Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 136 - 165

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

Qiaozhen Zhu, Jian Xu, Engui Fan*
School of Mathematical Sciences, Fudan University, NO.220 Handan Road, Shanghai, 200433, People’s Republic of China,
College of Science, University of Shanghai for Science and Technology, NO.334 Jungong Road, Shanghai 200093, People’s Republic of China,
School of Mathematical Sciences, Fudan University, NO.220 Handan Road, Shanghai, 200433, People’s Republic of China,
*Corresponding author.
Corresponding Author
Engui Fan
Received 12 October 2017, Accepted 28 October 2017, Available Online 6 January 2021.
10.1080/14029251.2018.1440747How to use a DOI?
Two-component Gerdjikov-Ivanov equation; initial-boundary value problem; Fokas unified method; Riemann-Hilbert problem

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and SL(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.

© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (

1. Introduction

The Gerdjikov-Ivanov (GI) equation takes in the form [10]

In these years, there has been much work on the GI equation, including Hamiltonian structures [2], Darboux transformation [3], rouge wave and breather soliton [19], algebro-geometric solutions [11], envelope bright and dark soliton solution [17]. Recently, Zhang, Cheng and He obtained the N-soliton solutions with Riemann-Hilbert method about the two-component (2-GI) equation [22]

In 1997, Fokas announced the unified transform for the analysis of initial boundary value (IBV) problems for linear and nonlinear integrable PDEs [4]. The Fokas method was usually used to analyze the IBV problem for integrable PDEs with 2 × 2 Lax pair on the half-line and the finite interval, such as nonlinear Schröding equation [5, 6], sine-Gordon equation [7,15], KdV equation [8], mKdV equation [1, 9], derivative nonlinear Schröding equation [12]. In 2012, Lenells extended this method to the IBV problem of integrable systems with 3 × 3 Lax pair on the half-line [13]. After that, several important integrable equations with 3 × 3 Lax pair have been investigated, including Degasperis-Procesi [14], Sasa-satuma [18]. However, there has been still less work on the IBV problems on the finite interval of integrable equations with 3 × 3 Lax pair except to the two-component NLS [20], general coupled NLS [16] and the integrable spin-1 Gross-Pitaevskii [21] equations.

In this paper, we apply Fokas method to consider 2-GI equation with the following initial boundary value data:

Initial value:q1(x,t=0)=q10(x),q2(x,t=0)=q20(x),Dirichlet boundary value:q1(x=0,t)=g01(t),q1(x=L,t)=f01(t),q2(x=0,t)=g02(t),q2(x=L,t)=f02(t),Neumann boundary value:q1x(x=0,t)=g11(t),q1x(x=L,t)=f11(t),q2x(x=0,t)=g12(t),q2x(x=L,t)=f12(t),(1.3)
where q1(x, t) and q2(x, t) are complex-valued functions of (x, t) ∈ Ω, and Ω denotes the finite interval domain
here L > 0 is a positive fixed constant and T > 0 being a fixed final time.

Comparing with two-component NLS equation [20], the IBV problem of the 2-GI equation (1.2) also presents some distinctive features in the use of Fokas method: (i) The order of spectral variable k in the Lax pair (2.1) is higher than that of 2-NLS equation. In order to make the results on the interval reduce to the ones on the half-line, we should first introduce transformation ψ(x,t,k)=k12Λϕ(x,t,k)k12Λ so that the Lax pairs are even functions of k. (ii) The 2-GI equation admits a generalized Wadati-Konno-Ichikawa (WKI) type Lax pair, which admits a gauge transformation to AKNS-type Lax pair, but this gauge transformation can not be used to analyze the IBV problem by mapping it into 2-NLS equation. We need to introduce a matrix-value function G(x, t) to transform the WKI-type Lax pair into AKNS-type Lax pair.

Organization of this paper is as follows. In the following section 2, we perform the spectral analysis of the associated Lax pair for the 2-GI equation (1.2). In the section 3, we give the corresponding matrix RH problem associated with the IBV problem of 2-GI equation. In section 4, we get the map between the Dirichlet and the Neumann boundary problem through analysising the global relation. Especially, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity.

2. Spectral analysis

2.1. Lax pair

The 2-GI equation admits a 3 × 3 Lax pair [22]

where ψ(x, t, k) is a 3 × 3 – matrix valued eigenfunction, k ∈ ℂ is the spectral parameter, and U1(x, t), U2(x, t) are 3 × 3 – matrix valued functions given by

There are both odd power and even power of k in the Lax pair (2.1), to make (2.1) are even functions of k for analyzing the large L limit, we introduce a transformation

and get an equivalent Lax pair

Let λ = k2, Lax pair (2.5) becomes

where U˜1,U˜2 are given by (2.6) with k2 replaced with λ.

2.2. The closed one-form

Defining a 3 × 3 matrix-value function

and making a transformation
then we get a new Lax pair for μ(x, t, λ)

Letting A^ denotes the operators which acts on a 3 × 3 matrix X by A^X=[A,X], then the equations (2.11) can be rewritten in a differential form

where the closed one-form W(x, t, k) is defined by

2.3. The eigenfunctions μj’s

We define four eigenfunctions {μj}14 of (2.11) by the Volterra integral equations

where Wj is given by (2.14) with μ replaced by μj, and the contours {γj}14 can be given by the following inequalities ( see Figure 1):
and the matrix equation (2.15) involves the exponentials
from which, we find that the functions {μj}14 are bounded and analytic for λ ∈ ℂ such that λ belongs to
where {Dn}14 denote four open, pairwisely disjoint subsets of the complex λ – plane showed in Figure 2.

And the sets {Dn}14 admit the following properties:

where li(λ) and zi(λ) are the diagonal entries of matrices Λ and 22Λ, respectively.

Fig. 1.

The four contours γ1, γ2, γ3 and γ4 in the (x, t) – domain.

Fig. 2.

The sets Dn, n = 1,…,4, which decompose the complex λ-plane.

2.4. The spectral functions Mn’s

For each n = 1, …, 4, a solution Mn(x, t, λ) of (2.11) can be defined by the following system of integral equations:

where Wn is given by (2.14) with μ replaced with Mn, and the contours γijn,n=1,,4,i,j=1,2,3 are defined by
γijn={γ1 if Reli(λ)<Relj(λ) and Rezi(λ)Rezj(λ),γ2 if Reli(λ)<Relj(λ) and Rezi(λ)<Rezj(λ),γ3 if Reli(λ)Relj(λ) and Rezi(λ)Rezj(λ),γ4 if Reli(λ)Relj(λ) and Rezi(λ)Rezj(λ), for λDn.(2.20)
Here, we make a distinction between the contours γ3 and γ4 as follows,
γijn={γ3, if Π1i<j3(Reli(λ)Relj(λ))(Rezi(λ)Rezj(λ))<0,γ4, if Π1i<j3(Reli(λ)Relj(λ))(Rezi(λ)Rezj(λ))>0.(2.21)
The rule chosen in the produce is if lm = ln, m may not equals n, we just choose the subscript is smaller one.

According to the definition of the γn, one find that


The following proposition ascertains that the Mn’s defined in this way have the properties required for the formulation of a Riemann-Hilbert problem.

Proposition 2.1.

For each n = 1,…,4, the function Mn(x, t, λ) is well-defined by equation (2.19) for λD¯n and (x,t)Ω. Moreover, Mn admits a bounded and contious extension to D¯n and


Proof. Analogous to the proof provided in [13] □

Remark 2.1.

Of course, for any fixed point (x, t), Mn is bounded and analytic as a function of kDn away from a possible discrete set of singularities {kj} at which the Fredholm determinant vanishes. The bounedness and analyticity properties are established in appendix B in [13]. □

2.5. The jump matrices

The spectral functions {Sn(λ)}14 can be defined by

Let M denote the sectionally analytic function on the Riemann λ – plane which equals Mn for λDn. Then M satisfies the jump conditions
where the jump matrices Jm, n(x, t, λ) are given by

2.6. The adjugated eigenfunctions

As the expressions of Sn(λ) will involve the adjugate matrix of {s(λ), S(λ), SL(λ)} defined in the next subsection. We will also need the analyticity and boundedness of the the matrices {μj(x,t,λ)}14. We recall that the adjugate matrix XA of a 3 × 3 matrix X is defined by

where mi j(X) denote the (i j) th minor of X.

It follows from (2.11) that the adjugated eigenfunction μA satisfies the Lax pair

where VT denotes the transform of a matrix V. Thus, the eigenfunctions {μjA}14 are solutions of the integral equations
Then we can get the following analyticity and boundedness properties:

2.7. Symmetries

We will show that the eigenfunctions μj(x, t, k) satisfy an important symmetry.

Proposition 2.2.

The eigenfunction ψ(x, t, k) of the Lax pair (2.1) satisfies the following symmetry:

here the superscript T denotes a matrix transpose.

Proof. The matrices U(x, t, k) and V(x, t, k) in the Lax pair (2.1) written in the form

satisfy the following symmetry relations
In turn, relations (2.31) and (2.32) imply

Remark 2.2.

From proposition 2.3, one can show that the eigenfunctions μj(x, t, λ) of Lax pair equations (2.11) satisfy the same symmetry.

2.8. The Jm, n’s computation

Let us define the 3 × 3 – matrix value spectral functions s(λ), S(λ) and SL(λ) by

And we can deduce from the properties of μj and μjA that {s(λ), S(λ), SL(λ)} and {sA(λ),SA(λ),SLA(λ)} have the following boundedness properties:

Proposition 2.3.

The Sn can be expressed in terms of the entries of s(λ), S(λ) and SL(λ) as follows:

where A=(Aij)i,j=13 is a 3 × 3 matrix, which is defined as A=s(λ)eiλLΛ^SL(λ). And the functions

Proof. Firstly, we define Rn(λ), Tn(λ) and Qn(λ) as follows:

Then, we have the following relations:

The relations (2.40) imply that

These equations constitute a matrix factorization problem which, given {s(λ), S(λ), SL(λ)} can be solved for the {Rn, Sn, Tn, Qn}. Indeed, the integral equations (2.19) together with the definitions of {Rn, Sn, Tn, Qn} imply that
{(Rn(λ))ij=0 if γijn=γ1,(Sn(λ))ij=0 if γijn=γ2,(Tn(λ))ij=δij if γijn=γ3,(Qn(λ))ij=δij if γijn=γ4.(2.42)
It follows that (2.41) are 27 scalar equations for 27 unknowns. By computing the explicit solution of this algebraic system, we arrive at (2.38). □

Remark 2.3.

Due to our symmetry, see Lemma 2.30, the representation of the functions Sn(λ) can be become simple. It leads to much more simple to compute the jump matrices Jm, n(x, t, λ).

2.9. The residue conditions

Since μ2 is an entire function, it follows from (2.37) that M can only have sigularities at the points where the Sn s have singularities. We denote the possible zeros by {λj}1N and assume they satisfy the following assumption. We assume that

  • m11()(λ) has n0 possible simple zeros in D1 denoted by {λj}1n0;

  • (ST sA)11(k) has n1n0 possible simple zeros in D2 denoted by {λj}n0+1n1;

  • (sT SA)11(k) has n2n1 possible simple zeros in D3 denoted by {λj}n1+1n2;

  • 11(k) has Nn2 possible simple zeros in D4 denoted by {λj}n2+1N;

and that none of these zeros coincide. Moreover, we assume that none of these functions have zeros on the boundaries of the Dns. We determine the residue conditions at these zeros in the following:

Proposition 2.4.

Let {Mn}14 be the eigenfunctions defined by (2.19) and assume that the set {λj}1N of singularities are as the above assumption. Then the following residue conditions hold:

Resλ=λj[M]1=S21(λj)s33(λj)S31(λj)s23(λj)(STsA)11(λj)m11(λj)e2θ(λj)[M(λj)]2+S31(λj)s22(λj)S21(λj)s32(λj)(STsA)11(λj)m11(λj)e2θ(λj)[M(λj)]3 n0+1jn1,λjD2,(2.43b)
Resλ=λj[M]2=m33(s)(λj)m21(S)(λj)m23(s)(λj)m31(S)(λj)(sTSA)11(λj)s11(λj)e2θ(λj)[M(λj)]1 n1+1jn2,λjD3,(2.43c)
Resλ=λj[M]3=m32(s)(λj)m21(S)(λj)m22(s)(λj)m31(S)(λj)(sTSA)11(λj)s11(λj)e2θ(λj)[M(λj)]1 n1+1jn2,λjD3.(2.43d)
where f˙=dfdλ, and θ is defined by

Proof. We will prove (2.43a),(2.43c), the other conditions follow by similar arguments. Equation (2.37) implies the relation


In view of the expressions for S1 and S3 given in (2.38), the three columns of (2.45a) read:

while the three columns of (2.45b) read:

We first suppose that λjD1 is a simple zero of m11()(λ). Solving (2.46b) and (2.46c) for [μ2]1,[μ2]3 and substituting the result in to (2.46a), we find

Taking the residue of this equation at λj, we find the condition (2.43a) in the case when λjD1.

In order to prove (2.43c), we solve (2.47a) for [μ2]1, then substituting the result into (2.47b) and (2.47c), we find

Taking the residue of this equation at λj, we find the condition (2.43c) in the case when λjD3. □

2.10. The global relation

The spectral functions S(λ), SL(λ) and s(λ) are not independent but satisfy an important relation. Indeed, it follows from (2.35) that

Since μ1(0, T, λ) = !!!x1D540;, evaluation at (0, T) yields the following global relation:
where c(T, λ) = μ4(0, T, λ)

3. The Riemann-Hilbert problem

The sectionally analytic function M(x, t, λ) defined in section 2 satisfies a Riemann-Hilbert problem which can be formulated in terms of the initial and boundary values of q1(x, t) and q2(x, t). By solving this Riemann-Hilbert problem, the solution of (1.2) can be recovered for all values of x, t.

Theorem 3.1.

Suppose that q1(x, t) and q2(x, t) are a pair of solutions of (1.2) in the interval domain Ω. Then q1(x, t) and q2(x, t) can be reconstructed from the initial value {q10(x), q20(x)} and boundary values {g01(t), g02(t), g11(t), g12(t)},{f01(t), f02(t), f11(t), f12(t)} defined as follows,


Use the initial and boundary data to define the jump matrices Jm, n(x, t, λ) in terms of the spectral functions s(λ) and S(λ), SL(λ) by equation (2.35).

Assume that the possible zeros {λj}1N of the functions m11()(λ),(ST sA)11(λ),(sT SA)11(λ) and11(λ) are as the assumption in subsection 2.8.

Then the solution {q1(x, t), q2(x, t)} is given by

where M(x, t, λ) satisfies the following 3 × 3 matrix Riemann-Hilbert problem:
  • M is sectionally meromorphic on the Riemann λsphere with jumps across the contours D¯nD¯m,n,m=1,,4, see Figure 2.

  • Across the contours D¯nD¯m, M satisfies the jump condition


  • M(x,t,λ)=I+O(1λ), λ.

  • The residue condition of M is showed in Proposition 2.4.

Proof. It only remains to prove (3.2) and this equation follows from the large λ asymptotics of the eigenfunctions. □

4. Non-linearizable Boundary Conditions

A key difficulty of initial-boundary value problems is that some of the boundary values are unkown for a well-posed problem. While we need all boundary values to define the spectral functions S(λ) and SL(λ), and hence for the formulation of the Riemann-Hilbert problem. Our main result, Theorem 4.3, expresses the unknown boundary data in terms of the prescribed boundary data and the initial data in terms of the solution of a system of nonlinear integral equations.

4.1. Asymptotics

An analysis of (2.11) shows that the eigenfunctions {μj}14 have the following asymptotics as λ → ∞ :


Remark 4.1.

The explicit formulas of μ11(2) and μij(2),i,j=2,3 are not presented in the following analysis, we do not write down the asymptotic expressions of these functions.

Next, we define functions {Φij(t,λ)}i,j=13 and ϕij(t,λ)i,j=13 by:

From the asymptotic of μj(x, t, λ) in (4.1) we have
Recalling the definition of the boundary data at x = 0, we have
In particular, we find the following expressions for the boundary values at x = 0:
Similarly, we have the asymptotic formulas for μ3(L,t,λ)=ϕij(t,λ)i,j=13,
Recalling that the definition of the boundary data at x = L, we have
In particular, we find the following expressions for the boundary values at x = L:
From the global relation (2.50) and replacing T by t, we find

Lemma 4.1.

We assuming that the initial value and boundary value are compatible at x = 0 and x = L, then in the vanishing initial value case, the global relation (A.3) implies that the large λ behavior of cj 1(t, λ), j = 2,3 satisfy


Proof. The global relation shows that under the assumption of vanishing initial value


Recalling the equation

From the first column of the equation (4.15) we get
From the second column of the equation (4.15) we get
From the third column of the equation (4.15) we get
where the coefficients αj(t) and βj(t), j = 0,1,2, ⋯, are independent of k and are 3 × 1 matrix functions.

To determine these coefficients, we substitute the above equation into equation (4.16a) and use the initial conditions

Then we get

Similarly, suppose

where the coefficients αj(t) and βj(t), j = 0,1,2, ⋯, are independent of k and are 3 × 1 matrix functions.

To determine these coefficients,we substitute the above equation into equation (4.16b) and use the initial conditions

Then we get

Similar to the derivation of Φi 2, i = 1,2,3, from (4.16 c) we can get the asymptotic formulas of Φi 3, i = 1,2,3


Similar to (4.16), we also know that {ϕij}i,j=13 satisfy the similar partial derivative equations. Substituting these formulas into the equation (4.14a) and noticing that we assume that the initial value and boundary value are compatible at x = 0 and x = L, we get the asymptotic behavior (4.13a) of cj 1(t, λ) as λ → ∞. Similar to prove the formula (4.13b). □

4.2. The Dirichlet and Neumann problems

In what follows, we can derive the effective characterizations of spectral function S(λ), SL(λ) for the Dirichlet ({g01(t),g02(t)} and {f01(t),f02(t)} prescribed), the Neumann({g11(t),g12(t)} and {f11(t),f12(t)} prescribed) problems.

Define the following new functions as

Denoting D30 as the boundary contour which is not included the zeros of Δ(λ).

Theorem 4.1.

Let T < ∞. Let q0(x) = (q10(x), q20(x)), 0 ≤ xL, be two initial functions.

For the Dirichlet problem it is assumed that the function {g01(t),g02(t)},0t<T, has sufficient smoothness and is compatible with {q10(x), q20(x). at x = t = 0, that is

the function.{f01(t), f) 02(t)}, 0 ≤ t<T, has sufficient smoothness and is compatible with q10(x), q20(x) at x = L, that is

For the Neumann problem it is assumed that the functions {g11(t), g 12(t)}, 0 ≤ t <T, has sufficient smoothness and is compatible with q0(x) at x = t = 0. The functions {f11(t), f12(t)}, 0 ≤ t<T, has sufficient smoothness and is compatible with q0(x) at x = L.

Then the spectral function S(λ), SL(λ) is given by

and the complex-value functions {Φl3(t,λ)}l=13 satisfy the following system of integral equations:
and {Φl1(t,λ)}l=13,{Φl2(t,λ)}l=13 satisfy the following system of integral equations:
  1. (i)

    For the Dirichlet problem, the unknown Neumann boundary value {g11(t), g12(t). and {f11(t), f12(t). are given by

    where the conjugate of a function h denotes h¯=h(λ¯)¯.

  2. (ii)

    For the Neumann problem, the unknown boundary values {g01(t), g02(t)} and {f01(t), f02(t)} are given by



The representations (4.24) follow from the relation S(k)=e2iλ2TΛ^μ21(0,T,k). And the system (4.26) is the direct result of the Volteral integral equations of μ2(0, t, k).

  1. (i)

    In order to derive (4.29 a) we note that equation (4.8 ~b) expresses g11 in terms of Φ12(2) and Φ22(1),Φ32(1). Furthermore, equation (A.4) and Cauchy theorem imply

    where I(t) is defined by
    The last step involves using the global relation (4.14a) to compute I(t), that is
    Using the asymptotic (4.13a) and Cauchy theorem to compute the first term on the righthand side of equation (A.13), we find

    Equations (4.34) and (A.14) imply

    Equations (4.33) and(4.37) together with (4.8b) yield (4.29a). Similarly, we can prove (4.29b).

    The expressions (4.30a) for f11(t) can be derived in a similar way. Indeed, we note that equation (4.11b) expresses f11 in terms of ϕ12(2) and ϕ22(1),ϕ32(1). These three equations satisfy the analog of equations (4.33) and (4.34). In particular, ϕ21(2) satisfies


    Then using the global relation to compute J(t), that is


    The equation (4.38) and (4.39) together with the asymptotics of c12(t, λ) yield (4.30 a) The proof of (4.30b) is similar.

  2. (ii)

    In order to derive the representations (4.31a) relevant for the Neumann problem, we note that equation (4.8 a) expresses g01 and g02 in terms of Φ12(1) and Φ13(1), respectively. Furthermore, equation (A.4) and Cauchy's theorem imply

    using the global relation and the asymptotic formulas of c21(t, λ), we have
    Equations (4.8 a),(4.40) and (4.41) yields (4.31 a). The proof of the other formulas is similar. □

4.3. Effective characterizations

Substituting into the system (4.26),(4.27) and (4.28) the expressions

where ε > 0 is a small parameter, we find that the terms of O(1) give
Moreover, the terms of O(ε) give
the terms of O(ε2) give

Similarly, we will have the analogue formulas for {ϕij,l}i,j=13,l=0,1,2 expressed in terms of the boundary data at x = L, that is {fij(l)}i=0,1j=1,2,l=1,2.

On the other hand, expanding (4.29),(4.30) and assuming for simplicity that m11()(λ) has no zeros, we find

we also find that

The Dirichlet problem can now be solved perturbatively as follows: assuming for simplicity that m11()(λ) has no zeros and given g01(1),g02(1) and f¯11(1),f¯12(1), we can use equation (4.47) to determine Φ1j,1,ϕj1,1,j=2,3. We can then compute g11(1),g12(1) from (4.46a),(4.46b) and then Φ1 j, 1, j = 2,3 from (4.44) and the analogue results for ϕj 1,1, j = 2,3. In the same way we can determine Φ1 j, 2, j = 2,3 from (4.45) and the analogue results for ϕj 1,2, j = 2,3, then compute g11(2),g12(2) and f11(2),f12(2)

These arguments can be extended to the higher order and also can be extended to the systems (4.26), (4.27) and (4.28) thus yields a constructive scheme for computing S(k) to all orders. The construction of SL(λ) is similar.

Similarly, these arguments also can be used to the Neumann problem. That is to say, in all cases, the system can be solved perturbatively to all orders.

4.4. The large Llimit

In the limit L → ∞, the representations for g11(t), g12(t) and g01(t), g02(t) of theorem 4.3 reduce to the corresponding representations on the half-line. Indeed, as L → ∞,

f010,f020,f110,f120,ϕijδij,ΣΔ1 as λ in D3
Thus, the L → ∞ limits of the representations (4.29a),(4.29b) and (4.31a),(4.31b) are
respectively. And these formulas coincide with the corresponding half-line formulas, see (A.9), (A.10).

Appendix A. Some formulas on the half-line

For the convenience of reader, we show the half-line formulas of g11(t), g12(t) and g01(t), g02(t) on the λ-plane.

From the global relation (2.50) and replacing T by t, we find

We partition matrix as following,
where Φ2 × 2 denotes a 2 × 2 matrix, Φ1 j denotes a 1 × 2 vector, Φj 1 denotes a 2 × 1 vector. Then, we can write the second column of the global relation, undering the matrix partitioned as (A.2), as
The functions c1 j(t, λ), c2 × 2(t, λ) are analytic and bounded in D1D2 away from the possible zeros of m11(λ) and of order O(1λ) as k → ∞.

From the asymptotic of μj(x, t, λ) in (4.1) we have

where Q = (q1, q2), Δ11 is defined by first identities of (4.3 a), η11 is defined by (4.3b), Δ and η are 2 × 2 matrices defined as following,
Also, we have
here g0(t) and g1(t) are vector boundary functions defined by the boundary data of (1.3) as g0(t) = (g01(t), g02(t)) and g1(t) = (g11(t), g12(t)).

In particular, we find the following expressions for the boudary values:

We will also need the asymptotic of c1 j(t, λ),

Lemma A.1.

The global relation (A.3) implies that the large λ behavior of c1 j(t, λ), c2 × 2(t, λ) satisfies


Proof. Analogous to the proof provided in Lemma 4.2. □

We can now derive the maps between Dirichlet boundary condition and the Neumann boundary condition as follows:

  1. (i)

    For the Dirichlet problem, the unknown Neumann boundary value g1(t) is given by


  2. (ii)

    For the Neumann problem, the unknown boundary values g0(t) is given by



  1. (i)

    In order to derive (A.9) we note that equation (A.7b) expresses g1 in terms of Φ2×2(1) and Φ1j(2). Furthermore, equation (A.6) and Cauchy theorem imply

    where I(t) is defined by
    The last step involves using the global relation to compute I(t)
    Using the asymptotic (A.8) and Cauchy theorem to compute the first term on the right-hand side of equation (A.13), we find
    Equations (A.12) and (A.14) imply
    This equation together with (A.7b) and (A.11) yields (A.9).

  2. (ii)

    In order to derive the representations (A.10) relevant for the Neumann problem, we note that equation (A.7a) expresses g0 in terms of Φ1j(1). Furthermore, equation (A.6a) and Cauchy’s theorem imply

    and using the global relation, we have
    Equations (A.7a), (A.16) and (A.17) yields (A.10). □


Fan was support by grants from the National Science Foundation of China under Project No. 11671095. Xu was supported by National Science Foundation of China under project No. 11501365, Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No.15YF1408100 and the Hujiang Foundation of China (B 14005 ).


[9]Fokas Fokas and Its Its, The mKdV equation on the half-line, J. Inst. Math. Jussieu, Vol. 3, 2004, pp. 139-164.
[10]Gerdjikov Gerdjikov and Ivanov Ivanov, A quadratic pencil of general type and nonlinear evolution equations, Bulg. J. Phys., Vol. 10, 1983, pp. 130-143.
[19]Xu Xu and He He, The rouge wave and breather solution of the Gerdjikov-Ivanov equation, J. Math. Phys., Vol. 53, 2012, pp. 1-17. 063507
[21]Yan Yan, An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4 × 4 Lax pair on a finite interval, Chaos, Vol. 27, 2017, pp. 1-20. 053117
Journal of Nonlinear Mathematical Physics
25 - 1
136 - 165
Publication Date
ISSN (Online)
ISSN (Print)
10.1080/14029251.2018.1440747How to use a DOI?
© 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (

Cite this article

AU  - Qiaozhen Zhu
AU  - Jian Xu
AU  - Engui Fan
PY  - 2021
DA  - 2021/01/06
TI  - Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval
JO  - Journal of Nonlinear Mathematical Physics
SP  - 136
EP  - 165
VL  - 25
IS  - 1
SN  - 1776-0852
UR  -
DO  - 10.1080/14029251.2018.1440747
ID  - Zhu2021
ER  -