Riemann–Hilbert problem

Suppose that $\backslash Sigma$ is a smooth, simple, closed contour in the complex plane.{{sfn | Ablowitz | Fokas | 2003 |pp=71-72}} Divide the plane into two parts denoted by $\backslash Sigma\_\{+\}$ (the inside) and $\backslash Sigma\_\{-\}$ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation, was that of finding a function

:$M\_+(t)\; =\; u(t)\; +\; i\; v(t),$

analytic inside $\backslash Sigma\_\{+\}$, such that the boundary values of $M\_+$ along $\backslash Sigma$ satisfy the equation

:$a(t)u(t)\; -\; b(t)v(t)\; =\; c(t),$

for $t\; \backslash in\; \backslash Sigma$, where $a(t)$, $b(t)$ and $c(t)$ are given real-valued functions.{{sfn|Bitsadze|2001}}{{sfn|Pandey|1996}} For example, in the special case where $a\; =\; 1,\; b=0$ and $\backslash Sigma$ is a circle, the problem reduces to deriving the Poisson formula.{{sfn | Ablowitz | Fokas | 2003 |pp=514}}

By the Riemann mapping theorem, it suffices to consider the case when $\backslash Sigma$ is the circle group $\backslash mathbb\; T\; =\; \backslash \{\; z\; \backslash in\; \backslash mathbb\; C\; :\; |z|\; =\; 1\; \backslash \}$.{{sfn|Pandey|1996|loc=§2.2}} In this case, one may seek $M\_+(z)$ along with its Schwarz reflection

:$M\_-(z)\; =\; \backslash overline\{M\_+\backslash left(\backslash bar\{z\}^\{-1\}\backslash right)\}.$

For $z\backslash in\; \backslash mathbb\{T\}$, one has $z\; =\; 1/\backslash bar\{z\}$ and so

:$M\_-(z)\; =\; \backslash overline\{M\_+(z)\}.$

Hence the problem reduces to finding a pair of analytic functions $M\_+(z)$ and $M\_-(z)$ on the inside and outside, respectively, of the unit disk, so that on the unit circle

:$\backslash frac\{a(z)+ib(z)\}\{2\}M\_+(z)\; +\; \backslash frac\{a(z)-ib(z)\}\{2\}M\_-(z)\; =\; c(z),$

and, moreover, so that the condition at infinity holds:

:$\backslash lim\_\{z\backslash to\backslash infty\}M\_-(z)\; =\; \backslash overline\{\{M\}\_+(0)\}.$

{{see also|Wiener-Hopf method}}

Hilbert's generalization of the problem attempted to find a pair of analytic functions $M\_+(t)$ and $M\_-(t)$ on the inside and outside, respectively, of the curve $\backslash Sigma$, such that for $t\; \backslash in\; \backslash Sigma$ one has

:$\backslash alpha(t)\; M\_+(t)\; +\; \backslash beta(t)\; M\_-(t)\; =\; \backslash gamma(t),$

where $\backslash alpha(t)$, $\backslash beta(t)$ and $\backslash gamma(t)$ are given complex-valued functions (no longer just complex conjugates).{{sfn|Noble|1958|loc=§4.2}}

In the Riemann problem as well as Hilbert's generalization, the contour $\backslash Sigma$ was simple. A full Riemann–Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The "+" and "−" sides of the "contour" may then be determined according to the index of a point with respect to $\backslash Sigma$. The Riemann–Hilbert problem is to find a pair of analytic functions $M\_+(t)$ and $M\_-(t)$ on the "+" and "−" side of $\backslash Sigma$, respectively, such that for $t\; \backslash in\; \backslash Sigma$ one has

:$\backslash alpha(t)\; M\_+(t)\; +\; \backslash beta(t)\; M\_-(t)\; =\; \backslash gamma(t),$

where $\backslash alpha(t)$, $\backslash beta(t)$ and $\backslash gamma(t)$ are given complex-valued functions.

Given an oriented contour $\backslash Sigma$ (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane), a Riemann–Hilbert factorization problem is the following.

Given a matrix function $G(t)$ defined on the contour $\backslash Sigma$, find a holomorphic matrix function $M(z)$ defined on the complement of $\backslash Sigma$, such that the following two conditions are satisfied{{sfn | Ablowitz | Fokas | 2003 |loc=§7.5}}

1. If $M\_+$ and $M\_-$ denote the non-tangential limits of $M$ as we approach $\backslash Sigma$, then $M\_+(t)=G(t)M\_-(t)$, at all points of non-intersection in $\backslash Sigma$.
2. $M(z)$ tends to the identity matrix $I\_N$ as $z\; \backslash to\; \backslash infty$ along any direction outside $\backslash Sigma$.

In the simplest case $G(t)$ is smooth and integrable. In more complicated cases it could have singularities. The limits $M\_+$ and $M\_-$ could be classical and continuous or they could be taken in the $L^2$-sense.

At end-points or intersection points of the contour $\backslash Sigma$, the jump condition is not defined; constraints on the growth of $M$ near those points have to be posed to ensure uniqueness (see the scalar problem below).

Suppose $G\; =\; 2$ and $\backslash Sigma=[-1,1]$. Assuming $M$ is bounded, what is the solution $M$?

To solve this, let's take the logarithm of equation $M\_+=GM\_-$.

:$\backslash log\; M\_+(z)\; =\; \backslash log\; M\_-(z)\; +\; \backslash log\; 2.$

Since $M(z)$ tends to $1$, $\backslash log\; M\; \backslash to\; 0$ as $z\; \backslash to\; \backslash infty$.

A standard fact about the Cauchy transform is that $C\_+\; -C\_-\; =\; I$ where $C\_+$ and $C\_-$ are the limits of the Cauchy transform from above and below $\backslash Sigma$; therefore, we get

:$\backslash frac\{1\}\{2\backslash pi\; i\}\backslash int\_\{\backslash Sigma\_+\}\; \backslash frac\{\backslash log\; 2\}\{\backslash zeta-z\}\; \backslash ,\; d\backslash zeta\; -\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash int\_\{\backslash Sigma\_-\}\; \backslash frac\{\backslash log\{2\}\}\{\backslash zeta-z\}\; \backslash ,\; d\backslash zeta\; =\; \backslash log\; 2$

when $z\backslash in\backslash Sigma$. Because the solution of a Riemann–Hilbert factorization problem is unique (an easy application of Liouville's theorem (complex analysis)), the Sokhotski–Plemelj theorem gives the solution. We get

:$\backslash log\; M\; =\; \backslash frac\{1\}\{2\backslash pi\; i\}\backslash int\_\{\backslash Sigma\}\backslash frac\{\backslash log\{2\}\}\{\backslash zeta-z\}d\backslash zeta\; =\; \backslash frac\{\backslash log\; 2\}\{2\backslash pi\; i\}\backslash int^\{1-z\}\_\{-1-z\}\backslash frac\{1\}\{\backslash zeta\}d\backslash zeta\; =\; \backslash frac\{\backslash log\; 2\}\{2\backslash pi\; i\}\; \backslash log\{\backslash frac\{z-1\}\{z+1\}\},$

and therefore

:$M(z)=\backslash left(\; \backslash frac\{z-1\}\{z+1\}\; \backslash right)^\{\backslash frac\{\backslash log\{2\}\}\{2\backslash pi\; i\}\},$

which has a branch cut at contour $\backslash Sigma$.

Check:

:$\backslash begin\{align\}$

M_+(0) &=(e^{i\pi} )^{\frac{\log 2}{2\pi i}} = e^{\frac{\log 2}{2}} \\

M_-(0) &=(e^{-i\pi})^{\frac{\log 2}{2\pi i}} = e^{-\frac{\log 2}{2}}

\end{align}

therefore,

:$M\_+(0)=M\_-(0)e^\{\backslash log\{2\}\}=M\_-(0)2.$

CAVEAT 1: If the problem is not scalar one cannot easily take logarithms. In general explicit solutions are very rare.

CAVEAT 2: The boundedness (or at least a constraint on the blow-up) of $M$ near the special points $1$ and $-1$ is crucial. Otherwise any function of the form

:$M(z)=\backslash left(\; \backslash frac\{z-1\}\{z+1\}\; \backslash right)^\{\backslash frac\{\backslash log\{2\}\}\{2\backslash pi\; i\}\}\; +\; \backslash frac\{a\}\{z-1\}+\; \backslash frac\{b\}\{z+1\}$

is also a solution. In general, conditions on growth are necessary at special points (the end-points of the jump contour or intersection point) to ensure that the problem is well-posed.

=== DBAR problem ===

{{main|DBAR problem}}

Suppose $D$ is some simply connected domain of the complex $z$ plane. Then the scalar equation

:$\backslash frac\{\backslash partial\; M(z,\backslash bar\{z\})\}\{\backslash partial\; \backslash bar\{z\}\}=f(z,\backslash bar\{z\}),\; \backslash quad\; z\; \backslash in\; D,$

is a generalization of a Riemann-Hilbert problem, called the DBAR problem (or $\backslash overline\{\backslash partial\}$ problem). It is the complex form of the nonhomogeneous Cauchy-Riemann equations. To show this, let

:$M\; =\; u\; +\; i\; v,\; \backslash quad\; f\; =\; \backslash frac\{g\; +\; ih\}\{2\},\; \backslash quad\; z\; =\; x\; +\; iy,$

with $u(x,y)$, $v(x,y)$, $g(x,y)$ and $h(x,y)$ all real-valued functions of real variables $x$ and $y$. Then, using

:$\backslash frac\{\backslash partial\}\{\backslash partial\backslash bar\{z\}\}\; =\; \backslash frac\{1\}\{2\}\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; i\backslash frac\{\backslash partial\}\{\backslash partial\; y\}\backslash right),$

the DBAR problem yields

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; -\backslash frac\{\backslash partial\; v\}\{\backslash partial\; y\}\; =\; g(x,y),\; \backslash quad\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; +\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}\; =\; h(x,y).$

As such, if $M$ is holomorphic for $z\; \backslash in\; D$, then the Cauchy-Riemann equations must be satisfied.{{sfn | Ablowitz | Fokas | 2003 |p=598}}

In case $M\; \backslash to\; 1$ as $z\; \backslash to\; \backslash infty$ and $D\; :=\; \backslash mathbb\{C\}$, the solution of the DBAR problem is{{sfn | Ablowitz | Fokas | 2003 |p=600}}

:$M(z,\backslash bar\{z\})\; =\; 1\; +\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash iint\_\{\backslash mathbb\{R\}^2\}\; \backslash frac\{f(\backslash zeta,\backslash bar\{\backslash zeta\})\}\{\backslash zeta\; -z\}\backslash ,\; d\backslash zeta\; \backslash wedge\; d\backslash bar\{\backslash zeta\},$

integrated over the entire complex plane; denoted by $\backslash mathbb\{R\}^2$, and where the wedge product is defined as

:$d\backslash zeta\; \backslash wedge\; d\backslash bar\{\backslash zeta\}\; =\; (d\backslash xi\; +\; i\; d\backslash eta)\backslash wedge(d\backslash xi\; -\; i\; d\backslash eta)\; =\; -2i\; d\backslash xi\; d\backslash eta.$

If a function $M(z)$ is holomorphic in some complex region $R$, then

:$\backslash frac\{\backslash partial\; M\}\{\backslash partial\; \backslash bar\{z\}\}\; =\; 0,$

in $R$. For generalized analytic functions, this equation is replaced by

:$\backslash frac\{\backslash partial\; M\}\{\backslash partial\; \backslash bar\{z\}\}\; =\; A(z,\backslash bar\{z\})M\; +\; B(z,\backslash bar\{z\})\backslash overline\{M\},$

in a region $R$, where $\backslash overline\{M\}$ is the complex conjugate of $M$ and $A(z,\backslash bar\{z\})$ and $B(z,\backslash bar\{z\})$ are functions of $z$ and $\backslash bar\{z\}$.

{{sfn | Vekua | 2014 }}

Generalized analytic functions have applications in differential geometry, in solving certain type of multidimensional nonlinear partial differential equations and multidimensional inverse scattering.{{sfn | Ablowitz | Fokas | 2003 |p=601}}

Riemann–Hilbert problems have applications to several related classes of problems.

;A. Integrable models: The inverse scattering or inverse spectral problem associated to the Cauchy problems for 1+1 dimensional partial differential equations on the line, or to periodic problems, or even to initial-boundary value problems ({{harvtxt|Fokas|2002}}), can be stated as a Riemann–Hilbert problem. Likewise the inverse monodromy problem for Painlevé equations can be stated as a Riemann–Hilbert problem.

;B. Orthogonal polynomials, Random matrices: Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a Riemann–Hilbert factorization problem ({{harvtxt|Fokas|Its|Kitaev|1992}}). Furthermore, the distribution of eigenvalues of random matrices in several classical ensembles is reduced to computations involving orthogonal polynomials (see e.g. {{harvtxt|Deift|2000}}).

;C. Combinatorial probability: The most celebrated example is the theorem of {{harvtxt|Baik|Deift|Johansson|1999}} on the distribution of the length of the longest increasing subsequence of a random permutation. Together with the study of B above, it is one of the original rigorous investigations of so-called "integrable probability". But the connection between the theory of integrability and various classical ensembles of random matrices goes back to the work of Dyson (see e.g. {{harvtxt|Dyson|1976}}).

;D. Connection to Donaldson-Thomas theory: The work of Bridgeland {{harvtxt|Bridgeland|2019}} studies a class of Riemann-Hilbert problems coming from Donaldson-Thomas theory and makes connections with Gromov-Witten theory and exact WKB.

The numerical analysis of Riemann–Hilbert problems can provide an effective way for numerically solving integrable PDEs (see e.g. {{harvtxt|Trogdon|Olver|2016}}).

In particular, Riemann–Hilbert factorization problems are used to extract asymptotic values for the three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity). There exists a method for extracting the asymptotic behavior of solutions of Riemann–Hilbert problems, analogous to the method of stationary phase and the method of steepest descent applicable to exponential integrals.

By analogy with the classical asymptotic methods, one "deforms" Riemann–Hilbert problems which are not explicitly solvable to problems that are. The so-called "nonlinear" method of stationary phase is due to {{harvtxt|Deift|Zhou|1993}}, expanding on a previous idea by {{harvtxt|Its|1982}} and {{harvtxt|Manakov|1974}} and using technical background results from {{harvtxt|Beals|Coifman|1984}} and {{harvtxt|Zhou|1989}}. A crucial ingredient of the Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard Cauchy kernel (see {{harvtxt|Gakhov|2001}}; also cf. the scalar example below).

An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by {{harvtxt|Deift|Venakides|Zhou|1997}}, which has been crucial in most applications. This was inspired by work of Lax, Levermore and Venakides, who reduced the analysis of the small dispersion limit of the KdV equation to the analysis of a maximization problem for a logarithmic potential under some external field: a variational problem of "electrostatic" type (see {{harvtxt|Lax|Levermore|1983}}). The g-function is the logarithmic transform of the maximizing "equilibrium" measure. The analysis of the small dispersion limit of KdV equation has in fact provided the basis for the analysis of most of the work concerning "real" orthogonal polynomials (i.e. with the orthogonality condition defined on the real line) and Hermitian random matrices.

Perhaps the most sophisticated extension of the theory so far is the one applied to the "non self-adjoint" case, i.e. when the underlying Lax operator (the first component of the Lax pair) is not self-adjoint, by {{harvtxt|Kamvissis|McLaughlin|Miller|2003}}. In that case, actual "steepest descent contours" are defined and computed. The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on the external field, was done in {{harvtxt|Kamvissis|Rakhmanov|2005}}; the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov.

An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in {{harvtxt|McLaughlin|Miller|2006}}, especially convenient when jump matrices do not have analytic extensions. Their method is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours. An alternative way of dealing with jump matrices with no analytic extensions was introduced in {{harvtxt|Varzugin|1996}}.

Another extension of the theory appears in {{harvtxt|Kamvissis|Teschl|2012}} where the underlying space of the Riemann–Hilbert problem is a compact hyperelliptic Riemann surface. The correct factorization problem is no more holomorphic, but rather meromorphic, by reason of the Riemann–Roch theorem. The related singular kernel is not the usual Cauchy kernel, but rather a more general kernel involving meromorphic differentials defined naturally on the surface (see e.g. the appendix in {{harvtxt|Kamvissis|Teschl|2012}}). The Riemann–Hilbert problem deformation theory is applied to the problem of stability of the infinite periodic Toda lattice under a "short range" perturbation (for example a perturbation of a finite number of particles).

Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by Arno Kuijlaars and collaborators, see e.g. {{harvtxt|Kuijlaars|López|2015}}.

• Riemann–Hilbert correspondence
• Wiener-Hopf method

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