ON THE QUESTION OF ANALYTICITY OF DATA FOR SEMICLASSICAL NLS

Spyridon Kamvissis

ABSTRACT

In a recent book [KMM], Kamvissis, K. McLaughlin, and P. Miller analyze the semiclassical behavior of solutions to the focusing, completely integrable nonlinear Schrödinger equation, under the assumption of real analytic initial data. In this article, we show that the constraint of analyticity is unnecessary.

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In a recent study [KMM], the authors have analyzed the semiclassical behavior of the solution u(x, t) to the initial value problem for the focusing NLS equation:

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

where h is small, and A(x) is a real valued function, sufficiently decaying at infinity so that the theory of inverse scattering applies. Furthermore A(x) was assumed to be ”bell-shaped”, i.e. even, with a single maximum at x = 0, and not too flat at x = 0, i.e. A”(0) < 0. Finally, A(x) was assumed to be analytic.

Under these assumptions, several results have been proved. Most importantly,

1. The existence of strong limits for the densities ρ = |ψ|^{2} and μ = hIm(ψ^{∗}ψ_{x}), has been proved, as h → 0, for short times.

2. Asymptotics for the solution u have been provided as h → 0, for any x, t, as long as the so-called ”finite gap ansatz” is valid. The existence of weak limits for ρ, μ, as h → 0, for any x, t, follows.

3. The validity of the finite gap ansatz has been justified, under assumptions of genericity.

In this article, we show that the assumption of analyticity can be dropped.

THEOREM. The results of [KMM] are valid for the semiclassical analysis of (1), under the assumption that A(x) is a C^{2} bell-shaped function decaying rapidly enough at infinity. The first result has to be qualified; it may not hold at points of non-analyticity of the initial data.

SKETCH OF PROOF:

It is essential for the proofs in [KMM] that the ”density of eigenvalues” ρ^{0}(η) (see (3.2) of [KMM]), derived by WKB theory and a priori defined in the straight line interval connecting 0 to i, be analytically extensible to the closed upper half-plane ℍ. The main issue is whether the function

(2)  R^{0}(η) = ∫_{x_{−}(η)}^{x_{+}(η)}(A(x)^{2} + η^{2})^{1/2}dx,

where the turning points are defined by

(3)  \begin{matrix}A(x_{±}(η)) = −iη, 0 < −iη < 1, \\ −1 < x_{−}(η) < 0 < x_{+}(η) < 1, \end{matrix}

admits an analytic extension. We note here that we choose the branch of the square root that is positive for x_{−} < x < x_{+}.

We will show that even if R^{0} does not admit an analytic extension in ℍ, the analysis of Chapter 5 in [KMM] can be amended via the solution of a scalar Riemann-Hilbert problem.

Indeed, consider the following scalar additive Riemann-Hilbert problem, with jump on the linear segment Σ = [−i, i]. Let p be a function analytic in ℂ \ [−i, i], such that

p_{+}(η) + p_{−}(η) = ρ_{0}(η) = \frac{dR^{0}}{dη} , η ∈ Σ.

Here R^{0}(η) is extended to the lower half of Σ by the relation R^{0}(η^{∗})^{∗} = R^{0}(η). The ”+” side is to the left of Σ and the ”-” side is to the right of Σ.

Note that p is not defined uniquely because we are not specifying a normalizing condition (e.g. at infinity). Also, note that if R^{0} is entire, then we can choose p = ρ^{0} = \frac{dR^{0}}{dη} .

Now, the analysis of Chapter 5 in [KMM] can be amended as follows. First, let’s amend the definition of X in Chapter 3, which describes the interpolant of the norming constants. We simply set

X(λ) = iπ(2K + 1) ∫_{λ}^{iA}(p_{+}(η) + p_{−}(η))dη,

for λ in the linear segment [0, iA]. Then, the discussion of Chapter 5 in [KMM], in particular from relation (5.4) to (5.8), is amended by substitutitng ρ̄^{σ} = p − ρ^{0}. More precisely, taking σ = 1,

∫_{0}^{iA} L_{η}^{0}(λ)p_{−}(η)dη = ∫_{C_{I}} L_{η−}^{C}(λ)p(η)dη,

and similarly, by symmetry,

∫_{−iA}^{0} L_{η}^{0}(λ)p_{−}(η^{∗})^{∗} dη = ∫_{C_{I}^{∗}} L_{η−}^{C}(λ)p(η^{∗})^{∗} dη.

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(Recall here that L_{η}^{0}(λ) = log(λ − η), with a cut along the imaginary axis from η to −i∞. In the above integral we integrate over the ”-” side, while in the integral just following we integrate over the ”+” side.) Also

∫_{0}^{iA} L_{η}^{0}(λ)p_{+}(η)dη = ∫_{C_{F}} L_{η−}^{C}(λ)p(η)dη,

and similarly, by symmetry,

∫_{−iA}^{0} L_{η}^{0}(λ)p_{+}(η^{∗})^{∗}dη = ∫_{C_{F}^{∗}} L_{η−}^{C}(λ)p(η^{∗})^{∗}dη.

Next, note that L_{η+}^{C}(λ) = L_{η−}^{C}(λ) for all η ∈ C_{I} ∪ C_{I}^{∗} “below” λ ∈ C_{I} and at the same time L_{η_{+}}^{C}(λ) = 2πi + L_{η−}^{C}(λ) for η ∈ C_{I} “above” λ. This means that for λ ∈ C,

\begin{matrix}∫_{C} L_{η±}^{C}(λ)p(η)dη + ∫_{C^{∗}} L_{η±}^{C}(λ)p(η^{∗})^{∗}dη = \\ ∫_{C} \overline{L_{η}^{C}}(λ)p(η)dη + ∫_{C^{∗}} \overline{L_{η}^{C}}(λ)p(η^{∗})^{∗}dη ± πi/2 ∫_{C_{I}} p(η)dη ± πi/2 ∫_{C_{F}} p(η)dη, \end{matrix}

with \overline{L_{η}^{C}}(λ) = \frac{L_{η+}^{C}(λ)+L_{η−}^{C}(λ)}{2}. Assembling these results gives the expression

\begin{matrix}ϕ̃(λ) = ∫_{C} \overline{L_{η}^{C}}(λ)\overline{ρ}(η)dη + ∫_{C^{∗}} \overline{L_{η}^{C}}(λ)\overline{ρ}(η^{∗})^{∗}dη \\ +J(2iλx + 2iλ^{2}t) − (J(2K + 1) + 1) (±πi/2 ∫_{C_{I}} p(η)dη ± πi/2 ∫_{C_{F}} p(η)dη), \end{matrix}

valid for λ ∈ C, where we have introduced the complementary density for η ∈ C : \overline{ρ}(η) := p(η) − ρ(η). Choosing K so that J(2K + 1) + 1 = 0, the last term vanishes and we simply have

(4)  ϕ̃(λ) = ∫_{C} \overline{L_{η}^{C,σ}}(λ)\overline{ρ}(η)dη + ∫_{C^{∗}} \overline{L_{η}^{C}}(λ)\overline{ρ}(η^{∗})^{∗}dη + J(2iλx + 2iλ^{2}t).

Compare with (5.11) of [KMM]. Formula (4) is less awkward, since it does not depend on the a priori constraint that the contour C has to go through iA, a constraint that is eventually suspended anyway.

The rest of the proofs of [KMM] go through, with p substituting ρ^{0}. We omit the detailed discussion, but we do stress one major point on the variational problem of Chapter 8 of [KMM].

The contour C and the measure ρdη are characterized by a solution of a Green’s variational problem of electrostatic kind. Indeed

E_{ϕ}(ρdη) = max_{C′}min_{μ:supp(μ)∈C}E_{ϕ}(μ),

where the contours C′ are a priori supported in the upper half-plane minus the linear segment [0, iA], and E_{ϕ} is the weighted energy of a measure with respect to the external field given by

ϕ(z) = ∫ log \frac{|z − η^{∗}|}{|z − η|}ρ^{0}(η)dη − Re(iπJ ∫_{z}^{iA} p(η)dη + 2iJ(zx + z^{2}t)).

The harmonicity of ϕ is important to the structure of C, supp(ρ). But again, even if ρ is not analytically extended, it can be written as a sum of two terms that are.

One could write ϕ as

(5) ϕ(z) = ∫ log \frac{|z − η^{∗}|}{|z − η|}(p_{+} + p_{−})(η)dη − Re(iπJ ∫_{z}^{iA} p(η)dη + 2iJ(zx + z^{2}t)).

Again, this representation is perhaps more natural, since in setting the variational problem it is more appropriate to think of the ”left” and ”right” sides of the linear segment [0, iA] as distinct.

CONCLUSION: The moral of the story is that if ρ^{0} does not admit an entire extension, we can write it as the average of two functions p_{−}, p_{+} that can be extended to the left and right of the segment [0, iA] respectively, and proceed as before, with ρ^{0} substituted by p.

REMARKS: 1. Because of the analysis of [KMM] it is now believed that the behavior of the solutions of the focusing NLS equation, under any analytic initial data, is described semiclassically by algebro-geometric solutions of the same equation, corresponding to Riemann Surfaces of slowly modulated moduli. In this respect, focusing NLS is surprisingly not very different to defocusing NLS. However, the geometry of the phase space of the periodic focusing NLS is much richer, since it also contains complicated homoclinic and heteroclinic manifolds, apart from the invariant tori where algebro-geometric solutions live. For this reason, it has been

6 SPYRIDON KAMVISSIS

conjectured ([L], [V]) that, at least for non-analytic data, the semiclassical focusing NLS should be qualitatively different from the defocusing NLS, and that modulated algebro-geometric solutions should only give part of the semicalssical picture.

The above result shows that this is not the case. We believe that the full geometry of the phase space of the periodic focusing NLS should only be involved in a study of small perturbations of the semiclassical problem. A numerical study of the perturbed problem has been initiated in [BKB].

2. In the case of real A(x) it has proved crucial that the non-self-adjoint Dirac problem can be reduced (as h → 0) to a self-adjoint Schrödinger problem. If we allow more general initial data u(x, 0) = A(x)exp(\frac{iS(x)}{h}), the direct scattering problem for the associated Dirac operator, as h → 0, is open. A formula like (2) does not exist, so it is conceivable that analyticity of the data is important.

3. If the initial data A(x) are not analytic, then no solution of the Euler system that appears as a formal limit of (1) is guaranteed. So, no strong limit can in general exist. However, the finite gap ansatz may well hold, albeit with a higher number of ”gaps” than one would attain with analyticity.

BIBLIOGRAPHY

[BKB] J.Bergamin, S.Kamvissis, T.Bountis, A Numerical Analysis of the Perturbed Semiclassical Nonlinear Schrödinger Equation, Physics Letters A, v.304, n.3-4, 2002, pp.85-94.

[KMM] S.Kamvissis, K.McLaughlin, P.Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, nlin.SI/0012034, 2000; Annals of Mathematics Studies, v.154, Princeton University Press, 2003.

[L] C.D.Levermore, private communication, 1999.

[V] S.Venakides, private communication, 2001.