RIEMANN-HILBERT FACTORIZATION AND INTEGRABILITY

Spyridon Kamvissis

The present is based on a Seminar given at the Max Planck Institute on September 12, 2002.

ABSTRACT

I will try to review some ideas that have first appeared in the work of Soviet mathematicians in the 1970s-1980s and have further been developed in the 1980s-1990s ”in the West”, concerning the connection of so-called completely integrable systems (KdV, NLS, Toda, etc.) and Riemann-Hilbert factorization problems. At first, such ideas have been most fruitful in enabling us to compute rigorous asymptotics (long time, semiclassical, etc.) for integrable equations. But Riemann-Hilbert problems have also enriched the theory of orthogonal polynomials, random matrices, and combinatorics (and not only concerning asymptotics).

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I. INTEGRABILITY

Archetypal examples of integrable equations are:

1. The KdV equation

(1)  u_{t} − 6uu_{x} + ϵ^{2}u_{xxx} = 0,

where ϵ > 0.

2. The nonlinear Schrödinger equation (focusing (+) and defocusing (-) cases)

(2)  iϵu_{t} + \frac{ϵ^{2}}{2} u_{xx} ± |u|^{2}u = 0.

Integrable equations always admit a ”Lax pair” representaion: they can be written in the form

(3)  L_{t} = [L, B] = LB − BL,

where L, B are operators on a Hilbert space. The existence of a Lax pair does not guarantee integrability. It is only the first step.

Integrability machinery is based on the scattering and inverse scattering theory for equations like (1) and (2). The Lax operator L associated to (1) is the Schrödinger operator

(4)  L = −ϵ^{2} \frac{d^{2}}{dx^{2}} + u(x)

while

B = 4ϵ^{3} \frac{d^{3}}{dx^{3}} − 3ϵ[u\frac{d}{dx} + \frac{d}{dx}u].

The NLS equations are associated to the Dirac operators

(5)  ( \begin{matrix}iϵ∂_{x} & ∓iu(x) \\ −iu^{∗}(x) & −iϵ∂_{x} \end{matrix} ) ,

where sign ”+” corresponds to the defocusing case, while sign ”-” corresponds to the focusing case. * denotes complex conjugation. It is important to note that, although (4) gives a self-adjoint operator (here u is always real), (5) is self-adjoint in the defocusing case and non-self-adjoint in the focusing case.

II. RIEMANN-HILBERT FACTORIZATION PROBLEMS

The solution of an integrable system via the inverse scattering transform can always be reduced to the solution of a Riemann-Hilbert factorization problem. In particular, the asymptotic analysis of the given equation is thus reduced to the asymptotic analysis of a Riemann-Hilbert problem.

Consider, for example, the nonlinear Schrödinger equation (defocusing case), that is equation (2) with the ”-” sign. Assume ϵ = 1 for the moment. Suppose we want to solve (2) under some initial data u_{0}(x) that, for simplicity, are considered to belong to the Schwartz class.

Let r(λ) be the reflection coefficient corrsponding to u_{0}. It is a fact of direct scattering theory that r is also Schwartz, and that |r(λ)| < 1, for all λ ∈ ℝ.

Let Q solve

(8) Q_{+}(λ) = Q_{−}(λ) ( \begin{matrix}1 − |r(λ)|^{2} & −r^{∗}(λ)e^{−2iλx−4iλ^{2}t} \\ r(λ)e^{2iλx+4iλ^{2}t} & 1 \end{matrix} ) , Imλ = 0,

Note that the jump matrix has determinant 1. Also

(9)  Q(λ = ∞) = I.

This is a Riemann-Hilbert factorization problem: Q is a matrix function, analytic in the complement of the real line, satisfying the jump condition and the asymptotics above.

The solution of the Riemann-Hilbert problem enables us to recover u(x, t). Indeed

(10)  u(x, t) = −2lim_{λ→∞}λQ_{12}.

Thus, the initial value problem for (2) is reduced to the above Riemann-Hilbert factorization problem.

Conversely, suppose we are given a Riemann-Hilbert problem of the form

(11)  Q_{+}(λ) = Q_{−}(λ)e^{iθ(x,t,λ)adσ_{3}}v(λ), λ ∈ Σ,

4 SPYRIDON KAMVISSIS

where v(λ) is independent of x, t. In (8) above, for example, θ(x, t, λ) = 2iλx+4iλ^{2}t and

v(λ) = ( \begin{matrix}1 − |r(λ)|^{2} & −r^{∗}(λ) \\ r(λ) & 1 \end{matrix} ) .

Define m(x, t, λ) = Q(λ)e^{iθ(x,t,λ)σ_{3}}. Then m_{+} = m_{−}v for λ ∈ Σ. Differentiating with respect to x or t, we also get ∂_{x}m_{+} = ∂_{x}m_{−}v and ∂_{t}m_{+} = ∂_{t}m_{−}v, for λ ∈ Σ. Hence both ∂_{x}m · m^{−1} and ∂_{t}m · m^{−1} have no jumps, so they are entire functions. By calculating ∂_{x}m and ∂_{t}m at infinity, we find

\begin{matrix}m_{x} = U(x, t, λ)m, \\ m_{t} = V (x, t, λ)m, \end{matrix}

where U, V are polynomials in λ [quadratic in the NLS case].

This is essentially the Lax pair arising from the Riemann-Hilbert problem (11). The compatibility condition m_{xt} = m_{tx} leads to an integrable equation.

III. FREDHOLM DETERMINANTS AND RANDOM MATRICES

PROPOSITION (Its, Izergin, Korepin, Slavnov). Let

τ = det(I + K_{x}),

where K_{x} is the integral operator on L^{2}(Σ) with kernel

K_{x}(z, z′) = \frac{A(z′)B(z) + A′(z)B′(z′)}{z − z′},

such that \frac{d}{dx}K_{x}(Z, z′) = A(z)A′(z′) + A′(z)A(z′)

and A(z)B(z) + A′(z)B′(z) = 0.

The determinant above is a Fredholm determinant.

Consider the following Riemann-Hilbert problem on the contour Σ. Let m be a 2x2 matrix function analytic in the complement of Σ, and let m_{+}, m_{−}, the limits of m on Σ from left and right, respectively, satisfy

\begin{matrix}m_{+}(z) = m_{−}(z) ( \begin{matrix}1 + 2πiA′(z)B′(z) & −2πiA(z)B′(z) \\ −2πiA′(z)B(z) & 1 + 2πiA(z)B(z) \end{matrix} ) , for z ∈ Σ, \\ and lim_{z→∞}m(z) = I. \end{matrix}

RIEMANN-HILBERT FACTORIZATION AND INTEGRABILITY 5

Then

(logτ)_{x} = i(m_{11}^{1} − m_{22}^{1}),

where m(z) = I + \frac{m^{1}}{z} + O(z^{−2}) is the expansion of m(z) near infinity and the suffixes 11 and 22 denote the diagonal entries of m^{1}.

PROPOSITION (Mehta, Gaudin, Dyson). Let J = ∪_{k=0}^{n}J_{k}, where J_{k} = (a_{k}, b_{k}) are n + 1 disjoint intervals in ℝ. Let K(z, z′) = \frac{sinx(z−z′)}{z−z′} be the kernel of the operator K acting on L^{2}(J, dz). Then the determinant τ = det(I + K) is the probability of finding no eigenvalues in the union of intervals \frac{xJ}{π} = ∪_{k=0}^{n}(\frac{xa_{k}}{π} , \frac{xb_{k}}{π}) for a random Hermitian matrix chosen from the so-called Gaussian unitary ensemble (GUE), in the bulk scaling limit with mean spacing 1.

PROPOSITION (des Cloizeaux, Mehta). If J = (−1, 1), then as x → ∞, then logP = −x^{2}/2 − logx/4 + c + o(1).

REMARK. The Gaussian Unitary Ensemble consists of Hermitian matrices M with the following probability distribution

P_{n}(M)dM = c · exp(−tr(M^{2}))Π_{i}M_{ii}Π_{i<j}Re(M_{ij})Im(M_{ij}),

such that c is a normalization constant.

In general one can consider distributions

P_{n}(M)dM = c · exp(−F (M))Π_{i}M_{ii}Π_{i<j}Re(M_{ij})Im(M_{ij}),

where F (M) is not too rapidly growing at infinity. If we impose the symmetry P_{n}(UMU^{∗}) = P_{n}(M) for any unitary U, then F (M) must be a symmetric function of the eigenvalues of M. For GUE we actually choose F (M) to be the trace of a polynomial in M. Then, it must actually be a quadratic polynomial, and through renormalization, F (M) = tr(M^{2}).

By the way the Universality results in Random Matrix Theory imply that the distribution of the eigenvalues of large matrices with distribution as above is independent of F (for a fairly general class of F).

6 SPYRIDON KAMVISSIS

One of the most famous applications of the theory of GUE matrices and in particular the asymptotic distribution of their eigenvalues (for large matrix dimension) is the so-called Montgomery-Odlyzko law for the distribution of zeros of the Riemann zeta function.

Montgomery found that on assuming the Riemann Hypothesis one could partially determine the pair correlation function for the zeros, and the result matched the pair correlation function for eigenvalues of GUE random hermitian matrices.

More precisely, let ⋯ ≤ γ_{−1} < 0 < γ_{1} ≤ ⋯ ≤ γ_{n} ≤ ⋯ be the sequence of the imaginary parts of the nontrivial zeros of ζ (γ_{−i} = −γ_{i}). For i ≥ 1, put γ̂_{i} = γ_{i} log γ_{i}/2π, so that the γ̂_{i} have mean spacing 1 (renormalization process). Assuming the Riemann hypothesis (RH), Montgomery, in order to study the statistics of the normalized spacings, computed the limiting pair correlation distribution and got the formula

lim_{N→+∞} \frac{1}{N} ∑_{1≤i≠j≤N} f(γ̂_{j} − γ̂_{i}) = ∫_{−∞}^{+∞} f(x)(1 − ((sin πx)/πx)^{2})dx,

valid for any f whose Fourier transform is compactly supported in (−1, 1); he also conjectured that the above equality should hold without restriction on the size of the support of f̂ (this was known as the pair correlation conjecture: it was later largely confirmed numerically by Odlyzko). The density 1−((sin πx)/πx)^{2} matches the one found for the pair correlation distribution of the eigenvalues of a typical large GUE matrix!

IV. RIEMANN-HILBERT DEFORMATION. A SINGULAR INTEGRAL EQUA-TION AND THE BEALS-COIFMAN FORMULA

In the scalar case, a Riemann-Hilbert problem can be explicitly solved. For example, let d solve

\begin{matrix}d_{+}(z) = d_{−}(z)q(z), \\ lim_{z→∞}d(z) = 1, \end{matrix}

where q is a scalar 1 + L^{1} ∩ L^{∞} function on the real line. Taking logarithms

logd_{+}(z) = logd_{−}(z) + logq(z),

and by Plemelj’s formula

logd = ∫_{ℝ} \frac{logq(s)}{2πi(s − z)}ds,

hence

d = exp[∫_{ℝ} \frac{logq(s)}{2πi(s − z)}ds].

Note that indeed

lim_{z→∞}d(z) = 1.

However, it is not a priori clear that the integral ∫_{ℝ} \frac{logq(s)}{2πi(s−z)}ds is defined. For this we need to have ensured that the index of q is zero, i.e ∫_{ℝ} logq(s)ds = 0.

In any case, scalar problems, to the extent that they are solvable, are explicitly solvable. This is not so for matrix problems.

8 SPYRIDON KAMVISSIS

The solution of a Riemann-Hilbert problem can be reduced to the solution of a singular integral equation (see [B-C]). In our case Σ = ℝ. Define the matrix Cauchy operators as follows:

\begin{matrix}C_{±} : (L^{2}(Σ))_{2x2} → (L^{2}(Σ))_{2x2} \\ (C_{±}f)(z) = \frac{1}{2πi} ∫_{Σ} \frac{f(ζ)dζ}{ζ − z_{±}} \end{matrix}

where z ∈ Σ and the signs ”+” and ”-” conote normal limits as z → Σ, from the right and the left of Σ respectively (with respect to its orientation).

It is easy to see that C_{+} and C_{−} are bounded. It is a simple calculation to show that

C_{+} − C_{−} = I.

Note that the jump in (8) can be factorized as

(b_{−})^{−1}b_{+}

where

\begin{matrix}b_{+} = ( \begin{matrix}1 & −r^{∗}(λ)e^{2iλx+4iλ^{2}t} \\ 0 & 1 \end{matrix} ) , \\ b_{−} = ( \begin{matrix}1 & 0 \\ −r(λ)e^{−2iλx−4iλ^{2}t} & 1 \end{matrix} ) . \end{matrix}

Define the ”Cauchy” operators C:

for z ∉ Σ, let (Cf)(z) = (2πi)^{−1} ∫_{Σ} \frac{f(w)dw}{w − z} and

Let the operator C_{w} be defined by

C_{w}f = C_{+}(fw_{−}) + C_{−}(fw_{+})

for a 2x2 matrix valued f, where

w_{+} = b_{+} − I and w_{−} = I − b_{−}.

Let μ solve the singular integral equation

(12)  μ = I + C_{w}μ.

RIEMANN-HILBERT FACTORIZATION AND INTEGRABILITY 9

THEOREM Let Q be defined by the integral formulae Q_{+} = I + C_{+}(μw)

Q_{−} = I + C_{−}(μw)
Q = I + C(μw) on ℂ \ Σ,

where w = w_{+}+w_{−} . Suppose detb_{−} = detb_{+}. Then Q provides the unique solution of the following Riemann-Hilbert problem.

(*)  \begin{matrix}Q_{+}(z) = Q_{−}(z)b_{−}^{−1}(z)b_{+}(z), z ∈ Σ, \\ limQ(z) = I, \end{matrix}

as z → ∞ in compact subsets of ℂ \ Σ.

PROOF: Uniqueness: The jump matrix in has determinant 1. Hence det Q is holomorphic. It is also equal to 1 at infinity. By Liouville’s theorem it is constant, and hence equal to 1 everywhere. In particular Q^{−1} exists. Now, suppose there was another solution, say Q̃. Then Q̃Q^{−1} exists, is holomorphic, and is I at infinity. By Liouville, it has to be identically I.

To show that Q defined above solves (*) note that

Q_{+}(I + w_{+})^{−1} = (I + C_{+}(μw))(I + w_{+})^{−1} =

(I + C_{+}(μw_{+}) + C_{+}(μw_{−}))(I + w_{+})^{−1} =
(I + μw_{+} + C_{−}(μw_{+}) + C_{+}(μw_{−}))(I + w_{+})^{−1},
since C_{+} − C_{−} = I,
= (I + μw_{+} + C_{w}μ)(I + w_{+})^{−1}, by the definition of C_{w},
= (μw_{+} + μ)(I + w_{+})^{−1} = μ, by (12).
Similarly Q_{−}(I − w_{−})^{−1} = (I + C_{−}(μw))(I − w_{−})^{−1} =
(I + C_{−}(μw_{+}) + C_{−}(μw_{−}))(I − w_{−})^{−1} =
(I − μw_{−} + C_{−}(μw_{+}) + C_{+}(μw_{−}))(I − w_{−})^{−1} =
(I − μw_{−} + C_{w}μ)(I − w_{−})^{−1} = (μ − μw_{−})(I − w_{−})^{−1} = μ.

Hence Q_{+}(I + w_{+})^{−1} = Q_{−}(I − w_{−})^{−1} and thus Q_{−}^{−1}Q_{+} = (I − w_{−})^{−1}(I + w_{+}) = (b_{−})^{−1}b_{+} and thus Q_{+} = Q_{−}(b_{−})^{−1}b_{+}.

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The above theory is useful if one seeks asymptotics of the solution of a Riemann-Hilbert problem with respect to a parameter [e.g. time t in (8)-(9)]. The idea is to ”deform” the factorization problem (*) into one that can be solved explicitly, in a series of steps each of which involves either an extension or a deletion of a part of the factorization contour, or a substitution of w_{±} by approximate (for large times) matrices, or an appropriate conjugation. The choice of the contour extensions or deletions depends on the phase of the exponentials appearing in b_{±}. For each step, the final factorization problem is an approximation to the initial one in the sense that the solution as given by the corresponding integral formula is close to the exact solution of (*).

There are basically two kind of deformations used. Deformations of a ”geometric type” involve a contour deformation. This is allowed if the jump matrix on that contour is analytic. These deformations are exact, not approximate. There are also ”analytic” type deformations that involve perturbing the formula

Q = I + C(μw) = I + ∫_{Σ} \frac{(I − C_{w})^{−1}(I)(s)w(s)}{2πi(s − z)}ds

with respect to w while keeping the contour Σ fixed. Such deformations are only approximate, with respect to some underlying parameter (like t for example).

V. REFERENCES

For scalar Riemann-Hilbert problems:

[G] Gakhov, Boundary Value Problems, Pergamon Press 1966. For the complete rigorous treatment of the incerse scattering problem as a

Riemann-Hilbert problem:

[BC] R.Beals, R.Coifman, Scattering and Inverse Scattering for First Order Systems, Communications in Pure and Applied Mathematics, v.37, pp.39-90, 1984.

[BDZ] R.Beals, P.Deift, X.Zhou, Inverse Scattering on the Line, Important Developments in Soliton Theory, 1980-1990, ed. by A.Fokas and V.E.Zakharov, Springer-Verlag, 1993.

[Z] X.Zhou, Direct and Inverse Scattering Transforms with Arbitrary Spectral Dependence, Communications of Pure and Applied Mathematics, v.42, pp.895-938, 1989.

For one of the first applications of Riemann-Hilbert problems to long time asymptotics for soliton equations:

[IN] A.R.Its, V.Novokshenov, The Isomonodromic Deformation Method in the Theory of Painleve Equations, Lecture Notes in Math., v.1191, Springer 1986.

For the first rigorous exposition of the Riemann-Hilbert deformation method:

[DZ] P.Deift, X.Zhou, A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems, Annals of Mathematics, March 1993.

For the first application to a discrete lattice:

[K1] S. Kamvissis, Long Time Behavior of the Doubly Infinite Toda Lattice Under Initial Data Decaying at Infinity, Communications in Mathematical Physics, v.153, n.3, 1993.

An important extension of the original method applied to zero dispersion limits:

[DVZ] P.Deift, S.Venakides, X.Zhou, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems, IMRN, n.6, 1997, pp.286-299.

12 SPYRIDON KAMVISSIS

For Random Matrices, Fredholm Determinants and Integrable Operators:

[DIZ] P.Deift, A.R.Its, X.Zhou, A Riemann-Hilbert Approach to Asymptotic Problems Arising in the Theory of Random Matrix Models and also in the Theory of Integrable Statistical Mechanics, Annals of Mathematics, v.146, n.1, 1997, pp.149-235.

For the first application involving non-self-adjoint problems:

[K2] S.Kamvissis, Long Time Behavior for the Focusing Nonlinear Schrödinger Equation with Real Spectral Singularities, Communications in Mathematical Physics, v.180, n.2, 1996, pp.325-343.

For applications to orthogonal polynomials:

[DKMVZ] P.Deift, T.Kriecherbauer, K.McLaughlin, S.Venakides, X.Zhou, Uniform Asymptotics for Orthogonal Polynomials, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), Doc. Math. 1998, Extra Vol. III, pp.491–501.

The most general exposition of the Riemann-Hilbert deformation method, with several extensions of the original theory:

[KMM] S.Kamvissis, K.McLaughlin, P.Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Annals of Mathematics Studies, v.155, Princeton University Press, 2003.

For the first applications to a combinatorics problem:

[BDJ] J.Baik, P.Deift, K.Johansson, On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations. J. Amer. Math. Soc. v.12, no. 4, 1999, pp.1119–1178.

For an application to the representation theory of U(∞):

[BD] A.Borodin, P.Deift, Fredholm Determinants, Jimbo-Miwa-Ueno τ-functions, and Representation Theory, Comm. Pure Appl. Math. v.55, no. 9, 2002, pp.1160–