as a model that describes the bi-directional propagation of long waves in shallow water [1,2]. Eq. (1) is a coupled integrable system induced from the Boussinesq equation [1,3]. Being integrable, the BK system has been widely examined such as the Lax pair and the inverse scattering transform [2,4,5], extended Painlevé [6], perturbation theroy based on the inverse scattering transform [7], the Darboux transformation [8], a tri-Hamiltonian structure with an infinite number of conservation laws [9], the soliton solutions given by its trilinear form [10] and peaked solitary wave solutions [11,12]. In the context of the inverse scattering transform, the 2 × 2 matrix Lax pair has been analyzed in [4,5]. It has been shown in [5] that there are two cases of eigenvalues; purely imaginary eigenvalues and complex conjugate pair of eigenvalues. The former case reduces to the elastic interaction of the solitons and the latter leads to the blow up and breather solutions. The main purpose of the paper is to develop the inverse scattering transform for initial boundary value problems (IBVPs) of nonlinear integrable equations. More specifically, we are concerned with the IBVP for the BK system (1) formulated on the half-line for which the initial and boundary values satisfy where the functions v 0 (x) and w 0 (x) are assumed to be sufficiently smooth for x > 0 and to decay fast as x → ∞ . We also assume that the functions f j (t) and g j (t) ( j = 0, 1 ) are sufficiently smooth for t > 0 (and to decay rapidly as t → ∞ if T = ∞ ). This BK system posed on the half-line can be analyzed by using the Fokas method, which is considered as a generalization of the inverse scattering transform for IBVPs of integrable systems. We remark that the Fokas method has been extensively applied for analyzing a large class of boundary value problems such as nonlinear integrable equations [13][14][15][16], linear evolution equations [17,18], linear and nonlinear elliptic partial differential equations [19][20][21] and difference-differential equations [22,23] (see also [24][25][26] for recent applications of the method to coupled integrable IBVPs).
The Fokas method has several efficient advantages for analyzing IBVPs of integrable systems. In particular, we note that (1) the spectral functions satisfy a certain algebraic relation called the global relation involving all initial and boundary values. This global relation allows one not only to establish the existence of the unique solution for IBVPs, but to characterize unknown boundary values that enter in the spectral functions [15]. For example, for the Dirichlet boundary value problem, the Neumann boundary value is unknown. In this case, it is necessary to characterize the unknown boundary value. This characterization can be done by analyzing the global relation, known as the generalized Dirichlet-to-Neumann map [27][28][29] (see also [30,31] for further applications of the global relation). (2) The jump matrix of the Riemann-Hilbert problem has an explicit exponetial form of dependence on x and t. Thus, it is possible to study long time asymptotics of the solution by using the Deift-Zhou method [32] or to study the small dispersion limit by using the Deift-Venakides-Zhou method [33]. Moreover, it also provides an efficient way to characterize the long-time asymptotics for unknown boundary data by using the perturbative approach [34][35][36][37][38]. It should be also remarked that the Fokas method is relatively simple, but effective in solving IBVPs for linear partial differential equations. The Fokas method presents an explicit integral representation of the solution for linear IBVPs, which also leads to efficient new numerical scheme, called a hybrid analytical-numerical method [39][40][41].
In this paper, assuming that the solution for the BK system exists, we show that it can be represented by the solution of the matrix Riemann-Hilbert problem formulated in the complex plane with the jump matrix given by the spectral functions constructed from the initial and boundary values. We also derive the global relation for the BK system that relates the spectral functions.
The outline of the paper is as follows. In Sect. 2, the Lax pair for the BK system is analyzed so as to define the appropriate eigenfunctions and the spectral functions, which are used to formulate the basic Riemann-Hilbert for the BK system posed on the half-line. In Sect. 3, we define the spectral functions from the initial values and the boundary values and we investigate their associated Riemann-Hilbert problems as the inverse problems. Finally, we end with some concluding remarks in Sect. 4.

Lax Pair and Eigenfunctions
The BK system admits the following overdetermined linear systems, called the Lax pair [4,5] where k ∈ ℂ is a spectral parameter, Ψ(x, t, k) is a 2 × 2 matrix-valued eigenfunction, 3 denotes the third Pauli matrix, namely, 3 = diag(1, −1) and with r = v 2 − v x and q = w x + wv for simplicity. Here, we assume that the real-valued functions v and w decay rapidly for all t as x → ∞ . It is convenient to denote the matrix commutator simply by ̂3 ; then ê3 can be easily computed as where A is a 2 × 2 matrix. Note that the Lax pair given in Eqs. (2) is not a standard form for defining special eigenfunctions that are well-controlled for large k. It should require to transform the original Lax pair into the form that U int and V int vanish as x → ∞ and the leading order term for k is off-diagonal. In this respect, we first define ∞ with the closed differential one-form defined by For a simple calculation, we take (x 0 , t 0 ) = (0, 0) . The asymptotics for the eigenfunction Ψ ∞ given in Eq. (5) suggests to introduce a new function (x, t, k) [16] Then we have and Eqs. (3) can be written as where the closed differential one-form is defined by with We also note that Eq. (12) is equivalent to the following modified Lax pair It should be now remarked that U, V → 0 as x → ∞ and the leading order term for k is off-diagonal. As a result, we define the Jost eigenfunction as the simultaneous solution for the both parts of the Lax pair (15) .
and W j is the differential one-form defined by Eq. (13) with j . Note that since the one-form W is exact, the integration in Eq. (16) is path-independent. Hence, we choose three distinct normalization points (cf. Fig. 1) More precisely, we define the Jost eigenfunctions that solve the following integral equations Since v and w are real-valued, the potential functions U and V have the symmetry: the overline denotes the complex conjugation. Thus the eigenfunction has the symmetry (x, t, −k) = (x, t, k) . Note that the off-diagonal components of the matrixvalued eigenfunction involve the explicit exponential terms. Thus, we partition the complex plane into the domains D j ( j = 1, … , 4 ) defined as (see Fig. 2) We denote (1) and (2) the columns of 2 × 2 matrix (x, t, k) = (1) , (2) . We can determine regions, where the eigenfunctions are analytic and bounded as follows Moreover, the asymptotic behavior for the eigenfunction as k → ∞ leads to the reconstruction formula for the solution of the BK system. As shown in Appendix, expanding we find the asymptotic behavior of the eigenfunction : with where the closed differential one-form Δ is given in eq. (10) and Ω is defined by We then have the reconstruction formula for the solution We note that (2)  Hereafter, we write the spectral matrices as Note that s(k) and S(k) enjoy the same symmetry as the eigenfunction, namely, s(−k) = s(k) and S(−k) = S(k) . Similarly, we can determine the regions, where the spectral functions s(k) and S(k) are analytic and bounded: Furthermore, eqs. (27) and (28) imply the relation where the first column is defined for k ∈ D 2 ∪ D 4 and the second column holds for k ∈ D 1 ∪ D 3 , depending on the eigenfunction 2 . Evaluating the above equation at (x, t) = (0, T) , the spectral functions satisfy the following relation, known as the global relation

Riemann-Hilbert Problem
We will formulate the matrix Riemann-Hilbert problem for the BK system. For later reference, we introduce the quantities After some tedious but straightforward algebra, from Eqs. (26a) and (26b), we can define the following Riemann-Hilbert problem where the sectionally meromorphic functions M ± are defined by

M(x, t, k)
Journal of Nonlinear Mathematical Physics (2022) 29:457-476 S(−k) = S(k) . We assume that there are a finite number of simple zeros. More precisely, we assume that (i) a 11 (k) has a finite number of simple zeros in D 4 . There are two types of zeros: k j ≠ −k j and k j = −k j (purely imaginary). We assume that a 11 (k) has 2n 1 simple zeros at k j and −k j with k j ≠ −k j for j = 1, 2, … , n 1 and assume that a 11 (k) has n ′ 1 simple zeros at k = ib j ( b j < 0 ) for j = 1, 2, … , n � 1 . (ii) a 22 (k) has 2n 2 simple zeros in D 1 . We label such zeros k j and −k j for j = 1, 2, … , n 2 . (iii) d 1 (k) has a finite number of simple zeros in D 3 . We assume that d 1 (k) has 2N 1 simple zeros at z j and −z j with z j ≠ −z j for j = 1, 2, … , N 1 and assume that We label such zeros z j and −z j for j = 1, 2, … , N 2 .
We then find the residue conditions where the overdot denotes differentiation with respect to k. Indeed, recalling 1 , where we have suppressed the x, t and k dependence for simplicity, we can compute the residue which is the first equation of eq. (35a). For eq. (35b), using M − = M + J 1 yields a 22 (1) 2 . Thus, we find which is the first equation of Eq. (35b). Similarly, we can derive the second equations of Eqs. (35a) and (35b). We remark that for the purely imaginary zeros, the residue conditions given in Eqs. (35) are valid. , ,

3 3 Spectral Functions
Motivated by the analysis of Sect. 2, we define the spectral functions.

Proposition 3.1 The spectral function s(k) has the following properties:
(i) s (1) is analytic for Im k < 0 and bounded on Im k ≤ 0 except for k = 0 ; s (2) is analytic Im k > 0 and bounded on Im k ≥ 0 except for k = 0.

the map is defined by
where and M (x) (x, k) is the unique solution of the following Riemann-Hilbert problem: • is a meromorphic function for k ∈ ℂ�ℝ , where ℂ ± denote the upper/half plane of the complex plane, respectively. • where ℝ is oriented so that k ∈ ℝ is increasing and the jump matrix J (x) is given by • Assume that the first column of M (x) − has 2n 3 simple zeros, labeled by k j and −k j with k j ≠ −k j ( j = 1, 2, … , n 3 ) and has n ′ 3 simple zeros at k = ib j ( b j < 0 and j = 1, 2, … , n � 3 ). Furthermore, assume that the second column of M (x) + has 2n 4 simple zeros, labeled by k j and −k j with k j ≠ −k j ( j = 1, 2, … , n 4 ) and has n ′ 4 simple zeros at k = ic j ( c j > 0 and j = 1, 2, … , n 2 (x, k) a 22 (k) , Im k > 0,  where M (t) (t, k) is the unique solution of the following Riemann-Hilbert problem: • is a meromorphic function for k ∈ ℂ�(L 1 ∪ L 3 ).

Concluding Remarks
In this work, we studied the IBVP for the BK system posed on the half-line by using the Fokas method. The Lax pair first should be transformed into a standard form so that we can define the well-controlled eigenfunctions for large spectral parameter k. We remark that the solution can be represented in terms of the solution of the matrix Riemann-Hilbert problem with the jump matrix defined by the spectral functions. We derived the global relation for the BK system involving the given initial values and the boundary values. When it comes to being well-posed, all boundary values may not be require to specify as boundary conditions. The fact that the spectral functions satisfy the global relation provides a constraint on the initial and boundary values, which makes it possible to characterize the unknown boundary values. In general, this can be done by solving nonlinear Volterra integral equations for the unknown boundary values [15]. We will analyze the global relation for the BK system in the near future.
From Eq. (60c), it follows that Thus, letting the closed differential one-form Ω as the solutions to Eqs. (61b) and (62) can be found as Finally, we can determine the asymptotics for the eigenfunction as We remark that (2) 12 (x, t) is given by eq. (61a) and from eqs. (58b) and (60b), we find