SEMICLASSICAL NLS ON THE HALF LINE

Spyridon Kamvissis Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany spyros@mpim-bonn.mpg.de

ABSTRACT We are studying the semiclassical limit of the 1 + 1 dimensional integrable

nonlinear Schrödinger equation with defocusing cubic nonlinearity on the half line. Our analysis relies on the recent theory of Fokas et al. which reduces boundary value problems for soliton equations to Riemann-Hilbert factorisation problems. We employ the method of nonlinear steepest descent to asymptotically deform the given Riemann-Hilbert problem to an explicilty solvable one.

KEYWORDS Semiclassical NLS Riemann-Hilbert factorisation Boundary value problems

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1. AN INITIAL-BOUNDARY VALUE PROBLEM FOR THE NONLINEAR SCHRÖDINGER EQUATION

In recent years there has been a series of results by Fokas and others on boundary value problems for soliton equations (see [1] for a comprehensive review). The Fokas method goes beyond existence and uniqueness. In fact, it reduces such problems to Riemann-Hilbert factorisation problems in the complex plane, thus generalising the existing theory which reduces initial value problems to Riemann-Hilbert problems via the method of inverse scattering. One of the main advantages of the Riemann-Hilbert formulation is that one can use recent powerful results on the asymptotic behavior of solutions to these problems (as some parameter goes to infinity) to derive asymptotics for the solution of the associated soliton equation. Such methods were pioneered by A.Its and perfected by P.Deift and X.Zhou; the Deift-Zhou method is known as ”nolinear steepest descent” in analogy with the linear steepest descent method which is applicable to asymptotic problems for Fourier-type integrals (see e.g. [2]). A generalisation of the steepest descent method developed in [3] is able to give rigorous results for the so-called ”semiclassical” or ”zero dispersion” limit of the solution of the Cauchy problem for 1+1 dimensional integrable evolution equations, in the case where the Lax operator is self-adjoint. The method has been further extended in [4] for the ”non-self-adjoint” case.

In this paper we consider the most basic example, that is the defocusing nonlinear Schrödinger equation. (In a recent paper [5] we have dealt with the simple problem of so-called linearisable data, for both the defocusing NLS and KdV equations.) We make use of the recent results of [6] in order to study the so-called ”semiclassical” limit of a particular initial-boundary value problem. More precisely we consider the 1+1 dimensional, integrable, defocusing, nonlinear Schrödinger equation on the half-line

(1)  \begin{matrix}ihu_{t}(x, t) + h^{2}u_{xx}(x, t) − 2|u(x, t)|^{2}u(x, t) = 0, \\ u(x, 0) = 0, u(0, t) = f_{0}(t), u_{x}(0, t) = f_{1}(t), \\ x ≥ 0, t ≥ 0, \end{matrix}

SEMICLASSICAL NLS ON THE HALF LINE 3

where f_{0} is assumed smooth for simplicity. We also assume that all derivatives of f_{0}(t) vanish at t = 0. The special restriction of zero initial data is not essential but makes the computations and proofs somewhat easier.

The initial-boundary problem above is overdetermined. It is believed that the right (well-posed?) problem for the NLS equation on the half line should involve only u(x, 0) and u(0, t):

(1a)  \begin{matrix}ihu_{t}(x, t) + h^{2}u_{xx}(x, t) − 2|u(x, t)|^{2}u(x, t) = 0, \\ u(x, 0) = 0, u(0, t) = f_{0}(t), \\ x ≥ 0, t ≥ 0. \end{matrix}

However, following [6], we will eventually impose a compatibility condition (the socalled ”global relation”) on the data 0, f_{0}, f_{1} which will ensure the existence (and uniquness under such a condition) of a solution to (1). It is also worthwhile noting (see [6]) that given data u(x, 0) and u(0, t) only, the global relation implicitly selects a function f_{1}(t) = u_{x}(0, t) which complements the data f_{0} and the initial data, and which will then ensure the existence of a solution to (1a) and the validity of the Riemann-Hilbert formulation.

It is well known that the equation above admits a ”Lax-pair” formulation. It arises as the compatibility condition for the equations Lμ = 0 and Bμ = 0 where the operators L, B are given by

\begin{matrix}L = ( \begin{matrix}∂_{x} − ik & iu \\ −iū & ∂_{x} + ik \end{matrix} ) , \\ B = ( \begin{matrix}ih∂_{t} + 4ik^{2} + i|u|^{2} & −2ku − iu_{t} \\ −2kū + iū_{t} & ih∂_{t} − i|u|^{2} \end{matrix} ) . \end{matrix}

Here the bar denotes complex conjugation, k is the spectral variable and u = u(x, t) is the solution of (1).

The traditional method of solving initial value problems for soliton equations that admit a similar Lax-pair formulation is to focus on the L operator and apply the theory of scattering and inverse scattering to that very operator. On the other hand, one of the main ideas of the Fokas method is that for initial-boundary value problems the two operators L and B should be on an equal footing. The scattering transform should be applied to both operators simultaneously, while a global relation has to be imposed on the data to ensure compatibility.

2. THE RIEMANN-HILBERT PROBLEM

As shown in [6], problem (1) can be reduced to the following Riemann-Hilbert problem, under the special assumption that the so-called global relation holds (see relation (3.18) of [6]; see also relation (5) below). One way to look at the global relation is as a way of selecting a solution of problem (1a). In fact it is known by the work of Colliander and Kenig, using methods unrelated to inverse scattering theory, that (1a) has a solution. What is still unknown is whether this solution is unique. On the other hand, it has been shown in [6] that given data f_{0} in (1a), there exists a function f_{1}(t) such that the problem (1) admits a solution, which furthermore can be explicitly characterised via inverse scattering and a particular Riemann-Hilbert factorisation problem. Indeed, let Σ be the contour ℝ ∪ iℝ with the following orientation:

(i) the real axis is oriented from left to right,

(ii) the positive imaginary axis is oriented from infinity towards zero,

(iii) the negative imaginary axis is oriented from infinity towards zero.

We use the following convention: the +-side of an oriented contour is always to its left, according to the given orientation.

Letting M_{+} and M_{−} denote the limits of M on Σ from left and right respectively, we define the Riemann-Hilbert factorization problem

(2)  \begin{matrix}M_{+}(x, t, k) = M_{−}(x, t, k)J(x, t, k), \\ where \\ J(x, t, k) = J_{4}^{−1}, k ∈ ℝ^{+}, \\ J_{1}^{−1}, k ∈ iℝ^{+}, \\ J_{3}^{−1}, k ∈ iℝ^{−}, \\ J_{2} = J_{3}J_{4}^{−1}J_{1}, k ∈ ℝ^{−}, \\ with \\ J_{1} = ( \begin{matrix}1 & 0 \\ Γ(k)e^{2iΘ} & 1 \end{matrix} ) , \\ J_{4} = ( \begin{matrix}1 & −γ(k)e^{−2iΘ} \\ γ̄(k)e^{2iΘ} & 1 − |γ(k)|^{2} \end{matrix} ) , \\ J_{3} = ( \begin{matrix}1 & −Γ̄(k̄)e^{−2iΘ} \\ 0 & 1 \end{matrix} ) , \\ Θ(x, t, k) = \frac{θ}{h}, \\ where \\ θ = kx + 2k^{2}t. \end{matrix}

The functions γ, Γ are defined in terms of the spectral functions of the problem (see [6] (2.28), (2.25)), with important analyticity properties (see (2.21) and (2.22) of [6]). In particular note that

(3)  Γ(k) = \frac{1}{a(k)(a(k)\frac{Ā(k̄)}{B̄(k̄)} − b(k))},

where a, b are the spectral functions for the x-problem and A, B are the spectral functions for the t-problem. The functions a, b are analytic and bounded in the upper half-plane, while A, B are analytic and bounded in the first and third quadrants of the k-plane. For our special choice of zero initial data, b = 0, a = 1. Hence γ = 0 and Γ(k) = \frac{B̄(k̄)}{Ā(k̄)} .

The solution of (1) can be recovered from the solution of (2) as follows.

(4)  u(x, t) = 2i h lim_{k→∞}(kM^{12}(x, t, k)),

where the index 12 here denotes the (12)-entry of a matrix.

The following ”global relation” is imposed on the scattering data:

a(k)B(k) − b(k)A(k) = e^{4ik^{2}T} c(k),

where c(k) is analytic and bounded for Imk > 0, and c(k) = O(1/k) as k → ∞. Here T is the time up to which we solve the initial boundary value problem for NLS. In general A, B are functions of T.

If u_{x}(0, t) is denoted by f_{1}(t) then there is a complicated relation between the data f_{0} and f_{1}; the global relation is the expression of this in the spectral space.

In our particular case (problem (1)) T = ∞ and the global relation becomes

(5)  a(k)B(k) − b(k)A(k) = 0,

for arg(k) ∈ [0, π/2]. For the special choice of zero initial data, since b = 0, a = 1, one has B = 0 for arg(k) ∈ [0, π/2]. In particular, Γ(0) = 0.

We note that in both the negative half-line and the positive half-line the jump matrix is of the same form. For positive k,

J = ( \begin{matrix}1 − |γ(k)|^{2} & γ(k)e^{−2iΘ} \\ −γ̄(k)e^{2iΘ} & 1 \end{matrix} ) ,

while for negative k,

J = ( \begin{matrix}1 − |γ(k) − Γ̄|^{2} & (γ(k) − Γ̄)e^{−2iΘ} \\ −(γ̄(k) − Γ)e^{2iΘ} & 1 \end{matrix} ) ,

In fact, let

\begin{matrix}r(k) = γ(k) − Γ̄(k) = \frac{bA − aB}{āA − b̄B} , k < 0, \\ r(k) = γ = \frac{b(k)}{ā(k)}, k ≥ 0. \end{matrix}

Then, for all non-zero real k,

J = ( \begin{matrix}1 − |r(k)|^{2} & r(k)e^{−2iΘ} \\ −r̄(k)e^{2iΘ} & 1 \end{matrix} ) ,

In the special case of zero initial data, the jump reduces to the identity for positive k, while for negative k, r = \frac{−B}{A}, so

(6)  J = ( \begin{matrix}1 − |\frac{B}{A} |^{2} & \frac{−B}{A} e^{−2iΘ} \\ \frac{B̄}{Ā} e^{2iΘ} & 1 \end{matrix} ) .

3. DIRECT SCATTERING AS h → 0

It is important to have some information about the ”spectral” coefficent r = −B/A, for real values of k.

THEOREM 1. For k < 0, the spectral function r(k, h) has the following asymptotic expression. There exist functions r̃, R_{0} of k alone, such that

(7)  r(k, h) ∼ r̃(k)e^{\frac{2iR_{0}(k)}{h}} ,

as h → 0, where r̃(k) is analytic and bounded as k → ∞, and R_{0}(k) is analytic. When k ∈ iℝ^{+}, Re(iR_{0}) ≤ 0. Also |r̃| ≤ 1.

PROOF: The representation (7) follows from standard WKB theory. Indeed, A, B admit representations of the form s(k)exp(iR(k)/h). Formula (7) thus follows. The analyticity of r̃, R_{0} follows from the analyticity of A, B. The fact that |A(k)|^{2}− |B(k)|^{2} = 1 (for real k) implies |r̃| ≤ 1.

More detailed information about r can be recovered after a detailed analysis of the spectral problem for the second Lax operator. An easy calculation shows that the associated spectral problem reduces to a WKB problem of the type

h^{2}y_{tt} = S(t, k)y,

where S(t, k) is real. The spectral coefficients A, B can then be asymptotically estimated along the lines of [7] (section 10.6). Eventually one is able to show the following.

THEOREM 2. Let

f_{2}(t) = \frac{5}{8 · 2^{1/3}} [Re(if̄_{1}(t)f_{0}(t))]^{4/3} + |f_{0}(t)|^{4} − |f_{1}(t)|^{2}.

Let −f = min{f_{2}(t)} over the interval [0, ∞]. Without loss of generality, we assume −f < 0. (Otherwise the analysis becomes trivial; the coefficent r(k, h) is everywhere small.) On the real line, the following holds.

For −f < k < 0,

r(k, h) ∼ −ie^{\frac{2iσ(k)}{h}} ,

where σ is smooth in (−f, 0) and takes real values. Also σ can be extended analytically in a small neighbourhood of the segment (−f, 0).

For other real values of k, r(k, h) is either zero or uniformly exponentially small in h.

Furthermore one has an asymptotic formula for 1−|r(k)|^{2}, −f < k < 0. Indeed,

1 − |r(k)|^{2} ∼ exp(\frac{−2τ(k)}{h}),

where τ(k) is positive and can be extended analytically in a small neighbourhood of (−f, 0).

REMARK. An explicit integral formula for σ and τ in terms of the data can only be found assuming u_{x} = f_{1} is known. In general, even though the existence of f_{1} is guaranteed (given f_{0}) via the global relation, f_{1} is not effectively computable.

However, in the sepcial case of so-called ”linearisable data”, f_{1} and σ are indeed effectively computable (see [5]).

PROOF OF THEOREM 2. We simply note that the turning curve for the t-spectral problem is given by

S(t, k) = 4k^{4} + kRe(if̄_{1}(t)f_{0}(t)) + |f_{0}(t)|^{4} − |f_{1}(t)|^{2} = 0.

Indeed consider the t-problem

By = ( \begin{matrix}ih∂_{t} + 4ik^{2} + i|f_{0}|^{2} & −2kf_{0} − if_{1} \\ −2kf̄_{0} + if̄_{1} & ih∂_{t} − i|f_{0}|^{2} \end{matrix} ) y = 0.

Applying the operator

B^{0} = ( \begin{matrix}ih∂_{t} − i|f_{0}|^{2} & 2kf_{0} + if_{1} \\ −2kf̄_{0} − if̄_{1} & ih∂_{t} + 4ik^{2} + i|f_{0}|^{2} \end{matrix} )

we end up, up to errors of order O(h), with

(8)  h^{2}y_{tt} = S(t, k)y,

with S(t, k) as above.

Note here that real k for which there exist L_{2}-solutions y of By = 0, are a priori excluded (see [6] p.16). So we do not need to concern ourselves with the possibility

SEMICLASSICAL NLS ON THE HALF LINE 9

of real k for which the solutions to (8) are in L_{2}. We can then follow the WKB analysis of the semiclassical Schrödinger operator without essential changes (e.g.

[7] section 10.6). We can thus show that at all k such that S(t, k) = k for some t, the reflection and transmission coefficients are given by the formulae above, while otherwise r(k, h) is exponentially small (or zero). A short calculation shows that S(t, k) as a function of k has only one local minimum, at f_{2}(t), as defined in the statement of Theorem 2. The result follows immediately.

4. REDUCING TO A PROBLEM ON THE REAL LINE.

We next consider two Riemann-Hilbert problems with sole jumps given by

\begin{matrix}J_{1}^{−1}, k ∈ iℝ^{+}, \\ J_{3}^{−1}, k ∈ iℝ^{−}, \end{matrix}

respectively.

We want U to be a function analytic in the complex plane except the upper imaginary axis, with normalization lim_{k→∞}U(x, t, k) = I. The jump is prescribed by

(9)  \begin{matrix}U_{+}(x, t, k) = U_{−}(x, t, k)J_{1}^{−1}(x, t, k), k ∈ iℝ^{+}, \\ with \\ J_{1} = ( \begin{matrix}1 & 0 \\ Γ(k)e^{2iΘ} & 1 \end{matrix} ) , \\ Θ(x, t, k) = \frac{θ}{h}, \\ where \\ θ = kx + 2k^{2}t. \end{matrix}

(The symbol U stands for ”upper” since the jump is on the upper half-plane. We will immediately see however that U is a lower triangular matrix.)

Similarly, we want L to be a function analytic in the complex plane except the lower imaginary axis, with normalization lim_{k→∞}L(x, t, k) = I. The jump is prescribed by

(10)  \begin{matrix}L_{+}(x, t, k) = L_{−}(x, t, k)J_{3}^{−1}(x, t, k), k ∈ iℝ^{−}, \\ with \\ J_{3} = ( \begin{matrix}1 & −Γ̄(k̄)e^{−2iΘ} \\ 0 & 1 \end{matrix} ) . \end{matrix}

(The symbol L stands for ”lower” since the jump is on the upper half-plane. However L is an upper triangular matrix.)

The two Riemann-Hilbert problems above can be easily solved explicitly, since the jumps are triangular matrices. Indeed, direct calculations show that

(11)  \begin{matrix}U(x, t, k) = ( \begin{matrix}1 & 0 \\ u(x, t, k) & 1 \end{matrix} ) , \\ where \\ u(x, t, k) = \frac{1}{2πi} ∫_{iℝ^{+}} \frac{Γ(s)e^{2iΘ(x,t,s)}ds}{s − k} \end{matrix}

satisifies (9). Similarly,

(12)  \begin{matrix}L(x, t, k) = ( \begin{matrix}1 & l(x, t, k) \\ 0 & 1 \end{matrix} ) , \\ where \\ l(x, t, k) = \frac{1}{2πi} ∫_{iℝ^{−}} \frac{−Γ̄(s̄)e^{−2iΘ(x,t,s)}ds}{s − k} \end{matrix}

satisfies (10). The direction of the integral contours is as prescribed in section 2, i.e. from infinity to zero.

Note that u(k) = −l̄(k̄). Note also that the integrals in (11) and (12) are not singular at k = 0, as Γ(0) = 0.

We next show that the Riemann-Hilbert problem (2) is equivalent to a problem on the real line. Indeed, let

(13)  \begin{matrix}N(x, t, k) = M(x, t, k)U^{−1}(x, t, k), Imk > 0, \\ N(x, t, k) = M(x, t, k)L^{−1}(x, t, k), Imk < 0. \end{matrix}

Then N(x, t, k) is analytic in ℂ \ ℝ, with lim_{k→∞}N(x, t, k) = I, and across ℝ the jump is given by

(14)  N_{+}(x, t, k) = N_{−}(x, t, k)L(x, t, k)J(x, t, k)U^{−1}(x, t, k).

In fact, the new jump is given by

(15)  LJU^{−1} = ( \begin{matrix}1 − |l(k) + r(k)e^{−2iΘ}|^{2} & l(k) + r(k)e^{−2iΘ} \\ −l̄(k) − r̄(k)e^{2iΘ} & 1 \end{matrix} ) .

Note here that while r(k) depends only on k but not on x, t, l(k) depends on x, t, h via Θ by (12).

We have thus reduced the Riemann-Hilbert problem (2) to the problem (15) with only jump on the real line.

5. THE g-FUNCTION

We next show how the Riemann-Hilbert problem can be ”deformed” to a problem that is explicitly solvable. We are essentially following the ideas of [3] (see also [4]).

The first idea involves the so-called ”g-function”. We introduce a scalar function g(k) which is to be analytic in ℂ \ ℝ and decay like O(\frac{1}{k}) at infinity. This function will be uniquely specified eventually.

Let O(k) = N(k)exp(\frac{ig(k)σ_{3}}{h}). If N satisfies N_{+} = N_{−}J, k < 0, with J given by (6), then O solves a Riemann-Hilbert problem with jump matrix v_{O}, say, that is (16)

O_{+}(x, t, k) = O_{−}(x, t, k)v_{O}(x, t, k),

v_{O}(k) = ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} (1 − |l + \frac{B}{A} e^{−2iΘ}|^{2}) & (l − \frac{B}{A} e^{−2iΘ})e^{\frac{−i(g_{+}+g_{−})}{h}} \\ (−l̄− \frac{B̄}{Ā} e^{2iΘ})e^{\frac{i(g_{+}+g_{−})}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) , k < 0,

v_{O}(k) = ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} (1 − |l|^{2}) & l(k)e^{\frac{−i(g_{+}+g_{−})}{h}} \\ −l̄(k)e^{\frac{i(g_{+}+g_{−})}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) , k > 0,

lim_{k→∞}O(k) = I.

Here g_{+}, g_{−} denote the limits of g from above and below the negative real axis respectively.

Note that problem (16) is exactly (not just asymptotically) equivalent to the original Riemann-Hilbert problem (2). Formula (4) has to be replaced by

(17)  u(x, t) = 2i h lim_{k→∞}(kO^{12}(x, t, k)) + 2∂_{x}g̃,

where g̃ is the residue of g at infinity.

6. REDUCTION TO A SOLVABLE RIEMANN-HILBERT PROBLEM

Our first approximation involves getting rid of the functions l, u appearing in the jumps. The reason is simple. By formulae (11)-(12) u, l can be shown to be at worst O(h) by use of the Laplace method for asymptotic evaluation of integrals, since the phase iR_{0} of Theorem 1, has negative or zero real part. This means that l can be erased from formula (16).

More precisely, an easy calculation shows that the jump v_{O} of (16) can be written as (18)

v_{O} = ( \begin{matrix}1 & le^{\frac{2ig_{+}}{h}} \\ 0 & 1 \end{matrix} )

· ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} (1 − |B/A|^{2}) & −\frac{B}{A} e^{−2iΘ}e^{\frac{−i(g_{+}+g_{−})}{h}} \\ −\frac{B̄}{Ā} e^{2iΘ}e^{\frac{i(g_{+}+g_{−})}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) · ( \begin{matrix}1 & 0 \\ −l̄e^{\frac{−2ig_{−}}{h}} & 1 \end{matrix} ) , k < 0,

v_{O} = ( \begin{matrix}1 & le^{\frac{2ig_{+}}{h}} \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}1 & 0 \\ −l̄e^{\frac{−2ig_{−}}{h}} & 1 \end{matrix} ) , k > 0.

Since g takes real values on ℝ (this will be clear later, see formula (28)) and since l = O(h), it follows that the triangular factors in (18) can be taken as the identity plus a resulting error of order at worst O(h) in formula (18).

We have asymptotically reduced Riemann-Hilbert problem (16) to a new Riemann-Hilbert problem for a matrix function Q(z), say.

If Q is defined by

(19)  \begin{matrix}Q_{+}(x, t, k) = Q_{−}(x, t, k)v_{Q}(x, t, k), \\ v_{Q}(k) = ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} (1 − |B/A|^{2}) & −\frac{B}{A} e^{−2iΘ}e^{\frac{−i(g_{+}+g_{−})}{h}} \\ \frac{B̄}{Ā} e^{2iΘ}e^{\frac{i(g_{+}+g_{−})}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) , k < 0, \\ lim_{k→∞}Q(k) = I, \end{matrix}

then Q is asymptotically equivalent to O in a neihbourhood of ∞. In particular,

(20)  u(x, t) ∼ 2ihlim_{k→∞}(kO^{12}(x, t, k)) + 2∂_{x}g̃.

The matrix v_{Q} can be written as

(21) \begin{matrix}v_{Q} = ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} (1 − |r|^{2}) & r(k)e^{−2iΘ}e^{\frac{−i(g_{+}+g_{−})}{h}} \\ −r̄(k)e^{2iΘ}e^{\frac{i(g_{+}+g_{−})}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) = \\ ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})−2τ}{h}} & −ie^{\frac{−2iθ−i(g_{+}+g_{−})−2iσ}{h}} \\ −ie^{\frac{2iθ+i(g_{+}+g_{−})+2iσ}{h}} & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) , if − f < k < 0, \\ ∼ ( \begin{matrix}e^{\frac{i(g_{+}−g_{−})}{h}} & 0 \\ 0 & e^{\frac{−(ig_{+}−ig_{−})}{h}} \end{matrix} ) , otherwise. \end{matrix}

At this point it becomes obvious that we should also impose

g_{+} − g_{−} = 0, k > 0, or k < −f.

So g is to be analytic in ℂ \ [−f, 0].

Let

H = −g_{+} − g_{−} − 2θ − 2σ.

In the spirit of [3], we seek to reduce v_{Q} to a jump of one of the three following types:

(22) ( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 1 \end{matrix} ) , ( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 0 \end{matrix} ) , ( \begin{matrix}1 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 0 \end{matrix} ) .

The motivation is the following. We expect that (22) will be deformable to a RH problem that can be explicitly solvable in terms of finite genus theta functions. Such a problem will have a ”finite gap” structure. This means that the real line will be divided into a finite number of subintervals. In some of them the jump matrix has to look like

( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 0 \end{matrix} )

and in others it has to be the identity. We know however (through ”lens”-type arguments, see Appendix) that matrices of the first or third form in (22) can be reduced to the identity. Hence the ansatz (22).

To arrive from (21) to (22) we impose some seemingly artificial conditions on the functions g, H. For any given x, t, we will consider finite sequences of real numbers −1 ≤ k_{1} < k_{2} ≤ k_{3} < k_{4} ≤ .... ≤ k_{2G+1} < k_{2G+2} ≤ 1. We call the G + 1 intervals I_{1} = [k_{1}, k_{2}], ...., I_{G+1} = [k_{2G+1}, k_{2G+2}] the ”bands”. Both the integer G and the real numbers k_{j}, j = 1, ..., 2G + 2 are to be eventually determined for each x, t.

We make the ansatz that for each x, t the interval [−f, 0] can be subdivided into intervals by such k_{j}, such that on each of the arising intervals one of the three conditions holds:

(23)  \begin{matrix}−2iτ = g_{+} − g_{−} and H′ < 0, \\ or \\ −2iτ < g_{+} − g_{−} < 0, and H′ = 0, \\ or \\ g_{+} − g_{−} = 0, and H′ > 0. \end{matrix}

H′ denotes the derivative of H with respect to k. In particular, the intervals where H′ = 0 are to be the bands I_{j}, while on the intervals of which [−f, 0] \ ∪I_{j} consists, either the first or the third condition has to hold.

We will eventually see that conditions (23) amount to a scalar Riemann-Hilbert problem that can be solved explicitly, plus a set of algebraic conditions on the endpoints k_{j} defining the gap structure, plus a set of inequalities which essentially pick up the appropriate number of k_{j}’s.

Now differentiating (with respect to k) the scalar Riemann-Hilbert problem given by the equalites in (23) and solving for g′ leads to

g′(k) = (p(k))^{1/2}(∫_{∪I_{j}} \frac{2σ′(μ) − 2θ′(μ)}{(p(μ))_{+}^{1/2}(μ − k)} \frac{dμ}{2πi} + ∫_{(−f,0)\∪I_{j}} \frac{−2iτ′(μ)}{(p(μ))_{+}^{1/2}(μ − k)} \frac{dμ}{2πi}),

where

(24)  p(k) = Π_{j=1}^{G+1}(k − k_{2j−1})(k − k_{2j}).

We have imposed the condition g(k) = O(k^{−1}), as k → ∞. Easy calculations then show that g′ has to satisfy the moment conditions

(25)  \begin{matrix}∫_{∪I_{j}} \frac{σ′(k) − θ′(k)}{(p(k))_{+}^{1/2}}k^{l}dk + ∫_{(−f,0)\∪I_{j}} \frac{−iτ′(k)}{(p(k))_{+}^{1/2}} k^{l}dk = 0, \\ l = 0, 1, 2, ....., G. \end{matrix}

Also, integrating g′ around I_{j} and requiring H′ < 0, we obtain

(26)  ∫_{I_{j}}(g′_{+} − g′_{−})dλ = −2i(τ(k_{2j−1}) − τ(k_{2j})); j = 1, ......, G + 1.

Conditions (25)-(26) form a set of 2G+2 equations for 2G+2 unknowns. They enable us to solve for k_{j}.

At this point, we note that H is smooth in [−f, 0]. We also note that it admits analytic continuations in (possibly small) lens-like domains of the complex plane, not including the points k_{j}.

In fact, conditions (25)-(26) together with the inequalities in (23) reduce to the Euler-Lagrange conditions of a variational problem. This is virtually the same variational problem introduced by Lax and Levermore and the existence and uniqueness of its solution is guaranteed by the theory of variational problems of logarithmic potentials (see e.g. [8]).

Indeed, assuming appropriate decay and regularity conditions on the data f_{0}, for example having f_{0} real analytic and rapidly decaying, one can prove that for each x, t there is a G for which both equalities and inequalities in (23) have a solution. In other words, the ”finite genus ansatz” can be eventually justified.

Once the existence of an appropriate ”g-function” is guaranteed, it is straightforward to reduce our Riemann-Hilbert problem to its final form.

At the end of this procedure, and because of conditions (23), the jump contour consists of the bands I_{j}, j = 1, .., G + 1 and on each band, the jump matrix is given by

w_{j} = ( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 0 \end{matrix} ) .

Furthermore, H is constant on each band. We actually have

(27)  w_{j} = ( \begin{matrix}0 & −ie^{\frac{iΩ_{j}}{h}} \\ −ie^{\frac{−iΩ_{j}}{h}} & 0 \end{matrix} ) ,

where the Ω_{j} are real constants.

The Ω_{j} can be computed explicitly. But first, let us note that the Riemann-Hilbert problem with jumps along the intervals I_{j} given by (27) can be explicitly solved in terms of theta functions. To appropriately define those functions we first need to introduce an underlying Riemann Surface, together with some associated holomorphic differentials.

Let X be the two-sheeted Riemann surface of genus G associated to (p(k))^{1/2}, obtained by adjoining two copies of the slit plane ℂ \ ∪_{k}I_{k}. On the ”top” sheet (p(k))^{1/2} ∼ k^{G+1} and on the ”bottom” sheet (p(k))^{1/2} ∼ −k^{G+1}. The branch points of the surface will be the end-points of the ”bands”, that is, k_{1}, .....k_{2G+2}. The homology cycles are defined in a standard way as follows. The cycles A_{k} lie on the top sheet and encircle slits I_{k}. The cycles B_{k} pass from the top sheet through the slit I_{1} to the bottom sheet and back again through I_{k}.

The basis ω = (ω_{1}, ...., ω_{G}) of holomorphic differentials on X is determined by the normalization

∫_{A_{i}} ω_{j} = δ_{ij}, 1 ≤ i, j ≤ G.

The Riemann-matrix of periods is

τ = (τ_{ij}) = (∫_{B_{i}} ω_{j})_{1≤i,j≤G}.

By the Riemann bilinear relations, τ is symmetric and iτ is negative definite. We can thus define the associated theta function

θ_{G}(s) = Σ_{m∈ℤ^{G}}exp(2πi(m, s) + πi(m, τm)), s ∈ ℂ^{G},

where (,.,) is the real scalar product. Note that θ_{N} is an even function.

Now, solving the scalar Riemann-Hilbert for g (not its derivative) we get (28)

g(k) = (p(k))^{1/2}(∫_{∪I_{j}} \frac{2σ(μ) − 2θ(μ) − Ω_{j}}{(p(μ))_{+}^{1/2}(μ − k)} \frac{dμ}{2πi} + ∫_{(−f,0)\∪I_{j}} \frac{−2iτ(μ)}{(p(μ))_{+}^{1/2}(μ − k)} \frac{dμ}{2πi}).

Applying the condition that g(k) = O(k^{−1}), as k → ∞ once more, we get the conditions

\begin{matrix}∫_{∪I_{j}}(2σ(k) − 2θ(k) − Ω_{j})ω_{l} + ∫_{(−f,0)\∪I_{j}}(−2iτ(k))ω_{l} = 0, \\ l = 1, 2, ....., G. \end{matrix}

Recalling the definition of the normalized differentials ω_{l}, we immediately get the following formulae.

(29)  \begin{matrix}Ω_{l} = ∫_{∪I_{j}}(2σ(k) − 2θ(k))ω_{l} + ∫_{(−f,0)\∪I_{j}}(−2iτ(k))ω_{l}, \\ l = 1, 2, ....., G. \end{matrix}

We also define the following function

ζ(k) = [Π_{i=1}^{G+1} \frac{k − k_{2i}}{k − k_{2i−1}} ]^{1/4},

where ζ is meant to be analytic off the union of the ”gaps”, i.e. the intervals between the bands, and ζ(k) ∼ 1, as k → ∞. The function ζ has the important property that ζ ± ζ^{−1} has G + 1 roots (ζ_{j}^{±})_{j=1}^{G+1}, lying in the bands I_{j}, one root in each band. Note also that ζ_{+} = iζ_{−} across the gaps.

We next define the ‘Abel map’ integral, for k on the top sheet of the Riemann surface X. Let

u(k) = ∫_{0}^{k} ω,

where the integral is taken along any path on the top sheet. Note that it is welldefined modulo ℤ^{G}. Also define the constant vector

d = −K − Σ_{j=1}^{G} ∫ P_{2}(ζ_{j})_{0} ω,

where K is the vector of Riemann constants and P_{2}(z) denotes the preimage of a point z ∈ X in the ”bottom” sheet. We can now state the following theorem.

THEOREM 3. The function Q defined by problem (19) is asymptotically equivalent, as h → 0, to

(30)  \begin{matrix}diag(\frac{θ_{G}(u(∞) + d)}{θ_{G}(u(∞) + \frac{Θ_{G}}{2πh} + d)}, \frac{θ_{G}(−u(∞) + d)}{θ_{G}(−u(∞) + \frac{Θ_{G}}{2πh} − d)}) \\ · ( \begin{matrix}\frac{ζ+ζ^{−1}}{2} \frac{θ_{G}(u(k)+ \frac{Θ_{G}}{2πh} +d)}{θ_{G}(u(k)+d)} & e^{\frac{−iH_{G+1}}{h}} \frac{ζ−ζ^{−1}}{2i} \frac{θ_{G}(u(k)+ \frac{Θ_{G}}{2πh} −d)}{θ_{G}(u(k)−d)} \\ e^{\frac{iH_{G+1}}{h}} \frac{ζ−ζ^{−1}}{−2i} \frac{θ_{G}(−u(k)+ \frac{Θ_{G}}{2πh} +d)}{θ_{G}(−u(k)+d)} & \frac{ζ+ζ^{−1}}{2} \frac{θ_{G}(−u(k)+ \frac{Θ_{G}}{2πh} −d)}{θ_{G}(−u(k)+d)} \end{matrix} ) , \end{matrix}

where Θ_{G} = (Ω_{1}, ...., Ω_{G})^{T} , the Ω_{j} being given by (29). The asymptotics is uniform in compact subsets of the Riemann Sphere with the bands I_{j} deleted.

The proof consists of a straightforward check of the jump relations. The important fact is that (because of our choice of d) the zeroes of ζ ± ζ^{−1} exactly cancel the poles of the theta function quotients.

The semi-classical asymptotics for the solution of (1) follows from (30) and (4). THEOREM 4. The asymptotics for u(x, t, h), the solution of (1), as h → 0, is

given by

(31) u(x, t, h) ∼ [Σ_{j=1}^{2G+2}k_{j}]e^{\frac{−iH_{G+1}}{h}} · \frac{θ_{G}(u(∞) + d)θ_{G}(u(∞) + Θ_{G}/(2πh) − d)}{θ_{G}(u(∞) − d)θ_{G}(u(∞) + Θ_{G}/(2πℏ) + d)}.

Formula (31) expresses a slowly modulated wave.

7.CONCLUSION

Since we have been able to reduce our Riemann-Hilbert problem to one that arises in the full-line problem, the results of [9], [10], [3] on the phenomenology of the solution as h → 0 apply.

Semiclassically, the half-plane x, t ≥ 0 can be divided in two regions. In the first (”smooth”) region the strong semiclassical limit exists and satisfies the formallly limiting Euler system. In the second (”turbulent”) region fast oscillations appear that can be described in terms of slowly modulating finite-gap solutions. Weak limits of an infinite number of densities including |u|^{2}, ih(uū_{x} − u_{x}ū) exist.

We also note that the Whitham equations theory is still relevant. The functions k_{j}(x, t) are in fact the Riemann invariants of the Whitham equations. The equations themselves can be derived by differentiating (25) and (26) (see e.g. [4]).

Let us also note that, even though the assumption that the initial data are equal to zero makes the analysis somewhat easier, it is not essential. In particular the above qualitative discussion of the semiclassical limit is still valid.

Finally, let us comment on the long time asymptotics of the semiclassical limit. There are two ways of computing the long time semiclassical limit of the defo-

cusing NLS on the full line (see [11]). One is to use the existing theory for times of order 1 (as in [10]) and take the limit t → ∞. Alternatively, because the limits

SEMICLASSICAL NLS ON THE HALF LINE 19

ϵ → 0 and t → ∞ commute, one can simply look at the long time behaviour of the problem with fixed ϵ and then take ϵ → 0. The second choice is in fact easier and is applied in [11] to compute the long time semiclassical limit of the defocusing NLS on the full line.

On the half-line, it is already known what the long time of the problem with fixed ϵ is. As in the full-line case, any initial data degenerates into a sequence of finitely many separated solitons (see [12], [6]).

It then follows, in exact analogy with the full line case [11], that the long time asymptotics of the semiclassical limit in the half line case can be described by a sequence of solitons (in the turbulent region). The number of solitons is finite but increasing like O(1/ϵ) as ϵ → 0. Their width is O(ϵ) and they are separated by a distance of order O(ϵt). In the smooth region, the solution simply dies out.

A. APPENDIX. THE ”LENS” ARGUMENT

Suppose we have the following Riemann-Hilbert problem. We are seeking a matrix L, analytic in the complex plane except for a jump along the real interval [α, β], oriented from left to right. The normalization at infinity is to be lim_{k→∞}L = I, and the jump across [α, β] is

L_{+} = L_{−} ( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 1 \end{matrix} ) ,

where \frac{dH}{dk} < 0. We also assume that H is real on [α, β] and admits an analytic continuation in a small ”lens”-like domain bouned by two analytic arcs C_{u}, C_{l} joining the points α, β (in that direction) and lying entirely in the upper and lower half-planes respectively. We note the following factorization of the jump matrix

( \begin{matrix}0 & −ie^{\frac{iH}{h}} \\ −ie^{\frac{−iH}{h}} & 1 \end{matrix} ) = ( \begin{matrix}1 & −ie^{\frac{iH}{h}} \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}1 & 0 \\ −ie^{\frac{−iH}{h}} & 1 \end{matrix} ) .

This suggests the following definition. Let L′ = L, outside the domain bounded by C_{u} ∪ C_{l},

L′ = L · ( \begin{matrix}1 & 0 \\ ie^{\frac{−iH}{h}} & 1 \end{matrix} ) , between [α, β] and C_{u}.

L′ = ( \begin{matrix}1 & −ie^{\frac{iH}{h}} \\ 0 & 1 \end{matrix} ) · L, between [α, β] and C_{l}.

The Riemann-Hilbert problem for L′ is as follows.

\begin{matrix}L′_{+} = L′_{−} · ( \begin{matrix}1 & 0 \\ ie^{\frac{−iH}{h}} & 1 \end{matrix} ) , k ∈ C_{u}. \\ L′_{+} = L′_{−} · ( \begin{matrix}1 & −ie^{\frac{iH}{h}} \\ 0 & 1 \end{matrix} ) , k ∈ C_{l}. \end{matrix}

Now, since \frac{dReH}{dk} < 0 on the interval [α, β], by the Cauchy-Riemann relations \frac{dImH}{dk} < 0 across the interval [α, β], in the positive imaginary direction. This means that ImH < 0 on C_{u} if C_{u} is chosen to be close enough to [α, β], except at the endpoints α, β. Similarly ImH > 0 on C_{l} if C_{l} is chosen to be close enough to [α, β], except at the endpoints α, β. Hence,

\begin{matrix}Re(−iH) < 0, k ∈ C_{u}, \\ Re(−iH) > 0, k ∈ C_{l}, \end{matrix}

except at the endpoints α, β. In other words the jump matrix for L′ is the identity plus an exponentially small quantity, at least away from the endpoints α, β. This implies that the contour C_{u} ∪C_{l} can be erased, at least away from the the endpoints α, β.

Near the endpoints one can use a parametrix argument, which we omit (see for example [4] for details).

ACKNOWLEDGEMENTS

We thank Professor Fokas for showing us preprint [6] and for illuminating discussions and suggestions.

BIBLIOGRAPHY

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SEMICLASSICAL NLS ON THE HALF LINE 21

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[10] S. Jin, C. D. Levermore, and D. W. McLaughlin, The Behavior of Solutions of the NLS Equation in the Semiclassical Limit, in Singular Limits of Dispersive Waves, N. M. Ercolani et. al. editors, NATO ASI Series, volume 320, Plenum Press, New York, 1994; The Semiclassical Limit of the Defocusing NLS Hierarchy, Communications on Pure and Applied Mathematics, v.52, n.5, 1999, pp.613-654.

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