A new approach to integrable evolution equations on the circle

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.


Introduction
Following the seminal discovery that the Kortewegde Vries (KdV) equation can be solved analytically via a novel methodology [1], Peter Lax understood that the distinguished feature of this equation was that it can be written as the compatibility condition of two linear eigenvalue equations, later called a Lax pair [2]. An explosion of results regarding equations possessing a Lax pair, later called integrable, occurred after the decisive work of Zakharov & Shabat [3]  namely, an equation which is derived under natural asymptotic considerations from a large class of partial differential equations (PDEs), can also be linearized via the analysis of its associated Lax pair. In this way, a new method in mathematical physics was born known as the inverse scattering transform. This method, which as clearly understood by Ablowitz et al. [4] (see also [5,6]) can be thought of as the implementation of a 'nonlinear Fourier transform', consists of two steps: (1) The first step, often referred to as the solution of the direct problem, involves the construction of the so-called scattering data, usually denoted by a(k) and b(k). These functions, which are defined in the spectral (Fourier) space are expressed in terms of the initial datum via the solution of a linear Volterra integral equation. In the linear limit where the solution q(x, t) of the associated nonlinear PDE is assumed to be small, a(k) tends to 1 and b(k) tends to the usual Fourier transform of the initial datum, q 0 (x) = q(x, 0). (2) The second step, often referred to as the solution of the inverse problem, involves the construction of the solution q(x, t) in terms of time-dependent scattering data. By employing the t-part of the Lax pair it can be shown that the function a(k) is time-independent, whereas the time dependence of b(k, t) is simple, namely it is the same as the time dependence of the underlying linear Fourier transform. The solution q(x, t) can be expressed via the solution of a linear integral equation of the Fredholm type. In the case of the KdV, this is the so-called Gelfand-Levitan-Marchenko (GLM) equation, first obtained in connection with the scattering theory of the time-independent Schrödinger operator. Although it was realized by Zakharov and Shabat that the inverse problem of NLS can also be formulated as a classical problem in complex analysis called a Riemann-Hilbert (RH) problem, the GLM equation continued to dominate the theory of integrable systems until the works of one of the authors and Ablowitz: it was shown in [7] that the inverse problem associated with the Benjamin-Ono equation gives rise to a RH problem, which in contrast to the local nature of the RH problems arising in the usual integrable evolution equations in one space variable, is nonlocal. Actually, a non-local RH problem also characterizes the solution of the inverse problems associated with the initial value problem of many integrable nonlinear evolution equations in two space dimensions, including KPI [8], DSI [9] and the N-wave interactions [10]. The formulation of the inverse problems in terms of either local or non-local RH problems made it clear that the essence of the underlying mathematical structure relevant to the solution of these problems is the following: there exist eigenfunctions of the associated t-independent part of the Lax pair which are sectionally holomorphic, namely, they are holomorphic in different domains of the complex-plane. Interestingly, there exist many nonlinear integrable evolution PDEs in two space dimensions, like KPII and DSII, whose associated eigenfunctions are nowhere analytic in the complex k-plane. The analysis of these equations requires the formulation of a so-called dbar problem, instead of a RH problem. The d-bar methodology was introduced in the area of integrable systems by Beals & Coifman [11] who employed it for equations in one space dimensions (for which the RH formulation is actually preferable). The d-bar formulation was used in two space dimensions where its use is indispensable, by one of the authors, Ablowitz and others [12][13][14][15][16]. Following the above developments, it became clear that the initial value problem for nonlinear evolution PDEs in one and two space variables can be solved by employing a local RH formalism and either a non-local RH or a d-bar formalism, respectively. Furthermore, it was shown by one of the authors and Gelfand that Fourier transforms in one and two space dimensions can also be derived via a RH and a d-bar formalism, respectively [17]. Hence, the initial value problem for linear and for integrable nonlinear evolution equations in one and two space dimensions can be solved via linear and suitable nonlinear Fourier transforms, which can be constructed via RH and d-bar formalisms.
Solving one-dimensional evolution equations formulated on the half-line or on an interval is far more challenging than solving the associated initial value problem. The first such problem to be analysed for nonlinear integrable equations was the so-called periodic problem, namely the problem formulated on the finite interval 0 < x < L, with x-periodic initial conditions. In this direction, remarkable results were obtained by many authors using techniques of algebraic geometry. For almost 100 years the only known x-periodic solution of the KdV equation was the periodic analogue of the one-soliton solution known as the cnoidal wave solution, obtained by Korteweg and de Vries in 1895. The possibility of constructing the x-periodic analogue of the multi-soliton solution of the KdV equation became clear to Russian mathematicians in 1973-1974, following the discovery of the work of Akhiezer [18] by Matveev, and the pioneering ideas of Novikov [19]. The first explicit formulas expressed in terms of theta functions were obtained by Its & Matveev [20] and then by Dubrovin [21]. The first explicit expression for the associated eigenfunction obtained by Its was never published but was reported in [22]. Parallel developments took place in the USA with the works of Kac & van Moerbeke [23], Lax [24], McKean & van Moerbeke [25] and Flaschka & McLaughlin [26]. The NLS was analysed in [27,28]. The extension of the above results to multidimensions was achieved by Krichever [29]. The relevant approach was later called the Baker-Akhiezer formalism after the realization that some of the key mathematical structures needed for the finite-gap integration were introduced by Baker in 1897 and 1907 [30]. Hamiltonian aspects of the finite-gap formalism were developed by Novikov, Dubrovin, Bogoyavlenskij, and Gelfand and Dickey (see the collection of articles in [31]). Computational aspects of the finite-gap solutions have been discussed by several investigators; for example, Deconinck et al. [32] and Klein & Richter [33]. An excellent review of the above remarkable developments and their implications in various branches of mathematics and physics is [34] (for example, the construction of periodic solutions of the KP equation led to the solution of the famous Schottky problem in algebraic geometry). However, despite the above important results, the solution of the periodic problem for arbitrary initial conditions (as opposed to the particular initial conditions corresponding to the above exact solutions) remained open. The main difficulty of solving this problem using techniques of algebraic geometry is that it requires the construction of Riemann surfaces of infinite genus. Progress in this direction has been recently announced in [35], where the approach of McKean, Trubowitz and others is followed, namely a RH problem is defined on the spectrum of the bands, but now an infinite number of gaps is allowed (this approach would be even more problematic for the focusing NLS).
Following the solution of the initial value problem in one and two space dimensions, and the construction of periodic analogues of multi-soliton solutions, the main open problems in the theory of nonlinear integrable evolution equations, in addition to solving the general periodic problem, became the following: (1) the extension of integrability to three space dimensions and (2) the solution of general initial-boundary value problems. Despite the efforts of many researchers, it has not been possible to construct nonlinear integrable evolution equations in 3 + 1, i.e. in three space dimensions and one temporal dimension (one of the authors has constructed integrable equations in 4 + 2, and has shown that their initial value problem can be solved in terms of a non-local d-bar formalism [36]; although these equations, at least aesthetically, provide the correct multi-dimensional analogue of the usual integrable equations, the question of reducing them to 3 + 1-dimensional equations remains open [37]). Regarding (2) above, after the efforts of several researchers (see, e.g. [38]), a novel new method known as the unified transform (UTM) or the Fokas method, emerged in 1997 [39].
The UTM is based on two novel ideas: (1) Perform the simultaneous spectral analysis of both parts of the Lax pair. This should be contrasted with the inverse scattering transform, which analyses only the t-independent part. The inverse scattering transform corresponds to implementing a space-variable transform (like the x-Fourier transform). Hence, it implements a nonlinear version of the classical idea of the separation of variables, whereas the UTM, by analysing simultaneously both parts of the Lax pair, goes beyond separation of variables. Indeed, for linear PDEs, the UTM gives rise to a completely new transform based on the synthesis as opposed to separation of variables [40][41][42]. This method has also led to a new rigorous treatment of the question of well-posedness for nonlinear evolution PDEs [43,44]. (2) The second ingredient of the UTM is the analysis of the global relation; this is an equation coupling appropriate transforms of the initial datum and all boundary values. As an example, in the case of NLS with 0 < x < ∞, the implementation of (1) expresses the solution q(x, t) in terms of a RH problem whose only (x, t) dependence is in the explicit form of exp(2ikx + 4ik 2 t) [45]. The jump matrices of this RH problem depend on the spectral functions a(k), b(k), A(k) and B(k). The functions a(k) and b(k) can be computed in terms of the initial datum q 0 (x). However, A(k) and B(k) are defined in terms of q(0, t) and q x (0, t), and since only one of these functions can be specified from the given boundary conditions, the functions A(k) and B(k) cannot be directly determined from the given data. At this stage, the crucial importance of the global relation becomes evident: it can be used to characterize the unknown boundary value in terms of the given initial and boundary conditions; but, unfortunately this step, in general, is nonlinear [46][47][48]. However, for a particular class of initial-boundary value problems called linearizable, this nonlinear step can be bypassed. As an example, it is noted that the boundary condition q x (0, t) + cq(0, t) = 0, c real constant, belongs to this class; in this case, the functions A(k) and B(k) can be expressed in terms of a(k), b(k) and c [45].
(a) A new approach to the periodic problem In 2004, one of the authors and Alexander Its implemented the UTM to NLS on the finite interval 0 < x < L [49]. In this case, the associated RH problem involves the spectral functions a(k), b(k), A(k) and B(k), mentioned above, as well as the functions A(k) and B(k) that are defined in terms of q(L, t) and q x (L, t). For the periodic problem, A(k) = A(k) and B(k) = B(k). It was speculated by the authors of [49] that the periodic problem belongs to the linearizable class. It will be shown here that this is indeed the case: it is possible to construct the solution q(x, t) of the NLS in terms of a RH problem whose jumps are expressed explicitly in terms of the spectral functions a(k) and b(k) (which, as noted earlier, can be determined in terms of the initial datum q 0 (x)). In this sense, the periodic problem can be solved with the same level of efficiency as the initial value problem on the infinite line.
In more detail: (a) By using a novel transformation, it is possible to map the RH problem formulated in [49] to one whose jump matrices depend on a(k), b(k) and the ratio Γ (k) = B(k)/A(k). (b) A general initial-boundary value problem for a one-dimensional evolution equation like the NLS defined on the finite interval 0 < x < L is formulated for 0 < t < T; although the solution q(x, t) is independent of T, the functions A(k) and B(k) do depend on T. It turns out that by employing a suitable transformation it is possible to map the RH problem obtained in (a) above to one whose jump matrices depend on a(k), b(k) and the ratioΓ (k), where the functionΓ (k) is independent of T. (c) By using the global relation it is possible to expressΓ (k) in terms of a(k) and b(k). This formula involves a square root, which necessitates the introduction of suitable branch cuts, which in turn introduce additional jumps in the above RH problem. Actually, this square root involves the expression 4 − (k) 2 , where (k) denotes the trace of the monodromy matrix. It is well known that this function plays a decisive role in the classical approach of the x-periodic problem.
For definiteness, we will use the NLS equation as our model example, but it will be clear that the same steps can be implemented also for other integrable evolution equations, such as the KdV equation. Since the associated RH problems are somewhat different, we will treat both the defocusing (corresponding to λ = 1) and the focusing (corresponding to λ = −1) versions of (1.1).

(b) Outline of the paper
In §2, we review the application of the unified transform method to the NLS equation posed on a finite interval [0, L] ⊂ R. In §3, we restrict our attention to the class of spatially periodic solutions and formulate a RH problem from which the solution of (1.1) can be obtained. The solution formula is not yet effective, because the formulation of this RH problem involves a certain function Γ (k), which depends on the boundary values q(0, t) = q(L, t) and q x (0, t) = q x (L, t). However, in §4, we show that the function Γ (k) can be replaced by another functionΓ (k) without affecting the resulting expression for q(x, t). The definition ofΓ (k) only involves the initial datum q(x, 0), x ∈ [0, L], yields our main result, which is stated in theorem 4.6. In §5, the main result is illustrated by means of an example for which the associated RH problem can be solved explicitly.
In the terminology of the finite-gap approach, this example corresponds to one-gap solutions.

(c) Notation
The four open quadrants of the complex k-plane will be denoted by D j , j = 1, . . . , 4, and Σ will denote the contour R ∪ iR oriented as in figure 1. The boundary values of a function f on a contour from the left and right will be denoted by f + and f − , respectively. We will use {σ j } 3 1 to denote the three standard Pauli matrices. The first and second columns of a 2 × 2 matrix A will be denoted by [A] 1 and [A] 2 , respectively.

The NLS equation on a finite interval
Before turning to the periodic problem, we recall some aspects of the analysis of the NLS equation on a finite interval presented in [49], which will be needed in later sections.
The NLS equation (1.1) has a Lax pair given by where k ∈ C is the spectral parameter, μ(x, t, k) is a 2 × 2-matrix-valued eigenfunction, the matrices Q andQ are defined in terms of the solution q(x, t) of (1.1) by where 0 < L > ∞ and 0 < T < ∞ is some fixed final time. Following [49], we let μ j (x, t, k), j = 1, 2, 3, 4, denote the four solutions of (2.1), which are normalized to be the identity matrix at the points (0, T), (0, 0), (L, 0) and (L, T), respectively. The spectral functions a, b, A, B, A, B are defined for k ∈ C by where c + (k) is an entire function, which is of order 1/k as k → ∞ in the upper half-plane; in fact, Here, and in what follows, a bar over a function denotes that the complex conjugate is taken, not only of the function but also of its argument; this is called the Schwarz conjugate. The above functions satisfy the unit determinant relations The functions a and b satisfy whereas the functions A and B satisfy The eigenfunctions μ j are related by Using the entries of s, S, S L , we construct the following quantities: (a) RH problem for M It was shown in [49] that the sectionally meromorphic function M(x, t, k) defined by where Σ = R ∪ iR is oriented as in figure 1 (2.14) Note that the jump matrix depends on x and t only via the function θ(x, t, k) defined in (2.10). The solution q(x, t) of (1.1) can be recovered from M via the identity where the limit may be taken in any quadrant. If the functions α(k) and d(k) have no zeros, then the function M is analytic for k ∈ C\Σ and it can be characterized as the unique solution of the RH problem (2.12) with jump matrix J. The jump matrix J depends via the spectral functions on the initial datum q(x, 0) as well as on the boundary values q(0, t), q(L, t), q x (0, t) and q x (L, t). If all these boundary values are known, then the value of q(x, t) at any point (x, t) can be obtained by solving the RH problem (2.12) for M and using (2.15).
If the functions α(k) and d(k) have zeros, then M may have pole singularities. The generic case of a finite number of simple poles can be treated by supplementing the RH problem with appropriate residue conditions, see [49,Proposition 2.3].

The periodic problem
From now on, we restrict our attention to the periodic problem and assume that q(x, t) satisfies (3.1) In this case, we clearly have A = A and B = B. Hence the jump matrices J i defined in (2.14) become Our goal is to find a representation for the solution q(x, t) in terms of the initial datum q 0 (x) = q(x, 0). Since the expression (3.2) for the jump matrix depends via the spectral functions A(k) and B(k) on the two (unknown) functions q(0, t) and q x (0, t) given in (3.1), the representation (2.15) does not achieve this goal. However, in what follows, we will show that it is possible to eliminate A and B from the formulation of the RH problem by performing two steps. In the first step, which is presented in this section, we will transform the RH problem (2.12) to a new RH problem which In the second step, which is presented in §4, we will show that Γ (k) can be effectively replaced by a functionΓ (k), which only depends on the initial datum.
(a) RH problem for m Consider the sectionally meromorphic function m(x, t, k) defined in terms of the eigenfunctions The function m is related to the solution M of (2.12) by where the sectionally meromorphic function H is defined by here, H j denotes the restriction of H to D j , j = 1, 2, 3, 4. The function m may have poles at the possible zeros of the entire functions α, A and d. In order to express the locations of these possible poles and the associated residue conditions in terms of only a, b, Γ , we introduce the functions η(k) and ξ (k), which are defined by Clearly, Γ , η and ξ are meromorphic functions of k ∈ C. We will consider the generic situation in which the possible poles of these functions satisfy the following assumption. -In D 2 , ξ (k) has at most finitely many poles {κ j } N 3 1 ⊂ D 2 and these poles are all simple. 3 1 denote the set of poles and their complex conjugates. Then P is disjoint from the set of zeros of a and b.

Remark 3.2.
Regarding assumption 3.1 it is noted that it is not important to investigate whether these poles exist, since, remarkably, they cancel out in the formulation of the final RH problem. Actually, as shown in the main theorem (theorem 4.6), we only need to worry about the poles of Γ and the existence of these poles is discussed in remark 4.3.
We will show that m satisfies the following RH problem.
-At the points K j ∈ D 1 andK j ∈ D 4 , m has at most simple poles and the residues at these poles satisfy, for j = 1, . . . , N 2 , -At the points κ j ∈ D 2 andκ j ∈ D 3 , m has at most simple poles and the residues at these poles satisfy, for j = 1, . . . , N 3 , Note that the above RH problem depends on the functions A and B only via their quotient Γ = B/A. The next proposition shows that the RH problem for m can be used to determine the solution q of (1.1) on the circle of length L, assuming that the initial datum and the quotient B/A are known.

t). Define a, b, Γ by (2.3) and (3.5) and let η(k)
and ξ (k) be given by (3.7). Suppose assumption 3.1 holds. Then the RH problem 3.3 has a unique solution The solution q can be obtained from m via the relation Moreover, det m = 1 and m obeys the symmetries Proof. Let us first show that the solution of the RH problem 3.3 is unique. If the set P of poles is empty, then uniqueness follows by standard considerations because the jump matrix has unit determinant. The problem with a non-empty set P can be transformed into a problem for which P is empty. Indeed, if the set {k j } N 1 1 of poles of Γ is non-empty, then the function satisfies an analogous RH problem but with no singularities at the points {k j ,k j } N 1 1 . The residue conditions (3.11) and (3.12) are of a standard form, so the possible poles {K j } N 2 1 and {κ j } N 3 1 of η and ξ can be regularized in the standard way; see, e.g. [38]. This proves uniqueness. It also follows from these arguments that det m = 1.
The symmetries (3.14) can be expressed as m(x, t, k) = σ 1 m(x, t,k)σ 1 if λ = 1 and as m(x, t, k) = σ 3 σ 1 m(x, t,k)σ 1 σ 3 if λ = −1. These symmetries follow from the uniqueness of the solution and the fact that v obeys the symmetries Define m by (3.6). Long but straightforward computations using the unit determinant relations (2.6) show that the jump matrix v given in (3.9) satisfies Since m = MH, the jump relation (3.8) follows from (2.12a). Alternatively, (3.8) can be derived directly from (2.9) and (3.6).
We next show that m is analytic for k ∈ C\(Σ ∪ P) and establish the residue conditions (3.10)-(3.12). We consider the columns of m in each quadrant separately.
The second column of m in D 1 . The condition AĀ − λBB = 1 implies that A and B cannot have any common zeros. Hence, the set of zeros of A coincides with the set of poles of Γ , and at each such pole B is non-zero. Since [m] 2 = [μ 4 ] 2 /A, it follows that [m] 2 is analytic in D 1 except at the possible poles k j of Γ .
Equations (2.9a) and (2.9c) imply that Sinceb is non-zero at each point of P by assumption and since A(k j ) = 0 and B(k j ) = 0, it follows that α(k j ) = λb(k j )B(k j ) e 2ik j L = 0. Thus [m] 1 = A[μ 2 ] 1 /α has at least a simple zero at k j and it then follows from (3.19) that [m] 2 has (at most) a simple pole at k j . This proves (3.10a).
The first column of m in D 1 . equation (3.17) can be written as The first column of (3.22) can be written as

Main result
The RH problem 3.3 for m depends on the final time T via the function Γ = B/A. However, the solution q(x, t) is independent of T. This suggests that it should be possible to eliminate the T dependence from the RH problem 3.3. In this section, we define a new T-independent RH problem-henceforth called the RH problem form-by applying an appropriate deformation to the RH problem for m. The basic idea of this deformation is to replace Γ by a new T-independent functionΓ . SinceΓ is defined in terms of the spectral functions a(k) and b(k) alone, this will lead to our main result.

(a) Motivating remarks
As motivation for the definition of the RH problem form, we temporarily consider a solution q(x, t) of the NLS equation on the interval [0, L] whose boundary values have decay as t → ∞. In this case, it can be shown that the functions A, B, A and B have finite limits as T → ∞; we denote these limits byÃ,B,Ã andB: Taking the same limit in the global relation (2.4) leads to the equation which can be viewed as a relation between the two quotientsB/Ã andB/Ã. In the spatially periodic setting, these two quotients are equal and hence (4.1) can be solved forB/Ã with the result thatB where = a e −ikL +ā e ikL . This suggests that we define the new RH problem form by replacing Γ with the T-independent functionΓ , whereΓ ≡B/Ã is defined by the right-hand side of (4.2). Since the right-hand side of (4.2) is defined in terms of the initial datum alone, this leads to an effective solution of the x-periodic initial value problem. We first need to give a careful definition ofΓ in which the branch of the square root in (4.2) is fully specified.

(b) Definition ofΓ
We define the functionΓ (k) bỹ where (k) is given by = a e −ikL +ā e ikL , k ∈ C. In order to make the definition (4.3) ofΓ precise, we need to introduce branch cuts and fix the sign of 4 − 2 . To this end, consider the zero-set P of the entire function 4 − 2 : The function (k) is the trace of the so-called monodromy matrix and features heavily in the classical approach to the x-periodic problem. In that framework, P is the periodic spectrum 1 and is well studied. In what follows, we collect some well-known facts about P; see, e.g. [50], and define a set of branch cuts C such that 4 − 2 becomes single-valued on C\C.
In the case of vanishing initial datum q 0 ≡ 0, we have a = 1, thus (k) = 2 cos(kL), and hence 4 − 2 = 4 sin 2 (kL) has a double zero at each of the points nπ/L, n ∈ Z. According to the so-called Counting Lemma (see [50,Lemma 6.3]), the zero-set of 4 − 2 has a similar structure for large k for any initial datum q 0 ∈ L 2 ([0, L]) in the following sense: For n ∈ Z, let denote the open disk of radius π/(4L) centred at nπ/L. Then, there is an integer N > 0 such that, counted with multiplicity, 4 − 2 has exactly two roots in each disk D n with |n| > N and exactly 1 The periodic spectrum is defined as the union of all periodic and anti-periodic eigenvalues. where and λ ± n belong to D n for all large enough |n|. The symmetry (k) = (k) implies that the periodic spectrum P is invariant under complex conjugation and that is real-valued on R. To define C, we consider the two cases λ = 1 and λ = −1 separately.
In the defocusing case (i.e. λ = 1), the periodic spectrum P is purely real and The open interval (λ − n , λ + n ) is called the nth spectral gap whenever it is non-empty. For λ = 1, we define C as the union of all spectral gaps: Thus, in this case C is a union of open subintervals of R. The nth interval (λ − n , λ + n ) is contained in D n for all sufficiently large |n|.
In the focusing case (i.e. λ = −1), P is typically not a subset of R. If z 1 , z 2 ∈ C, we let (z 1 , z 2 ) denote the open straight-line segment from z 1 to z 2 , i.e.
Using this notation, it is possible to define C by (4.7), but this definition has the disadvantage that it may break the existing symmetry under complex conjugation. Therefore, we instead define C as follows. Let N > 0 be as in the Counting Lemma so that there are 4N + 2 roots counted with multiplicity in the disk {|k| < Nπ/L + π/(4L)}. An even number, say 2M, of these 4N + 2 roots have odd multiplicity; let {λ odd j } 2M 1 denote these roots. Since is real-valued on R and = 2 cos(kL) + o(1) as k → ±∞, an even number, say 2M, of the roots {λ odd j } 2M 1 are real; let {λ − j ,λ + j }M j=1 denote these real roots ordered so thatλ denote the remaining odd-order roots in the disk {|k| < Nπ/L + π/(4L)} ordered lexicographically:λ Note that there is an even number of rootsλ ± j on any vertical line Re k = constant. We define C as the union of the open real intervals (λ − j ,λ + j ), the vertical line segments (λ − j ,λ + j ), as well as the open real intervals (λ − n , λ + n ), |n| > N: In both the defocusing and the focusing case, we orient the contour C so that (i) any part of C that is contained in Σ = R ∪ iR is oriented in the same direction as Σ, i.e. C ∩ iR + , C ∩ R + , C ∩ iR − and C ∩ R − are oriented down, right, up and left, respectively, and (ii) the subcontours  We can now complete the definition ofΓ . The function 4 − (k) 2 in (4.3) is single-valued for k ∈ C\C up to a choice of sign. This sign can be fixed by considering the large k behaviour ofΓ . Indeed, as k → ∞ in R, (2.7) implies that a ∼ 1 andā ∼ 1, and so = a e −ikL +ā e ikL ∼ 2 cos(kL), 4 − 2 ∼ ±2 sin(kL).
Since we would like to haveΓ = O(1/k) as k → ∞ in R (at least if k stays away from the disks D n ), we fix the branch of 4 − 2 by requiring that (see (4.28) for a more detailed estimate) 4 − 2 ∼ 2 sin(kL), k → ∞, k ∈ R\ n∈Z D n . (4.9) In summary,Γ : C\C → C is defined by (4.3) with the branch of 4 − 2 fixed by (4.9).

(c) RH problem form
Define the function g(k) by where g j denotes the restriction of g to D j for j = 1, . . . , 4. We introducem(x, t, k) bỹ m = mg, (4.11) where m is the solution of RH problem 3.3. The function g is defined in such a way thatm satisfies the jump relationm − =m +ṽ on Σ\C, where the jump matrixṽ is given by the same expression (3.9) as v except that Γ is replaced byΓ . This follows by a direct computation using that However, because of the square root 4 − 2 ,ṽ is not given by the same expression as v on Σ ∩ C, and if λ = −1, thenm may also have jumps across the contours C ∩ D j , j = 1, . . . , 4. It is quite remarkable that all these jumps can be expressed completely in terms of a and b alone. Let B =C\C denote the set of branch points of 4 − 2 . The set B is always contained in the periodic spectrum P and it may be strictly smaller than P if 4 − 2 has roots of even multiplicity. LetΣ = Σ ∪C denote the union of the cross Σ = R ∪ iR and the set of branch cuts and branch points, see figure 2. Let S denote the set of self-intersections of the contourΣ. In the defocusing case, S only consists of the origin. In the focusing case, S consists of the origin together with any points at which vertical branch cuts intersect the real axis. LetΣ =Σ\(B ∪ S) denote the contour Σ with all branch points and all points of self-intersection removed. The jump matrixṽ is defined for k ∈Σ as follows: whereṽ j− denotes the boundary values of the matrixṽ j as k approaches C from the right. We emphasize that the jump matrixṽ can be computed from the knowledge of the initial datum alone. Whereas the function m, in general, has singularities at the poles of Γ , η and ξ , it turns out that m is analytic at these points. In fact,m can have singularities only ifΓ has poles in the first or third quadrant. We make the following assumption. Regarding assumption 4.2, we note that there exist large families of initial conditions for which it can be shown explicitly thatΓ does not have poles. For example, the single exponential families considered in §5 and in [51] are of this type. More precisely, the definition (4.3) ofΓ has the formΓ =γ /b, where the numeratorγ has no poles. Thus any possible poles of Γ are generated by the zeros ofb. However, in many cases the zeros ofb are cancelled by zeros of γ . Indeed, suppose k 0 is a zero ofb. Using the relation aā − λbb = 1, we can writẽ where the branch of the square root is fixed so that it tends to 1 as k tends to infinity in C\ n∈Z D n (it can be shown that 4λbb/(ā e ikL − a e −ikL ) 2 = O(k −2 ) in this limit as a consequence of (2.7)). Thus, assuming thatā e ikL − a e −ikL is non-zero at k 0 , we see thatγ vanishes at k 0 to the same order asb whenever the branch of the square root in (4.14) is such that it is close to 1 for k near k 0 . It follows, in particular, thatΓ (k) cannot have poles whenever k ∈ C\ n∈Z D n is large enough. The question of whetherΓ is always pole-free is under consideration. For the family of single exponential initial data considered in §5, we will see thatΓ has no poles even thoughb has infinitely many zeros.
LetP = {p j ,p j } n 1 1 ∪ {q j ,q j } n 2 1 denote the set of poles ofΓ in D 1 ∪ D 3 and their complex conjugates. We will show thatm is the unique solution of the following RH problem. -At the points p j ∈ D 1 andp j ∈ D 4 ,m has at most simple poles and the residues at these poles satisfy, for j = 1, . . . , n 1 , -At the points q j ∈ D 3 andq j ∈ D 2 ,m has at most simple poles and the residues at these poles satisfy, for j = 1, . . . , n 2 , In order to show thatm andṽ have the appropriate regularity properties, we need the following lemma, which shows that some of the denominators in the definition (4.10) of the matrices g j (k) are nowhere zero. In particular, the function a + λbΓ e 2ikL = a − λbΓ is non-zero for all k ∈ C\C.

Res
Proof. The identities follow by a direct computation using the definitions (4.3) and (4.4) ofΓ and . If − i 4 − 2 = 0 at some k, then 2 = 2 − 4 at k, which is a contradiction. Thus a + λbΓ e 2ikL = a − λbΓ has no zeros.

(d) Main result
The following theorem, which is the main result of the paper, provides an expression for the solution q(x, t) of the x-periodic NLS equation in terms of the solution of the RH problem 4.4. Since the formulation of this RH problem only involves quantities defined in terms of the initial datum, the theorem provides an effective solution of the IVP for the x-periodic NLS equation. where the limit is taken along any ray {k| arg k = φ} where φ ∈ R\{nπ/2 | n ∈ Z} (i.e. the ray is not contained in R ∪ iR).
Proof. The functionΓ is analytic in (D 1 ∪ D 3 )\(Σ ∪P) with continuous boundary values onΣ . Thus it follows from lemma 4.5 that each of the matrices appearing on the right-hand side of (4.13) is well-defined and continuous on its domain of definition. In particular,ṽ is well-defined and piecewise continuous onΣ.
Let us prove uniqueness ofm. The problem with a non-empty setP can be transformed into a problem for whichP is empty following a standard procedure; see e.g. [38]; we may therefore assume thatP is empty when proving uniqueness. Supposem is a solution of the RH problem 4.4. We will first show thatm has unit determinant. The jump matrixṽ has unit determinant everywhere onΣ . Thus detm is an entire function except for possible singularities at points in the discrete setΣ\Σ . However, the assumption thatm = O(1) as k →Σ\Σ implies that these singularities are removable. Thus detm is an entire function. The assumption thatm = I + O(k −1 ) as k ∈ C\ n∈Z D n tends to infinity implies that there is a constant C > 0 such that |m − I| ≤ C/k on each of the circles |k| = (n + 1/2)π/L, n = 1, 2, . . .. Hence detm = 1 + O(k −1 ) uniformly on these circles. By the maximum modulus principle, we conclude that detm = 1 + O(k −1 ) as k → ∞. Hence, by Liouville's theorem, detm = 1 for all k ∈ C. Thusmñ −1 is an entire function except for possible singularities at points in the discrete setΣ\Σ . The same arguments that led to detm = 1 show that these singularities are removable and that mñ −1 is in fact identically equal to the identity matrix. This proves uniqueness.
The function m obeys the symmetries (3.14) and it is easy to check that g satisfies the same symmetries: It follows thatm also obeys these symmetries: Moreover, since m and g have unit determinant, we have detm = 1. Let us show thatm is analytic for k ∈ C\(Σ ∪P). This can be established by considering the analyticity properties of m and using the conditions (3.10)-(3.12) to show thatm = mg has no poles at these points. However, then we would have to restrict ourselves to initial data for which assumption 3.1 holds. We therefore instead give a direct argument which takes the eigenfunctions μ j as its starting point.
In light of the symmetries (4.20), it is enough to establish analyticity ofm for k ∈ C + \(Σ ∪P), where C + = {Im k > 0} denotes the open upper half-plane. The definitions (3.6) and (4.11) imply thatm 1 andm 2 can be expressed in terms of the eigenfunctions μ j as follows:  note thatm = mg and that m has no jump across C ∩ D j . Hencem − =m +ṽ A direct computation using the definition (4.10) of g j shows that the matrixṽ cut D j is given by the expression in (4.13). On the other hand, on the part of C that is contained in It follows thatṽ This completes the proof of the jump relation (4.15).
It only remains to show thatm = I + O(k −1 ) as k ∈ C\ n∈Z D n approaches infinity. This follows from the fact that m = I + O(k −1 ) as k → ∞ provided that we can show that (4.23) In fact, due to the symmetry (4.19) of g, it is enough to prove for j = 1, 2 that The estimates (2.7) imply In particular, as k → ∞ in the closed upper half-planeC Recalling the identities in lemma 4.5, this shows that Let us consider g 1 . As k → ∞ inD 1 , we have Γ = O(k −1 ) andb e 2ikL = O(k −1 ), and hence a + λbΓ e 2ikL = 1 + O(k −1 ). Together with (4.31), this yields showing (4.24) for the diagonal elements of g 1 . As for the non-zero off-diagonal element we note that solving the global relation (3.4) where the branch of the square root is fixed by the requirement (cf. (4.9)) By ( = O e 4ik 2 T k 2 , k → ∞, k ∈ (D 1 ∪D 3 )\ n∈Z D n ; (4.33) thus the branch cuts and the values of the square root in (4.32) are close to those of 4 − 2 for large k ∈ (D 1 ∪D 3 )\ n∈Z D n . In particular, both of these roots are analytic for large k ∈ (D 1 ∪ D 3 )\ n∈Z D n . Subtracting (4.32) from (4.3), we find Utilizing (2.5), (4.28) and (4.26), we infer that, as k ∈D 1 \ n∈Z D n approaches infinity, Since x ≤ L and T > t, this yields This completes the proof of (4.24) for j = 1. We next consider g 2 . We have .

Example: A single exponential
We illustrate the approach of theorem 4.6 by considering the following initial datum involving a single exponential: q(x, 0) = q 0 e (2iπN/L)x , x ∈ [0, L], (5.1) where N is an integer and the constant q 0 > 0 can be taken to be positive due to the phase invariance of (1.1).

(a) RH problem form
Let N be an integer. Direct integration of the x-part of the Lax pair (2.1) with q given by (5.1) leads to the following expressions for the spectral functions a and b: where r(k) denotes the square root It follows that (k) = 2(−1) N cos(Lr(k)).
Note that a, b and are entire functions of k even though r(k) has a branch cut. The periodic spectrum P is given by the zeros of 4 − 2 = 4 sin 2 (Lr) and consists of the two simple zeros λ ± as well as the infinite sequence of double zeros − π N L ± n 2 π 2 L 2 + λq 2 0 , n ∈ Z\{0}.
If λ = 1, then all zeros are real; if λ = −1, then the zeros are real for |n| ≥ Lq 0 /π and non-real for |n| < Lq 0 /π . The function 4 − 2 is single-valued on C\C, where C defined in (4.7) and (4.8) consists of the single branch cut C = (λ − , λ + ). We fix the branch in the definition of r so that r : C\C → C is analytic and r(k) = k + π N/L + O(k −1 ) as k → ∞. Then, using (4.9) to fix the overall sign, and hence the functionΓ : C\C → C defined in (4.3) is given bỹ Note thatΓ (k) has no poles in spite of the fact thatb has infinitely many zeros, see The contourΣ is equal to R ∪ iR ∪ (λ − , λ + ) and is oriented as in figure 3. If λ = 1, then C = (−π N/L − q 0 , −π N/L + q 0 ) so the formulation of the RH problem also involves at least one of the jump matricesṽ cut 2 andṽ cut 4 . If λ = −1, then C = (−π N/L − iq 0 , −π N/L + iq 0 ) so the formulation involves the jump matricesṽ cut  Figure 3. The jump contour and jump matrices for RH problem 5.1 for λ = 1 (left) and λ = −1 (right).
where r(k) = q 2 0 + (k + π N/L) 2 ≥ 0. We conclude that in the case of the single exponential initial profile (5.1) with N ≥ 1, the RH problem 4.4 form can be formulated as follows.
The integrals in (5.13) involving x − L and t are easily computed by opening up the contour and performing a residue calculation (the only residue lies at infinity). This gives If λ = 1, then the substitutions s = −π N/L + σ and σ = q 0 sin θ give Substituting this expression for δ ∞ into (5.12), we find that the solution q(x, t) of (1.1) corresponding to the initial datum q(x, 0) = q 0 e (2iπN/L)x is given by It is easy to verify that this q indeed satisfies the correct initial value problem.

Remark 5.3 (Finite-gap solutions).
We have shown that the single exponential solutions (5.16) can be constructed by solving the RH problem 4.4 form directly. Associated to the solutions (5.16) is the genus zero Riemann surface defined by the square root 4 − 2 ; we have seen that this is a two-sheeted cover of the complex plane with a branch cut along the single gap (λ − , λ + ). More generally, whenever the Riemann surface defined by 4 − 2 has finite genus (i.e. whenever the given initial condition corresponds to a finite-gap solution), we expect that a representation for the solution in terms of theta functions associated with the compact Riemann surface defined by 4 − 2 can be obtained by solving the RH problem 4.4.
Data accessibility. This article has no additional data. Authors' contributions. A.S.F. and J.L. conceived the ideas and proved the mathematical results presented in this paper. Both authors gave final approval for publication and agree to be held accountable for all aspects of the work.
Competing interests. We declare we have no competing interests.