Initial Value Problems for Integrable Systems on a Semi-Strip

Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix solutions of the defocusing nonlinear Schr\"odinger equation with quasi-analytic boundary conditions is dealt with. (The result is new even for a scalar nonlinear Schr\"odinger equation.) Next, a special case of the nonlinear optics ($N$-wave) equation is considered.


Introduction
Cauchy problems for wave equations were actively studied using Inverse Scattering Transform, Laplace Transforms and some other methods (see, e.g., [3,5,15,33,57,58]). The theory of initialboundary value problems (and problems in a quarter-plane or semi-strip) is somewhat more complicated even for the case of linear differential equations, see some results, discussions and references on this topic in [6,19,20,34,45]. The mentioned above results and discussions are related also to the integrable nonlinear equations. Such problems for linear and nonlinear equations are often called forced or perturbed problems (see, e.g., [28]). They have numerous applications and are of great mathematical interest. Therefore, they are actively investigated and many interesting results are obtained in spite of remaining difficulties and open problems, see further discussion at the beginning of Section 3.
The cases, when an initial condition provides direct information about boundary conditions, or the solutions of nonlinear integrable equations in a semi-strip are uniquely defined by the initial conditions, are of special interest. (In particular, they are connected with the problems of wellposedness and bounds as well as with the construction of blow up solutions.) We consider such situations using the Inverse Spectral Transform approach [9,10,27]. More precisely, we follow the scheme introduced in [50,51,53], see also [54,Ch. 12] and references therein. Namely, we describe the evolution of Weyl-Titchmarsh (Weyl) function in terms of the linear-fractional transformations. The scheme is applicable to various integrable equations [39,41,43,45,46] and several interesting uniqueness and existence theorems were proved in this way (see [49,Ch. 6] and references therein for more details). Most of the mentioned above uniqueness and existence theorems were obtained for the equations with scalar solutions. Here we consider the matrix defocusing nonlinear Schrödinger (defocusing NLS or dNLS) equation which is equivalent [62,63] to the compatibility condition of the auxiliary linear systems where

5)
I m 1 is the m 1 × m 1 identity matrix and v is an m 1 × m 2 matrix function. We will consider dNLS equation on the semi-strip D = {(x, t) : 0 ≤ x < ∞, 0 ≤ t < a}, (1. 6) and we note that the auxiliary system y x = Gy = i(zj + jV )y (1.7) is (for each fixed t) a well-known self-adjoint Dirac system, also called AKNS or Zakharov-Shabat system. The matrix function v is called the potential of the system. Without changes in notations we speak about usual derivatives inside domains and about left or right (which should be clear from the context) derivatives on the boundaries, and boundaries of D in particular. Since v in dNLS (1.1) is an m 1 × m 2 matrix function, interesting matrix, vector and multicomponent dNLS equations from [2,Ch. 4] are included in the considered class. Another equation that we study in this paper is the nonlinear optics (or N-wave) equation: where ̺(x, t) is an m × m matrix function, diag{d 1 , d 2 , . . .} stands for a diagonal matrix with the entries d 1 , d 2 , ... on the main diagonal, and D > 0 is another diagonal matrix. First, we obtain evolution of the Weyl function for the equation (1.8). Next, we consider an interesting special case, where (similar to the inequalities for the entries of D) we have (1.10) In Section 2 we formulate some necessary results on Weyl functions (and their evolution for the dNLS case), in Section 3 we recover the boundary conditions for dNLS from an initial condition on a semi-axis (and in this way we solve the initial value problem for dNLS in a semi-strip), and Section 4 is dedicated to the evolution of the Weyl function and initial value problem in a semi-strip for the N-wave equation.
As usual, R stands for the real axis, R + = (0, ∞), C stands for the complex plain, and C + for the open semi-plane {z : ℑ(z) > 0}. We say that v(x) is locally summable if its entries are summable on all finite intervals of [0, ∞). We say that v is continuously differentiable if v is differentiable and its first derivatives are continuous. The notation · stands for the l 2 vector norm or the induced matrix norm. The partial derivative f xt stands for ∂f x /∂t and, correspondingly, f tx = ∂f t /∂x.

Dirac system and dNLS
We denote by u the fundamental solution of system (1.7) normalized by the condition u(0, z) = I m , m = m 1 + m 2 .
(2.1) Definition 2.1. Let Dirac system (1.7) on [0, ∞) be given and assume that v is locally summable. Then Weyl function is an m 2 × m 1 holomorphic matrix function, which satisfies the inequality The following proposition is proved in [22] (and in [49, Section 2.2]).

Proposition 2.2. The Weyl function always exists and it is unique.
In order to construct the Weyl function, we introduce a class of m × m 1 matrix functions P(z), which are an immediate analog of the classical pairs of parameter matrix functions. Namely, the matrix functions P(z) are meromorphic in C + and satisfy (excluding, possibly, a discrete set of points) the following relations P(z) * P(z) > 0, P(z) * jP(z) ≥ 0 (z ∈ C + ). (2. 3) It is said that P(z) are nonsingular (i.e., the first inequality in (2.3) holds) and with property-j (i.e., the second inequality in (2.3) is valid). Relations (2.3) imply (see, e.g., [22]) that where P(z) are nonsingular matrix functions with property-j.

Remark 2.4.
It was shown in [22] that a family N (x, z), where x increases to infinity and z is fixed (z ∈ C + ), is a family of embedded matrix balls such that the right semi-radii are uniformly bounded and the left semi-radii tend to zero.
Proposition 2.6. [45] Let some m × m matrix functions G and F and their derivatives G t and F x exist on the semi-strip D, let G, G t and F be continuous with respect to x and t on D, and let (1.2) hold. Then we have the equality where u(x, t, z) and R(x, t, z) are normalized fundamental solutions given, respectively, by: The equality (2.8) means that the matrix function v(x, s) ≤ M(t). (2.11) Then the evolution ϕ(t, z) of the Weyl functions of Dirac systems (1.7) is given (for ℑ(z) > 0) by the equality Remark 2.8. According to [47], the Dirac system y x = Gy (where G is given by (1.4) and (1. 5) and v is locally square summable) is uniquely recovered from the Weyl function ϕ. In other words, v is uniquely recovered from ϕ, see the procedure in [47,Theorem 4.4]. The case of a more smooth (i.e., locally bounded) v was dealt with in [49], see also references therein.

Auxiliary linear systems for nonlinear optics equation
see [61] for the case N = 3 and [1] for N > 3. We shall need some preliminary results on the Weyl theory of the auxiliary system y x = Gy from [49,Ch. 4] (see also [42,44]). The normalized fundamental solution w of such system is defined by the formula (2.14) Here and further we assume that D is a fixed matrix satisfying (1.9). We consider system (2.14) with locally bounded potentials ζ, that is, potentials satisfying (for each l < ∞) the inequality holds for each l < ∞.
The inverse spectral problem (ISpP) for system (2.14) is the problem to recover (from a GWfunction ϕ) a potential ζ(x) = −ζ * (x) (ζ kk ≡ 0) such that (2.16) is valid. Notation 2.10. The notation M stands for an operator mapping the pair D and ϕ into the corresponding potential ζ (i.e., M(D, ϕ) = ζ). That is, M(D, ϕ) stands for a solution of the ISpP.

Theorem 2.11. For any matrix function ϕ(z) which is analytic and bounded in C − M and has the property
there is at most one solution of the ISpP.
The situation becomes simpler when ζ(x) is uniformly bounded on [0, ∞), that is, We recall that a Weyl function of system (2.14) is introduced in another way than a GW-function. Namely, a Weyl function is an analytic m × m matrix function ϕ(z), satisfying for certain M > 0 and r > 0 and for all z from the domain 21) and the normalization conditions on the entries ϕ ij (z) : When (2.20) holds, a Weyl function of system (2.14) exists and is unique. Moreover, for that case ϕ is the unique GW-function (of system (2.14) with the given ζ) satisfying normalization conditions (2.22). In order to construct this Weyl (and simultaneously GW-) function we use matrices j of the form (1.5) for each 1 ≤ m 1 < m, that is, we set Now, the Weyl function is constructed [49, pp. 103-106] in the following way. First, for each k, we introduce a class of m × (m − k) matrix functions Q k , which are meromorphic in some semi-plane C − M k and satisfy the inequalities excluding, possibly, isolated points. These Q k are called nonsingular with property-J k . Assuming ) and using (2.24), one can show that the matrix function is well-defined for x ≥ 0, z ∈ C − M k , and satisfies the inequality The set of matrices ψ k (x, z) given by (2.25), where x and z are fixed and matrix functions Q k (z) are nonsingular with property-J k , is denoted by N k (x, z). These sets are embedded and have a point limit, that is, similar to (2.6) and (2.7) we havȇ In this way we recover the (k + 1)th column of the Weyl function ϕ. More precisely, we have There is also an inverse transformation [49,Remark 4.6], which expressesψ k via ϕ: (2.29) Finally, we will need a representation of

NLS with quasi-analytic boundary conditions
First publications on initial-boundary value problems for integrable systems (see, e.g., [27,29]) appeared only several years after the great breakthrough for Cauchy problems for such systems. Interesting numerical [17], uniqueness [14,60] and local existence [26,35] results followed. Special linearizable cases of boundary conditions were found using symmetrical reduction [56] or BT (Bäcklund transformation) method [12,24]. Global existence results for Dirichlet and Neumann initialboundary value problems (for cubic NLS equations) were obtained using PDE methods in [16] and [31], respectively. Interesting approaches were developed by D.J. Kaup and H. Steudel [30] and by P. Sabatier (elbow scattering) [37,38]. Finally, we should mention the well-known global relation method by A.S. Fokas, see [20] and references therein. (See also some discussions on the corresponding difficulties and open problems in [6,13,20].) Since many publications were dedicated to the initial-boundary value problems for NLS equations, it is of special interest that a wide class of solutions of NLS in a semi-strip is uniquely defined by the initial condition.
Recall that the domain D is defined in (1.6).
Notation 3.1. We consider m 1 ×m 2 matrix functions v(x, t), which are continuously differentiable and are such that v xx exists on the semi-strip D. Moreover, we require that for each k there is a value ε k = ε k (v) > 0 such that v is k times continuously differentiable with respect to x in the square The class of such functions v(x, t) is denoted by C ε (D).
Without loss of generality, we assume that the values ε k in (3.1) monotonically decrease.
hold, whereas both sides of these equalities are again continuous. For 0 ≤ k ≤ r and 0 ≤ ℓ ≤ s the functions ∂ ℓ ∂x ℓ ∂ k+1 ∂t k+1 v exist and are continuous in the domains D(ε s+4(r+1) ). In order to prove this proposition, we need a stronger version of the well-known Clairaut's (or Schwarz's) theorem on mixed derivatives. We need this version for the closed square D(ε) (as in Proposition 3.2 from [48]), which statement easily follows from the proofs of the mixed derivatives theorem for open domains (see, e.g., [55]). Proposition 3.3. [48] If the functions f , f t and f tx exist and are continuous on D(ε) and the derivative f x (x, 0) exists for 0 ≤ x ≤ ε, then f x and f xt exist on D(ε) and f xt = f tx .
Proof of Proposition 3.2. We prove Proposition 3.2 by induction. First, consider the case r = 0. Clearly, v(x, 0) and v x (x, 0) are given by the initial condition (3.2). Since the right-hand side of (1.1) is two times continuously differentiable with respect to x in D(ε 4 ), we derive that v t , v tx and v txx exist and are continuous in D(ε 4 ): Moreover, putting f = v we see that conditions of Proposition 3.3 are fulfilled and the first equality in (3.3) holds for k = 0. Putting f = v x and taking into account that the first equality in (3.3) yields v xt = v tx and v xtx = v txx , we see that conditions of Proposition 3.3 hold also for f = v x . That is, v xxt exists and equals v xtx = v txx . Thus, (3.3) is proved for k = 0 (i.e., for r = 0). In view of (1.1), it is immediate that the last statement of Proposition 3.2 is also valid for r = 0. Next, assuming that the statements of Proposition 3.2 hold for all 0 ≤ r ≤ r 0 , let us prove them for r = r 0 + 1 . Differentiating both sides of (1.1) r 0 times with respect to t and taking into account (for r 0 > 0 and k ≤ r 0 − 1) the second equality in (3.3), we express ∂ r 0 +1 ∂t r 0 +1 v (x, 0) via derivatives which we already know. Then, from the first equality in (3.3), we obtain the formula 0) and an expression for ∂ r 0 +1 ∂t r 0 +1 v x (x, 0) follows. Differentiating both sides of (1.1) r 0 + 1 times and using (3.3) for r = r 0 , we see also that the derivative ∂ r 0 +2 ∂t r 0 +2 v exists and is continuous. Furthermore, differentiating (1.1) r 0 + 1 times with respect to t and once or twice with respect to x, from the last statement of our proposition (for the case r = r 0 ) we derive that the derivatives ∂ r 0 +2 ∂t r 0 +2 v x and ∂ r 0 +2 ∂t r 0 +2 v xx exist and are continuous in D(ε 4(r 0 +2) ). Now, we see that the conditions of Proposition 3.3 are fulfilled for f = ∂ r 0 +1 ∂t r 0 +1 v, and so the first equality in (3.3) holds for k ≤ r 0 + 1. Using this first equality in (3.3), we derive that the conditions of Proposition 3.3 are fulfilled for f = ∂ r 0 +1 ∂t r 0 +1 v x and therefore the second equality in (3.3) holds for k ≤ r 0 + 1. Differentiating again both sides of (1.1), we show that the last statement in Proposition 3.2 holds for r = r 0 + 1.
The class C { M k } consists of all infinitely differentiable on [0, a) scalar functions f such that for some c(f ) ≥ 0 and for fixed constants M k > 0 (k ≥ 0) we have Here, we use the notation M k (as well M below) because the upper estimates M (without tilde) were already used in Section 2. Recall that C { M k } is called quasi-analytic if for the functions f from this class and for any 0 ≤ x < a the equalities d k f dx k (x) = 0 (k ≥ 0) yield f ≡ 0. According to the famous Denjoy-Carleman theorem, the inequality implies that the class C { M k } is quasi-analytic.

Corollary 3.4. If v(x, t) satisfies conditions of Proposition 3.2 and the entries of v(0, t)
or v x (0, t) are quasi-analytic, then the matrix functions v(0, t) or v x (0, t), respectively, are uniquely defined by the initial condition (3.2).
Let us consider the case, where both matrix functions v(0, t) and v x (0, t) are quasi-analytic. More precisely, we assume that the entries v ij (0, t) of v(0, t) belong to some quasi-analytic classes  The results of this section are related to the results of [48,Section 3], where initial condition was determined by the boundary conditions. Remark 3.6. We see that the scheme to recover v (in the semi-strip D) from the initial condition follows from Proposition 2.5, Theorem 2.7 and the proof of Proposition 3.2. The only step that we did not describe in detail is the recovery of the functions v(0, t) and v x (0, t) from their Taylor coefficients at t = 0. Although Taylor coefficients uniquely determine quasi-analytic functions v(0, t) and v x (0, t), the recovery of these functions presents an interesting problem, which is not solved completely so far. See [7] and [11,Section III.8] for some important results.

Remark 3.7. An interesting class of such solutions of a scalar dNLS that the Weyl functions ϕ(t, z)
may be presented as the series ϕ(t, z) = ∞ k=0 α k (t)/z k (for sufficiently large values of z) was treated in [52]. (We note that the Weyl functions ϕ(z) from [52] are Herglotz functions and can be easily mapped into the Weyl functions ϕ(z) considered here via a linear-fractional transformation with constant coefficients.) According to [52,Theorem 1

there is one and only one solution of dNLS from this class in some semi-strip.
We note that important results on the asymptotics of Weyl functions are given, for instance, in [18,25]. However, it would be fruitful to know, under which conditions the asymptotic series (for Weyl functions) from [18,25] converge or at least uniquely define the corresponding Weyl function.
In other words, we express in these terms the evolution of the Weyl function. For that purpose, following formulas (2.8) and (2.9) in Proposition 2.6, we introduce matrix function R(t, z) by the relations are well-defined for 1 ≤ k < m, and the evolution of the Weyl function is given by the formula and by the normalization conditions (2.22).

Proof. We set
Recall that G and F for the case of the nonlinear optics equation are given by (2.13). Since ̺ is continuously differentiable, the conditions of Proposition 2.6 are fulfilled. Taking into account that the fundamental solution of (2.14) is denoted by w (instead of u in Proposition 2.6), we rewrite (2.10) in the form w(x, t, z) = R(x, t, z)w(x, 0, z)R(t, z) −1 or, equivalently, Using (4.5), we express w(x, t, z) −1 via w(x, t, z) −1 (0 ≤ t, t < a): Recall that R t = F R, where F is given in (2.13) and D > 0. Hence, putting R(x, t, t, z) := R(x, t, z)R(x, t, z) −1 −1 we derive From the relations above it is immediate that which allows us to estimate the difference According to (4.4), (4.8) and (4.9), for each δ > 0 and c > M there is ε = ε(M 0 ) > 0 such that Modifying (2.25) (so that the functions ψ k and w depend there also on an additional variable t), in view of (4.6), we derive ψ k (x, t, z) = I k 0 R(t, z)R( t, z) −1 w(x, t, z) −1 R(x, t, t, z)Q k (z) we see that for sufficiently small δ the matrix function Q k (z) satisfies (2.24) in the domain (for z) given in (4.10). Substituting Q k (instead of Q k ) into (2.25), we obtain Using the first equality in (4.12), we can substitute (4.13) into (4.11). Thus, we derive and in view of (2.27), passing to the limit x → ∞, we have the following formula (4.14) Here we used the fact that, according to (2.26) and (4.10), for sufficiently small δ. Setting t = kε (k = 0, 1, . . .), recalling that R(0, z) = I m and applying each time (4.14), we easily prove (by induction) the equality for t on all intervals [0, (k + 1)ε] ∩ [0, a), that is, for t on [0, a). Although (4.15) is proved for z ∈ {z : |z| < c} ∩ {z : ℑ(z) < −M}, the analyticity of both sides of (4.15) implies that the equality holds in the semi-plane ℑ(z) < −M. Finally, we note that (2.29) at t = 0 yields (4.16) Substituting (4.16) into (4.15), we obtain (4.2). The procedure to recover ϕ(t, z) from {ψ k (t, z)} (for fixed values of t) is described in Subsection 2.2. (note that (4.3) coincides with (2.28)). Now, let us prove a uniqueness result for the case of D of the form (1.10). Let initial condition be given by the equality Denote the Weyl function of system where ρ(t) = ̺(0, t). First, we show that the inequality is valid. Indeed, according to (2.26) and (4.15) we have Clearly, (4.21) yields the inequality Taking into account that R t = F R and relations (1.10), (2.13), (4.19) and (4.22) hold, we derive for ℑ(z) < 0. Formulas (4.21) and (4.23) imply that Recall that ϕ 0 is given by (2.28). Hence, (4.20) follows from (4.24). Consider system (4.1). Since ϕ 0 is bounded and satisfies (2.17) and (4.20), according to Corollary 2.12, the matrix function ζ(0, t) = [ D, ρ(t)] (and so R) is uniquely defined by ϕ 0 . Thus, from Theorem 4.1, we see that ϕ(t, z) is uniquely defined by ϕ 0 .
In order to prove our theorem, it remains to show that ϕ(t, z) satisfies conditions of Theorem 2.11 for each t. Indeed, in view of (4.1), we can rewrite for R the representation (2.30): (4.25) By virtue of (4.20), the matrix function R(t, z) ϕ 0 (z) e −izt D − I m is bounded in the domain ℑ(z) ≤ −M. Since R(t, z) satisfies (4.25) and M is chosen so that ϕ 0 satisfies (2.17) for z = ξ−iM, we see that R(t, ξ−iM)ϕ 0 (ξ−iM)e −i(ξ−iM )t D −I m ∈ L 2 m×m (−∞, ∞) for each 0 ≤ t < a, where L 2 m×m (0, ∞) is the class of m×m matrix functions, the entries of which belong to L 2 (0, ∞). Hence, the well-known Theorems V and VIII ( §4 and §5 in [36], respectively) on Fourier transform in complex domains yield the representation (see [49, (E11)]): It is immediate also that formula (4.2) can be modified slightly: According to (4.26), the normalized GW-function ϕ(t, z) constructed via equalities (4.3) and (4.27) satisfies conditions of Theorem 2.11. In other words, there is no more then one solution of ISpP for ϕ(t, z), that is, ̺(x, t) is unique.
Although Theorem 4.2 was announced in [40], its proof is published for the first time. It is essential to know, for which initial conditions ρ(x), the restrictions on ϕ 0 (from Theorem 4. where M 1 (x) is absolutely continuous.
Recall that in Theorem 4.2 we speak about the Weyl function ϕ 0 or, equivalently for a bounded function ρ = ρ * , about the normalized GW-function. Thus, we should normalize M(0, z). For that purpose we construct a lower triangular matrix function M(z) via the right lower k × k blocks P k (z) of M(0, z). Namely, we construct M(z) columnwise via the equalities  Since M(z) is lower triangular and D satisfies (1.9), we see that e izxD M(z)e −izxD is bounded for ℑ(z) < −M. Hence, taking into account that M(0, z) is a GW-function, we derive that (2.16) is also valid for ϕ(z) = M(0, z) M(z) (i.e., ϕ 0 given by (4.32) is the normalized GW-function). Finally, relations (4.29), (4.31) and (4.32) show that ϕ 0 is bounded and that (2.17) also holds for ϕ 0 . Thus, we proved the statement below. An existence result for a solution of an initial value problem (for the nonlinear optics equation) is given in [44,Remark 4.7].