Isomonodromic deformation

Last updated

In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems.

Contents

Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. Inspired by results in statistical mechanics, a seminal contribution to the theory was made by Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, who studied cases involving irregular singularities.

Fuchsian systems and Schlesinger's equations

Fuchsian system

A Fuchsian system is the system of linear differential equations [1]

where x takes values in the complex projective line , the y takes values in and the Ai are constant n×n matrices. Solutions to this equation have polynomial growth in the limit x = λi. By placing n independent column solutions into a fundamental matrix then and one can regard as taking values in . For simplicity, assume that there is no further pole at infinity, which amounts to the condition that

Monodromy data

Now, fix a basepoint b on the Riemann sphere away from the poles. Analytic continuation of a fundamental solution around any pole λi and back to the basepoint will produce a new solution defined near b. The new and old solutions are linked by the monodromy matrix Mi as follows:

One therefore has the Riemann–Hilbert homomorphism from the fundamental group of the punctured sphere to the monodromy representation:

A change of basepoint merely results in a (simultaneous) conjugation of all the monodromy matrices. The monodromy matrices modulo conjugation define the monodromy data of the Fuchsian system.

Hilbert's twenty-first problem

Now, with given monodromy data, can a Fuchsian system be found which exhibits this monodromy? This is one form of Hilbert's twenty-first problem. One does not distinguish between coordinates x and which are related by Möbius transformations, and also do not distinguish between gauge equivalent Fuchsian systems - this means that A and

are regarded as being equivalent for any holomorphic gauge transformation g(x). (It is thus most natural to regard a Fuchsian system geometrically, as a connection with simple poles on a trivial rank n vector bundle over the Riemann sphere).

For generic monodromy data, the answer to Hilbert's twenty-first problem is 'yes'. The first proof was given by Josip Plemelj. [2] However, the proof only holds for generic data, and it was shown in 1989 by Andrei Bolibrukh that there are certain 'degenerate' cases when the answer is 'no'. [3] Here, the generic case is focused upon entirely.

Schlesinger's equations

There are generically many Fuchsian systems with the same monodromy data. Thus, given any such Fuchsian system with specified monodromy data, isomonodromic deformations can be performed of it. One therefore is led to study families of Fuchsian systems, where the matrices Ai depend on the positions of the poles.

In 1912 Ludwig Schlesinger proved that in general, the deformations which preserve the monodromy data of a generic Fuchsian system are governed by the integrable holonomic system of partial differential equations which now bear his name: [4]

The last equation is often written equivalently as

These are the isomonodromy equations for generic Fuchsian systems. The natural interpretation of these equations is as the flatness of a natural connection on a vector bundle over the 'deformation parameter space' which consists of the possible pole positions. For non-generic isomonodromic deformations, there will still be an integrable isomonodromy equation, but it will no longer be Schlesinger.

If one limits attention to the case when the Ai take values in the Lie algebra , the Garnier integrable systems are obtained. If one specializes further to the case when there are only four poles, then the Schlesinger/Garnier equations can be reduced to the famous sixth Painlevé equation.

Irregular singularities

Motivated by the appearance of Painlevé transcendents in correlation functions in the theory of Bose gases, Michio Jimbo, Tetsuji Miwa and Kimio Ueno extended the notion of isomonodromic deformation to the case of irregular singularities with any order poles, under the following assumption: the leading coefficient at each pole is generic, i.e. it is a diagonalisable matrix with simple spectrum. [5]

The linear system under study is now of the form

with n poles, with the pole at λi of order . The are constant matrices (and is generic for ).

Extended monodromy data

As well as the monodromy representation described in the Fuchsian setting, deformations of irregular systems of linear ordinary differential equations are required to preserve extended monodromy data. Roughly speaking, monodromy data is now regarded as data which glues together canonical solutions near the singularities. If one takes as a local coordinate near a pole λiof order , one can then solve term-by-term for a holomorphic gauge transformation g such that locally, the system looks like

where and the are diagonal matrices. If this were valid, it would be extremely useful, because then (at least locally), one has decoupled the system into n scalar differential equations which one can easily solve to find that (locally):

However, this does not work - because the power series solved term-for-term for g will not, in general, converge.

Jimbo, Miwa and Ueno showed that this approach nevertheless provides canonical solutions near the singularities, and can therefore be used to define extended monodromy data. This is due to a theorem of George Birkhoff [ citation needed ] which states that given such a formal series, there is a unique convergent function Gi such that in any sufficiently large sector[ clarification needed ] around the pole, Gi is asymptotic to gi, and

is a true solution of the differential equation. A canonical solution therefore appears in each such sector near each pole. The extended monodromy data consists of

Jimbo–Miwa–Ueno isomonodromic deformations

As before, one now considers families of systems of linear differential equations, all with the same (generic) singularity structure. One therefore allows the matrices to depend on parameters. One is allowed to vary the positions of the poles λi, but now, in addition, one also varies the entries of the diagonal matrices which appear in the canonical solution near each pole.

Jimbo, Miwa and Ueno proved that if one defines a one-form on the 'deformation parameter space' by

(where D denotes exterior differentiation with respect to the components of the only)

then deformations of the meromorphic linear system specified by A are isomonodromic if and only if

These are the Jimbo—Miwa—Ueno isomonodromy equations. As before, these equations can be interpreted as the flatness of a natural connection on the deformation parameter space.

Properties

The isomonodromy equations enjoy a number of properties that justify their status as nonlinear special functions.

Painlevé property

This is perhaps the most important property of a solution to the isomonodromic deformation equations. This means that all essential singularities of the solutions are fixed, although the positions of poles may move. It was proved by Bernard Malgrange for the case of Fuchsian systems, and by Tetsuji Miwa in the general setting.

Indeed, suppose that one is given a partial differential equation (or a system of them). Then, 'possessing a reduction to an isomonodromy equation' is more or less equivalent to the Painlevé property, and can therefore be used as a test for integrability.

Transcendence

In general, solutions of the isomonodromy equations cannot be expressed in terms of simpler functions such as solutions of linear differential equations. However, for particular (more precisely, reducible) choices of extended monodromy data, solutions can be expressed in terms of such functions (or at least, in terms of 'simpler' isomonodromy transcendents). The study of precisely what this transcendence means has been largely carried out by the invention of 'nonlinear differential Galois theory' by Hiroshi Umemura and Bernard Malgrange.

There are also very special solutions which are algebraic. The study of such algebraic solutions involves examining the topology of the deformation parameter space (and in particular, its mapping class group); for the case of simple poles, this amounts to the study of the action of braid groups. For the particularly important case of the sixth Painlevé equation, there has been a notable contribution by Boris Dubrovin and Marta Mazzocco, which has been recently extended to larger classes of monodromy data by Philip Boalch.

Rational solutions are often associated with special polynomials. Sometimes, as in the case of the sixth Painlevé equation, these are well-known orthogonal polynomials, but there are new classes of polynomials with an extremely interesting distribution of zeros and interlacing properties. The study of such polynomials has largely been carried out by Peter Clarkson and collaborators.

Symplectic structure

The isomonodromy equations can be rewritten using Hamiltonian formulations. This viewpoint was extensively pursued by Kazuo Okamoto in a series of papers on the Painlevé equations in the 1980s.

They can also be regarding as a natural extension of the Atiyah–Bott symplectic structure on spaces of flat connections on Riemann surfaces to the world of meromorphic geometry - a perspective pursued by Philip Boalch. Indeed, if one fixes the positions of the poles, one can even obtain complete hyperkähler manifolds; a result proved by Olivier Biquard and Philip Boalch. [6]

There is another description in terms of moment maps to (central extensions of) loop algebras - a viewpoint introduced by John Harnad and extended to the case of general singularity structure by Nick Woodhouse. This latter perspective is intimately related to a curious Laplace transform between isomonodromy equations with different pole structure and rank for the underlying equations.

Twistor structure

The isomonodromy equations arise as (generic) full dimensional reductions of (generalized) anti-self-dual Yang–Mills equations. By the Penrose–Ward transform they can therefore be interpreted in terms of holomorphic vector bundles on complex manifolds called twistor spaces. This allows the use of powerful techniques from algebraic geometry in studying the properties of transcendents. This approach has been pursued by Nigel Hitchin, Lionel Mason and Nick Woodhouse.

Gauss-Manin connections

By considering data associated with families of Riemann surfaces branched over the singularities, one can consider the isomonodromy equations as nonhomogeneous Gauss–Manin connections. This leads to alternative descriptions of the isomonodromy equations in terms of abelian functions - an approach known to Fuchs and Painlevé, but lost until rediscovery by Yuri Manin in 1996.

Asymptotics

Particular transcendents can be characterized by their asymptotic behaviour. The study of such behaviour goes back to the early days of isomonodromy, in work by Pierre Boutroux and others.

Applications

Their universality as some of the simplest nonlinear integrable systems means that the isomonodromy equations have a diverse range of applications. Perhaps of greatest practical importance is the field of random matrix theory. Here, the statistical properties of eigenvalues of large random matrices are described by particular transcendents.

The initial impetus for the resurgence of interest in isomonodromy in the 1970s was the appearance of transcendents in correlation functions in Bose gases. [7]

They provide generating functions for moduli spaces of two-dimensional topological quantum field theories and are thereby useful in the study of quantum cohomology and Gromov–Witten invariants.

'Higher-order' isomonodromy equations have recently been used to explain the mechanism and universality properties of shock formation for the dispersionless limit of the Korteweg–de Vries equation.

They are natural reductions of the Ernst equation and thereby provide solutions to the Einstein field equations of general relativity; they also give rise to other (quite distinct) solutions of the Einstein equations in terms of theta functions.

They have arisen in recent work in mirror symmetry - both in the geometric Langlands programme, and in work on the moduli spaces of stability conditions on derived categories.

Generalizations

The isomonodromy equations have been generalized for meromorphic connections on a general Riemann surface.

They can also easily be adapted to take values in any Lie group, by replacing the diagonal matrices by the maximal torus, and other similar modifications.

There is a burgeoning field studying discrete versions of isomonodromy equations.

Related Research Articles

<span class="mw-page-title-main">Monodromy</span> Mathematical behavior near singularities

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a monodromy group: a group of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called polydromy.

<span class="mw-page-title-main">Separation of variables</span> Technique for solving differential equations

In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.

In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.

The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group.

In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property, but which are not generally solvable in terms of elementary functions. They were discovered by Émile Picard , Paul Painlevé , Richard Fuchs, and Bertrand Gambier.

In linear algebra, it is often important to know which vectors have their directions unchanged by a given linear transformation. An eigenvector or characteristic vector is such a vector. More precisely, an eigenvector of a linear transformation is scaled by a constant factor when the linear transformation is applied to it: . The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor .

In mathematics, in the theory of ordinary differential equations in the complex plane , the points of are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. The theory is named in honour of Erik Ivar Fredholm.

In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R, where each block along the diagonal, called a Jordan block, has the following form:

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.

A differential equation can be homogeneous in either of two respects.

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point, in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.

The Gibbons–Tsarev equation is an integrable second order nonlinear partial differential equation. In its simplest form, in two dimensions, it may be written as follows:

In mathematics, a linear recurrence with constant coefficients sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree 0 or 1. A linear recurrence denotes the evolution of some variable over time, with the current time period or discrete moment in time denoted as t, one period earlier denoted as t − 1, one period later as t + 1, etc.

Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.

References

  1. Anosov, D. V.; Bolibruch, A. A. (1994). The Riemann-Hilbert problem. Braunschweig/Wiesbaden. p. 5. ISBN   978-3-322-92911-2.{{cite book}}: CS1 maint: location missing publisher (link)
  2. Plemelj, Josip (1964). Problems in the Sense of Riemann and Klein. Interscience Publishers. ISBN   978-0-470-69125-0.
  3. Bolibrukh, A. A. (February 1992). "On sufficient conditions for the positive solvability of the Riemann-Hilbert problem". Mathematical Notes. 51 (2): 110–117. doi:10.1007/BF02102113. S2CID   121743184.
  4. Schlesinger, Ludwig (1 January 1912). "Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten". Journal für die reine und angewandte Mathematik. 1912 (141): 96–145. doi:10.1515/crll.1912.141.96. S2CID   120990400.
  5. Jimbo, Michio; Miwa, Tetsuji; Ueno, Kimio (1981-04-01). "Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function". Physica D: Nonlinear Phenomena. 2 (2): 306–352. Bibcode:1981PhyD....2..306J. doi:10.1016/0167-2789(81)90013-0. ISSN   0167-2789.
  6. Biquard, Olivier; Boalch, Philip (January 2004). "Wild non-abelian Hodge theory on curves". Compositio Mathematica. 140 (1): 179–204. arXiv: math/0111098 . doi:10.1112/S0010437X03000010. ISSN   0010-437X. S2CID   119682616.
  7. Jimbo, Michio; Miwa, Tetsuji; Môri, Yasuko; Sato, Mikio (April 1980). "Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent". Physica D: Nonlinear Phenomena. 1 (1): 80–158. Bibcode:1980PhyD....1...80J. doi:10.1016/0167-2789(80)90006-8 . Retrieved 7 May 2023.

Sources