Admissible boundary values for the Gerdjikov-Ivanov equation with asymptotically time-periodic boundary data

We consider the Gerdjikov-Ivanov equation in the quarter plane with Dirichlet boundary data and Neumann value converging to single exponentials $\alpha e^{i\omega t}$ and $ce^{i\omega t}$ as $t\to\infty$, respectively. Under the assumption that the initial data decay as $x\to\infty$, we derive necessary conditions on the parameters $\alpha$, $\omega$, $c$ for the existence of a solution of the corresponding initial boundary value problem.


Introduction
Long time asymptotics of integrable nonlinear partial differential equations (PDEs) can be studied by means of the Riemann-Hilbert (RH) approach. In this approach, which has been successfully applied to several initial value problems on the line, both for decaying and nondecaying initial data, a RH problem is associated to the equation and the asymptotic behavior is computed with the aid of Deift-Zhou nonlinear steepest descent techniques.
For initial boundary value problems on the half-line, the RH approach involves additional steps compared to the case on the line, because, in general, not all boundary values are known for a well-posed problem. For instance, if one assumes that the Dirichlet data are given, then the Neumann value has to be computed. This is often referred to as the Dirichlet to Neumann map.
In the case of decaying boundary data, Antonopoulou and Kamvissis [1] showed for the defocusing nonlinear Schrödinger equation that if the Dirichlet data have sufficient decay as t → ∞, then the Neumann value also decays, thus successfully characterizing the large t limit of the Dirichlet to Neumann map for decaying boundary conditions.
In the setting of nondecaying boundary data, however, less is known. In this paper, we consider the special case of asymptotically periodic boundary values. More specifically, we consider solutions q(x, t) in the quarter plane {(x, t) ∈ R 2 | x ≥ 0, t ≥ 0} whose boundary values satisfy q(0, t) ∼ αe iωt , q x (0, t) ∼ ce iωt , t → ∞, (1.1) where α > 0, ω ∈ R, and c ∈ C are three parameters. For the focusing nonlinear Schrödinger equation Boutet de Monvel and coauthors [2][3][4][5][6] were able to show that equation (1.2) has a solution with boundary values satisfying (1.1) and with decay as x → ∞, if and only if the parameters (α, ω, c) satisfy either They also computed the long time asymptotics of any such solution using the Deift-Zhou nonlinear steepest descent method. The first step in the study of initial boundary value problems whose leading order long-time behaviour is described by a single exponential consists of determining those tuples (α, ω, c) which are admissible. Here we call a tuple (α, ω, c) admissible if there is a solution of the corresponding initial boundary value problem with boundary values of the form (1.2) (see Definition 2.2 for the precise definition). In the case of the focusing NLS equation the admissible parameter triples are precisely those determined by (1.3) and (1.4).
The defocusing nonlinear Schrödinger equation with boundary values satisfying (1.1) has been studied by Lenells [10] and Lenells and Fokas [11,12]. In [10] it was shown that every admissible parameter triple belongs to one of five families. Note that the corresponding result for the focusing case only leads to two admissible families (cf. (1.3) and (1.4)). Thus the defocusing case seems to be richer, although it is still unclear if all of the five families determined in [10] are indeed admissible. In this paper we aim to implement the first step in the program initiated by Boutet de Monvel and coauthors described above for the Gerdjikov-Ivanov (GI) equation [8] iq t + q xx + iq 2q x + 1 2 |q| 4 q = 0.
(1.5) Equation (1.5) is related to the derivative nonlinear Schrödinger (DNLS) equation We will not give a complete classification of the admissible parameter triples for (1.5) but instead focus on two particularly interesting families of parameters. The first family arises as a generalization of a two-parameter family of stationary solitons. The second family arises from the plane wave solutions q b (x, t) = αe iωt+ibx for suitable parameters α > 0, ω ∈ R and b ∈ R. Within each of these families we give necessary conditions for admissibility. The proof is inspired by the proof of the corresponding results in [5] and [10].

Main Result
Before stating our main result, we give the definition of an admissible triple (see Definition 1.2 in [5] or Definitions 2.1-2.3 in [10]) and introduce two special families of parameters. Let S([0, ∞)) denote the Schwartz space Definition 2.2. A parameter triple (α, ω, c) with α > 0, ω ∈ R, and c ∈ C, is admissible for the GI equation if there exists a solution q(x, t) of (1.5) in the quarter plane such that (2.1) 2.1. The soliton solution. Equation (1.6) admits a two-parameter family of solitons [9] (see also for example (1.2) in [7]) Letting d = 0, applying the gauge transform (1.7) and multiplying the resulting function by e −iπ/4 , we obtain a one-parameter family of solutions of the GI equation with periodic boundary values. More precisely, we obtain that for every ω > 0, the function is a solution of (1.5) (in the sense of Definition 2.1) with boundary values where α = 2ω 1/4 and c = −2ω 3/4 i. In particular, it follows that the family of parameters is admissible for the GI equation. We note that the parameters associated with the soliton solution q ω satisfy The plane wave. Equation (1.5) admits the plane wave solution where α > 0, ω ∈ R, and b ∈ R satisfy The boundary values of (2.4) are given by Substituting the latter expression into (2.5), we find that the parameters associated with the plane wave satisfy the conditions Note that the plane wave (2.4) itself does not decay as x → ∞ and hence is not a solution of (1.5) in the sense of Definition 2.1.

2.3.
Statement of the result. The following theorem classifies all potentially admissible parameter triples within the families corresponding to the stationary soliton and the plane wave given in (2.3) and (2.6), respectively.
(b) If (α, ω, c) satisfies (2.6), then it belongs to one of the two families

Eigenfunctions
Equation (1.5) is the compatibility condition of the Lax pair Here k ∈ C denotes the spectral parameter, µ(x, t, k) is a 2 × 2-matrix valued eigenfunction and The above Lax pair arises from the Lax pair for the DNLS equation discovered by Kaup and Newell [9] by applying the gauge transformation (1.7) (for details see for instance the appendix of [13]). Occasionally it is convenient to consider the rescaled Lax-pair For the remainder of the paper let (α, ω, c) be an admissible triple and let q(x, t) be an associated solution of the GI equation in the quarter plane satisfying (2.1).
3.1. The background eigenfunction. Consider the background t-part equation where the matrix V b is given by V with q and q x replaced by αe iωt and ce iωt , respectively. We define a solution φ b (t, k) of (3.1) by where Ω(k) and E(k) are defined by We view the functions Ω and (2Ω − H)/(2Ω) as being defined on the cut complex plane C \ X 1 and C \ X 2 , respectively, were X i contains the branch cuts connecting the zeroes and poles of the respective function.
Assumption 3.1. We will assume that X 1 and X 2 are invariant under the involutions k → −k and k →k, that C \ X i , i = 1, 2, is connected and that the branch cuts only intersect transversely in at most finitely many points.
We will see that in our case the above assumptions are always satisfied. We fix the branches of Ω and (2Ω − H)/(2Ω) by their asymptotics as k → ∞ as follows: The symmetries of the branch cuts together with the asymptotics of Ω at infinity imply that Ω satisfies the identities and where Ω * (k) := Ω(k) denotes the Schwartz conjugate of Ω(k). Similar identities are valid for (2Ω − H)/(2Ω) on C \ X 2 . In particular, we find where X = X 1 ∪ X 2 and Note also that det E(k) = 1 for k ∈ C \ X and that E(k) approaches the identity matrix as k → ∞. Finally, we note that the identity implies that zero is not a branch point of (2Ω − H)/(2Ω) so that E(k) is analytic near zero, assuming 0 ∈ X .
The eigenfunctions {µ j (x, t, k)} 3 j=1 have the following properties: • The first (resp. second) column of µ 1 (0, t, k) is defined and analytic for k ∈ D − \X (resp. D + \X ). Furthermore, the second column of µ 1 has a continuous extension to the boundary of D + \ X , in the sense that away from the branch points the limits from the right and left onto every branch cut in D + and onto each part of the boundary of D + exist and are continuous. Note that if a branch cut can be approached from both right and left from within D + \ X , then the right and left limits are, in general, different. • µ 2 (x, t, k) is defined and analytic for all k ∈ C. • The first (resp. second) column of µ 3 (x, t, k) is defined and analytic for Im k 2 < 0 (resp. Im k 2 > 0) with a continuous extension to Im k 2 ≤ 0 (resp. Im k 2 ≥ 0).
• The µ j 's are normalized so that where k ∈ (A 1 , A 2 ) indicates that the first and second columns are valid for k ∈ A 1 and k ∈ A 2 , respectively. Furthermore, if K ± are compact subsets of D ± \ X \ P, where P is the set of branch points, then (using (2.1))

Spectral Functions
We define the spectral functions s(k) and S(k) by In view of the identities σ 1 µ * j σ 1 = µ j , j = 1, 2, 3, we may write Then Note that the analyticity properties of µ 1 and µ 2 carry over to s and S and thus to a, b, A and B. In particular, the functions A and B are defined and analytic in D + \ X with a continuous extension toD + \ X . Furthermore, away from the branch cuts they also have continuous extensions onto any branch cut intersectingD + . The functions a and b are defined and analytic in Im k 2 > 0 with a continuous extension to Im k 2 ≥ 0.

Global relation
Consider the (12) entry of the equation Using the decay of e i(Ω(k)+2k 4 )T , we find In any unbounded component of D 1 \X , we can remove the condition Im(Ω(k)+2k 4 ) > 0 by analytic continuation. Letting this yields the global relation:

Inadmissible triples
The global relation leads to the following lemma, which is the basis for the proof of Theorem 2.3 (see [5] and [10] for the corresponding result for the focusing and defocusing nonlinear Schrödinger equation, respectively). Lemma 6.1. Assume that D 1 \ X is connected and that there exists an open set U ⊆D 1 such that one of the four branch cuts connecting the eight zeroes of Ω(k) 2 intersects U . Then the triple (α, ω, c) is inadmissible.
Proof. The proof is standard, see Lemma 3.1 in [10] for the proof of the corresponding result in the case of the defocusing nonlinear Schrödinger equation.

Proof of Theorem 2.3
Lemma 6.1 enables us to perform a classification of potentially admissible parameter families. We do not perform a complete classification as has been done in [5] and [10] for the focusing and defocusing nonlinear Schrödinger equation, respectively, but instead focus our attention on the two parameter ranges introduced in Section 2.1 and 2.2.
We note that in the cases below one can directly verify that Assumption 3.1 is satisfied by choosing the branch cuts appropriately.
7.1. The solition solution case. In the following we assume that the triple (α, ω, c) satisfies (2.3). We write Then condition (2.3) is equivalent to X 2 = 0. This implies that .
By choosing the branch cut appropriately, it follows that all parameter triples in this case are inadmissible by Lemma 6.1. Figure 3. The qualitative structure of the contour Im Ω(k) = 0 (without branch cuts) in the case − α 2 2 < b < (2 + √ 6)α 2 (left) and (2 + √ 6)α 2 ≤ b (right). The branch points of Ω are marked with a dot.