INITIAL-BOUNDARY VALUE PROBLEMS FOR THE COUPLED MODIFIED KORTEWEG-DE VRIES EQUATION ON THE INTERVAL

. In this paper, we study the initial-boundary value problems of the coupled modiﬁed Korteweg-de Vries equation formulated on the ﬁnite interval with Lax pairs involving 3 × 3 matrices via the Fokas method. We write the solution in terms of the solution of a 3 × 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s ( k ), S ( k ), and S L ( k ), which are determined by the initial values, boundary values at x = 0, and at x = L , respectively. Some of the boundary values are known for a well-posed problem, however, the remai- ning boundary data are unknown. By using the so-called global relation , the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.


1.
Introduction. Several important partial differential equations (PDEs) in mathematics and physics are integrable, which can be rewritten in terms of two linear eigenvalue equations, called a Lax pair [22]. In 1967, Gardner, Greene, Kruskal, Miura [16] solved the initial value problems for Korteweg-de Vries (KdV) equation by using the Inverse Scattering Transform (IST) formalism. From then on, the IST method can be often used to study the initial value problems for integrable evolution equations on the line [1,5,6,16]. In this case, the solution at time t can be recovered by using the solution of an inverse problem. This inverse problem is most conveniently formulated as a Riemann-Hilbert problem (RHP). However, boundary conditions play a very important role in some specific mathematical physics problems. Therefore, it is more meaningful to study the initial-boundary value (IBV) problem than simply studying the pure initial value problem.
In 1997, Fokas [7] (also see [8,9]) introduced a new method, the so-called Fokas method, for analyzing boundary value problems for linear and for integrable nonlinear PDEs. The Fokas method is the extension of the IST formalism from initial value to IBV problems. It is well-known that the IST method is usually used to find the scattering data via analyzing the x-part of the Lax pairs. It is based on the t-part of the Lax pairs to recover the time evolution of the scattering data. However, the Fokas method is based on the simultaneous spectral analysis of the Lax pair, as well as on the analysis of an algebraic relation coupling the initial conditions with all boundary values, which is called by Fokas as the global relation. For a well-posed problem, some of the boundary values are known, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation. The Fokas method can be used to study the boundary value problems of several important integrable equations including 2 × 2 Lax pair equations, such as the KdV equation [10,11], the nonlinear Schrödinger equation [12,13,14,15] and other PDEs [2,3,4,17,21,23,24,25,29,30,34].
In this paper, we will consider IBV problems of the following coupled modified Korteweg-de Vries (cmKdV) equation: on the plane wave background, where p(x, t) and q(x, t) are complex-valued functions of (x, t) ∈ Ω, with Ω denoting the following domain here L > 0 is a fixed constant and T > 0 is a fixed final time. The functions p(x, t) and q(x, t) are slowly varying pulse envelopes. The cmKdV equation (1.1) can be considered as the generalization of the modified KdV equation investigated by many researchers [19]. For instance, based on the bilinear method, Hirota and Iwao obtained 'molecule solution' and multi-soliton solutions for the cmKdV equation [20]. Tsuchida and Wadati solved the initial value problem of the cmKdV equation by using the inverse scattering transformation [35]. Recently, Geng et al derived algebro-geometric solutions of the cmKdV hierarchy associated with a 3 × 3 matrix spectral problem on the basis of the theory of algebraic curves [18]. Moreover, the published information shows that equation (1.1) is among a soliton hierarchy, which is also a special reduction of the multiple-component AKNS systems or more generally, the multiple-component interaction systems [28,39]. Recently, we studied the IBV problems of Eq. (1.1) on the half-line [33]. However, to the best of author's knowledge, the IBV problems of Eq. (1.1) on the interval have not been investigated before.
In [31], the general coupled nonlinear Schrödinger equations was successfully studied on the interval. The main purpose of the present study is try to solve the IBV problems of Eq. (1.1) on the interval. Here, by using the Gelfand-Levitan-Marchenko representation, the global relation is analyzed to find the expressions of the unknown boundary values. Because the order of the derivative is higher than the general coupled nonlinear Schrödinger equations case, it will be more complicate to analyze the global relation of the cmKdV equation (1.1).
Throughout this paper, we will consider the following IBV problems for the cmKdV equation Initial values: p 0 (x) = p 0 (x, t = 0), q 0 (x) = q 0 (x, t = 0), Dirichlet boundary values: g 01 (t) = p(x = 0, t), g 02 (t) = q(x = 0, t), f 01 (t) = p(x = L, t), f 02 (t) = q(x = L, t), First Neumann boundary values: g 11 (t) = p x (x = 0, t), g 12 (t) = q x (x = 0, t), f 11 (t) = p x (x = L, t), f 12 (t) = q x (x = L, t), Second Neumann boundary values: Organization of this paper. Section 2 contains the spectral analysis of the associated Lax pair equations. The main Riemann-Hilbert problem is formulated in Section 3. In Section 4, we study the the asymptotic analysis of the spectral functions. Finally, in Section 5, we derive the Gelfand-Levitan-Marchenko representations of {Φ ij } 3 i,j=1 and {φ ij } 3 i,j=1 , based on which we further analyze the global relation to obtain the expressions of the unknown boundary values.
2. Spectral analysis. System (1.1) is still integrable. Its Lax pair reads and where k is a spectral parameter, and Ψ(x, t, k) is a vector or a matrix function. The compatibility condition of Lax pair (2.1a) and (2.1b) gives the cmKdV equation (1.1).
2.1. The closed one-form. Let p(x, t) and q(x, t) be two sufficiently smooth functions of (x, t) in the interval domain Ω, which decay as x → ∞. By introducing a new eigenfunction µ(x, t, k) one has the new Lax pair equations Supposing that Λ satisfies ΛX = [Λ, X] which acts on a 3 × 3 matrix X, then one can rewrite the equations in (2.8) as the following form where the closed one-form W (x, t, k) can be defined as The spectral function µ j s definition. Based on the Volterra integral equation, four eigenfunctions {µ j } 4 1 of (2.8) can be defined as where W j is determined by (2.10) with µ replaced with µ j , and the contours {γ j } 4 1 are shown in Figure 1. The first, second, and third columns of the matrix equation (2.11) involves the exponentials (2.12) Figure 2. The domains D 1 , D 2 , D 3 and D 4 in the complex k−plane.
The contours {γ j } 4 1 can be given by the following inequalities From (2.13), one can show that the functions {µ j } 4 1 are bounded and analytic for k ∈ C such that k belongs to µ 1 is bounded and analytic for k ∈ (D 2 , D 4 , D 4 ), µ 2 is bounded and analytic for k ∈ (D 1 , D 3 , D 3 ), µ 3 is bounded and analytic for k ∈ (D 4 , D 2 , D 2 ), µ 4 is bounded and analytic for k ∈ (D 3 , D 1 , D 1 ), (2.14) where {D n } 4 1 denote four open, pairwisely disjoint subsets of the Riemann k-sphere shown in Figure 2.
It should notice that the sets {D n } 4 1 admit the following properties where l i (k) and z i (k) are the diagonal entries of matrices ikΛ and 4ik 3 Λ, respectively.
We note that µ 1 (x, t, k) and µ 2 (x, t, k) are entire functions of k. Moreover, in their corresponding regions of boundedness, In fact, for x = 0, the function µ 1 (0, t, k) can be enlarged the domain of boundedness: (D 2 ∪ D 3 , D 1 ∪ D 4 , D 1 ∪ D 4 ), the function µ 2 (0, t, k) can be enlarged the domain of boundedness:  1 of (2.8) can be defined by the following system of integral equations 1 are determined by (2.10) with µ replaced with M n , and the contours γ n ij , n = 1, . . . , 4, i, j = 1, 2, 3 are defined by It is remarked that the distinction between the contours γ 3 and γ 4 is given by It implies that if l m = l n , m may not equal n, one can just choose the subscript is smaller one by considering the rule (2.19).
Based on the definition of the γ n , one can show that (2.20) In order to present the formulation of a Riemann-Hilbert problem, we provide the following proposition ascertains that the M n s defined in this way.
Proof. The bounedness and analyticity properties are established in appendix B in [26]. Substituting the expansion into the Lax pair equations (2.8) and comparing the terms of the same order of k yield the result (2.21).

Remark 1.
Here, two sets of eigenfunctions: {µ j } 3 j=1 and {M n } 4 n=1 are introduced. The two types of eigenfunctions are also used in the unified approach introduced by Fokas [7] for Lax pairs involving 2 × 2 matrices. The µ j is introduced for the spectral analysis, whereas the other set of eigenfunctions can be used to formulate the Riemann-Hilbert problem. Here M n s is the analogues of eigenfunctions. Remark 3. The M n (0, 0, k) is defined by the integral equations (2.17) involving only integration along the boundary {x = 0, 0 < t < T } and the interval {0 < x < L, t = 0}. Similarly, the S n (and also the J m,n ) can be derived from the initial and boundary data alone. Thus, the solution {p(x, t), q(x, t)} from the initial and boundary data can be reconstructed by the jump condition provided by the relation (2.24) for a Riemann-Hilbert problem in the absence of singularities. However, if the M n admit pole singularities at some points {k j }, k j ∈ C, the Riemann-Hilbert problem needs to involve the residue conditions at these points. In order to determine the correct residue conditions (and also for analysing the nonlinearizable boundary conditions in Section 4), four eigenfunctions {µ j (x, t, k)} 4 j=1 should be introduced in addition to the M n .
2.5. The adjugated eigenfunctions. In order to obtain the analyticity and boundedness properties of the minors of the matrices {µ j (x, t, k)} 4 1 . Let's consider the cofactor matrix X A of a 3 × 3 matrix X given by where m ij (X) is the (ij) th minor of X. From (2.8), one can show that the adjugated eigenfunction µ A admits the following Lax pair equations where V T is the transform of a matrix V . Thus, the eigenfunctions {µ j } 4 1 satisfy the following integral equations (2.28)

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For µ A j 4 1 , one can obtain the following analyticity and boundedness properties µ A 1 is bounded and analytic for k ∈ (D 4 , D 2 , D 2 ), µ A 2 is bounded and analytic for k ∈ (D 3 , D 1 , D 1 ), µ A 3 is bounded and analytic for k ∈ (D 2 , D 4 , D 4 ), µ A 4 is bounded and analytic for k ∈ (D 1 , D 3 , D 3 ). (2.29) In fact, for x = 0, µ A 1 (0, t, k) can be enlarged the domain of boundedness: can be enlarged the domain of boundedness: (D 2 ∪ D 3 , D 1 ∪ D 4 , D 1 ∪ D 4 ), and µ A 4 (0, t, k) can be enlarged the domain of boundedness: 2.6. Symmetries. By the following Lemma, we show that the eigenfunctions µ j (x, t, k) admit an important symmetry.
Lemma 2.1. The eigenfunction Ψ(x, t, k) of the Lax pair equations (2.1a) and (2.1b) admits the following symmetry where the superscript T denotes a matrix transpose.
Proof. From the following equations and 2.7. The J m,n s computation. The 3 × 3-matrix value spectral functions s(k), S(k), and S L (k) can be defined by Thus, From the properties of µ j and µ A j , one can derive that {s(k), S(k), S L (k)} and s A (k), S A (k), S A L (k) admit the following boundedness properties (2.36) We also notice that Proposition 2. The functions {S n } 4 1 can be expressed in terms of the entries of s(k), S(k), and S L (k) as follows (2.39) Proof. In order to derive the expressions of {S n } 4 1 , we can introduce three new functions R n (k), T n (k) and Q n (k) given by

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Then, one can derive the following relations (2.42) From system (2.42), one can obtain a matrix factorization problem. For the given {s(k), S(k), S L (k)}, they can be solved for the {R n , S n , T n , Q n }. From the definitions of {R n , S n , T n , Q n } and the integral equations (2.17), one has In what follows, we will show that the spectral functions S(k), S L (k) and s(k) admit a very important relationship. They are dependent with each other. From the system (2.34a)-(2.34c), one can show that (2.44) By considering the point (0, T ), we have the following global relationship In fact, for each t ∈ (0, T ), suppose that R(x, t, k) is the solution of the x-part of the Lax pair of (1.1), such that R(L, t, k) = I, i.e. R is the unique solution of the Volterra integral equation (2.46) It implies that R can be connected with µ 3 by From the equations (2.34a) and (2.47), we have (2.48) 2.9. The residue conditions. From (2.37), one can show that M only has singularities at the points where the {S n s} 4 1 have singularities since µ 2 is an entire function. We introduce the symbols {k j } N 1 to denote the possible zeros and assume {k j } N 1 satisfy the following assumption. 11 (k) have zeros on the boundaries of the {D n s} 4 1 . The residue conditions at these zeros {k j } N 1 are determined by the following Proposition.
1 are the eigenfunctions defined by (2.17) and the zeros {k j } N 1 admit the above assumption, we have the following residue conditions In what follows, we will derive the results (2.49a), (2.49b), (2.49c) and (2.49e), the other conditions follow similar arguments. From equation (2.37), we have the following relations By considering the expression for S 1 determined by (2.38a), the three columns of M 1 (2.50a) can be derived as Similarly, one can also obtain the three columns of M 2 (2.50b) and the three columns of M 4 (2.50d) (2.54c) Let k j ∈ D 1 be a simple zero of m 11 . Solving (2.51b) and (2.51c) for [µ 2 ] 1 and [µ 2 ] 2 , and substituting the result of [µ 2 ] 1 into (2.51a), one can obtain Taking the residue of (2.55) at k j , it implies that (2.49a) holds in the case when k j ∈ D 1 . In order to derive (2.49b), we solve (2.52b) and (2.52c) for [µ 2 ] 2 and [µ 2 ] 3 . Substituting the results into (2.52a), one can obtain (2.57) Taking the residue of Equation (2.57) at k j , one can show that the condition (2.49c) holds in the case when k j ∈ D 3 . By similar arguments, solving (2.54a) for [µ 2 ] 1 , and substituting the result into (2.54b), one can obtain Taking the residue of Equation (2.58) at k j , one can show that the condition (2.49e) hold in the case when k j ∈ D 4 .
3. The Riemann-Hilbert problem. In Section 2, we introduce the sectionally analytic function M (x, t, k). It admits a Riemann-Hilbert problem, which can be formulated in terms of the initial and boundary values of the functions p(x, t) and q(x, t). By solving the Riemann-Hilbert problem, one can recover the solution of (1.1) for all values of the independent variables x, t.
Theorem 3.1. Let p(x, t) and q(x, t) be a pair of solutions of (1.1) in the interval domain Ω. Then p(x, t) and q(x, t) can be reformulated by the initial values {p 0 (x), q 0 (x)} and boundary values {g 01 (t), g 02 (t), g 11 (t), g 12 (t), g 21 (t),  By using the initial and boundary data, the jump matrices J m,n (x, t, k) can be defined in terms of the spectral functions s(k) and S(k), S L (k) by the system (2.34a)-(2.34c).
Supposing that the possible zeros {k j } N 1 of the functions m 11 (A)(k), (S T s A ) 11 (k), A 11 (k) and (s T S A ) 11 (k) satisfy Assumption 2.2 in {D n } 4 1 . Then the solution {p(x, t), q(x, t)} of (1.1) with initial and boundary values problem (3.1) is given by
Substituting equation (4.18) into system (4.15) and using the initial conditions we can determine these coefficients α j (t) and β j (t) by (4.20) By the same way to the derivation of (Φ 11 Φ 21 Φ 31 ) T , under the following initial conditions we can determine the asymptotic formulas of ( from the system (4.16).
Substituting the equation (4.23) into the system (4.17) and using the initial conditions we can determine these coefficients α j (t) and β j (t) by From the first column of Equation (4.14), one can obtain (4.26) From the second column of Equation (4.14), one can obtain

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From the third column of Equation (4.14), one can obtain (4.28) Then, substituting these formulas into Equation (4.13a) and noticing that we assume that the initial value and boundary value are compatible at x = 0 and x = L, respectively, one can derive the asymptotic behavior (4.12a) of c j1 (t, k) as k → ∞. By the same way, one can show that the formula (4.12b) is also hold.

5.
Nonlinearizable boundary conditions. It is very difficult to study IBV problems since some of the boundary values are unknown for a well-posed problem. All boundary dates are very important to define of S(k), S L (k), and hence to formulate the Riemann-Hilbert problem. In what follows, the effective characterizations of spectral functions S(k), S L (k) can be derived in our main result, theo- by For the eigenfunctions Φ ij and φ ij , we will first derive their Gelfand-Levitan-Marchenko (GLM) representations, then analyze the solution of the global relation. By considering the analysis of the global relation, the unknown boundary values can be expressed in terms of the known ones via the GLM representations.

The GLM representation.
Theorem 5.1. The eigenfunctions Φ ij and φ ij have the following GLM representations 2 ΛN (t, s) + k 2 N (t, s) e −4ik 3 (t−sΛ) ij ds, where the symbol δ ij given by 4) and the 3 × 3 matrices L(t, s), M (t, s) and N (t, s) admit the following initial con- and the ODE systems where the matrices A , B and C are determined by Proof. We just show that the GLM representations of the eigenfunctions Φ ij are hold. Following the same way, one can also derive the GLM representations of the eigenfunctions φ ij . Let's consider the following expression where L, M and N are 3 × 3 matrices. Substituting the above expression (5.7) into the t-part of the Lax pair (2.1b), one has the following systems 2 N s (t, s)Λ.

5.2.
The analysis of the global relation. In order to avoid routine technical complications, we will analyze the global relation in the case s(k) = I with respect to the zero initial conditions p 0 (x) = p(x, 0) = 0 and q 0 (x) = q(x, 0) = 0. For this case, the global relation (2.45) admits the following form From the Lemma 4.1, we have the following properties In what follows, we will consider the solution of the global relation. The expressions for {g 11 (t), g 21 (t), f 21 (t)} and {g 12 (t), g 22 (t), f 22 (t)} can be written in terms of the given boundary conditions {g 01 (t), g 02 (t), f 01 (t), f 02 (t), f 11 (t), f 12 (t)}, which are provided by Theorem 5.2 below in terms of the GLM representations.
In order to simplify our formulas, some notations are introduced. Let F (t, k) be a scalar function, then we define the functions F (t, k) and F (t, k) as follows.
• Define F (t, k) and F (t,k) as follows (5.12) • Define F (t, k) and F (t,k) as follows where G 12 (t, k) and G 22 (t, k) are given by Here E −1 (k) j , j = 1, 2, 3 imply the j-th row of the inverse matrix of E(k), which is of the form The symbol ∂D 0 is the boundary contour of D deformed to pass the zeros of the det E(k), where D = k|π < arg k < 4π 3 . Proof. By the substitution of the GLM representations of Φ ij and φ ij into c 21 (t, k) (4.13a) and c 31 (t, k) (4.13b), we have the following system where {G 1j } 2 j=1 and {G 2j } 2 j=1 are given by Replacing k by αk and α 2 k, respectively, in (5.17a) and (5.17b) for k ∈ D, one has six equations. One can further write these equations in the following vectors where with l = 1, 2. From above analysis, one can show that det E(k) → α − 1 = 0, for |k| → ∞ and k ∈ D.
By multiplying the following factor   and by integrating along the contour ∂D 0 , one can show that the terms including H cj vanishes from (5.11), by Jordan's lemma. For analyzing the other terms, the following identities (see [2] or [25]) should be considered where F(t, s) is an arbitrary function such that the integral is well defined, and ∂D 0 where m = 3, 4 and γ(τ ) is a smooth function for 0 < τ < t. Then by using Jordan's lemma and by considering the integration by parts, one can show that one can arrive at the limit as t → t in the right-hand side of (5.23b). By considering the integral term including H 1j with (5.23b), we find where the functions { G 1j (t, t , k)} 2 j=1 are given by , { N 2,j+1 (t, t)} 2 j=1 and { N 3,j+1 (t, t)} 2 j=1 appearing in (5.27a)-(5.27c) in terms of the spectral functions Φ ij (t, k) and φ ij (t, k).