Analytic theory of differential equations

$\def\eqref#1{(#1)}$ The branch of the theory of ordinary differential equations in which the solutions are studied from the point of view of the theory of analytic functions. A typical formulation of a problem in the analytic theory of differential equations is this: Given a certain class of differential equations, the solutions of which are all analytic functions of one variable, find the specific properties of the analytic functions that are solutions of this class of equations. In this wide sense, the analytic theory of differential equations includes the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — Bessel functions, Airy functions, Legendre functions, Laguerre functions, Hermite functions (cf. Hermite function), Chebyshev functions (cf. Chebyshev function), Whittaker functions, Weber functions (cf. Weber function), Mathieu functions, hypergeometric functions (cf. Hypergeometric function), Sonin functions and many other functions — are solutions of linear differential equations with analytic coefficients.

Linear theory.

Consider a system of $n$ equations in matrix notation:

$$ \dot{x} = A(t) x + f(t) . \tag{1} $$

1) Let the matrices $A(t)$, $f(t)$ be holomorphic in a region $G \subset \mathbf{C}(t)$, where $\mathbf{C}(t)$ is the complex $t$-plane. Any solution of the system \eqref{1} will then be analytic in $G$ (but will not, in general, be single-valued if $G$ is not simply-connected). It is assumed that $A(t)$ is meromorphic in $G$, and one considers the homogeneous system

$$ \dot{x} = A(t) x . \tag{2} $$

(The matrix $A(t)$ is called holomorphic (meromorphic) in $G$ if all its elements are holomorphic (meromorphic) in $G$.) A point $t_0\in G$ is called a pole of the matrix $A(t)$ of order $\nu\ge 1$ if, in a given neighbourhood of this point,

$$ A(t) = A_{-\nu} (t-t_0)^{-\nu} + \cdots + A_{-1} (t-t_0)^{-1} + B(t) , $$

where $A_{-j}$ are constant matrices, $A_{-\nu}\ne 0$, and the matrix $B(t)$ is holomorphic at $t_0$. A pole $t_0\ne \infty$ of order $\nu$ is called a regular singular point if $\nu=1$ and an irregular singular point if $\nu\ge 1$. The case $t_0 = \infty$ is reduced to the case $t_0=0$ by the change of variables $t\mapsto t^{-1}$. In what follows, $t_0 \ne 0$.

2) Let $t_0$ be a pole of $A(t)$. Then there exists a fundamental matrix $X(t)$ for the system \eqref{2} of the form

$$ X(t) = \Phi(t) (t-t_0)^D \tag{3} $$

where $D$ is a constant matrix, $\Phi(t)$ is holomorphic for $|t-t_0|<\rho$ when $t_0$ is a regular singular point, and $\Phi(t)$ is holomorphic for $0<|t-t_0|<\rho$ when $t_0$ is an irregular singular point, for some $\rho > 0$. (Here, $(t-t_0)^D = \exp(D\ln(t-t_0))$, by definition.) For a regular singular point the matrix $D$ can be expressed in terms of $A(t)$ in an explicit form [1], [2]; this is not the case for irregular singular points.

A similar classification of singular points is introduced for differential equations of order $n$ with meromorphic coefficients. Differential equations and systems of differential equations with only regular singular points are known as Fuchsian (systems of) differential equations. The general form of $A(t)$ for such a system is:

$$ A(t) = \sum_{j=1}^{k} (t-t_j)^{-1} A_j, \quad A_j = \text{const}, \quad k < \infty . $$

An example of a Fuchsian differential equation is the hypergeometric equation.

3) Let $A(t) = t^q B(t)$ where $q\ge 0$ is an integer and $B(t)$ is holomorphic at $t=\infty$ ($\infty$ is an irregular singular point if $B(\infty) \ne 0$). If $S$ is a sufficiently narrow sector of the form $|t| > R$, $\alpha < \arg t < \beta$, then there exists a fundamental matrix of the form

$$ X(t) = P(t) t^Q \exp(R(t)) , \tag{4} $$

where $Q$ is a constant matrix, $R(t)$ is a diagonal matrix whose elements are polynomials in $t^{1/p}$, $p\ge 1$ is an integer and

$$ P(t) \sim \sum_{j=0}^\infty P_j t^{-j/p} $$

as $|t| \to \infty$, $t\in S$. The plane $\mathbf{C}(t)$ is subdivided into a finite number of sectors, and in each of them there exists a fundamental matrix of the form \eqref{4} ([3], [4]; see also [1], [2]).

4) As a result of analytic continuation along a closed path $\gamma$ the fundamental matrix $X(t)$ is multiplied by a constant matrix $B_\gamma$: $X(t) \mapsto X(t) B_\gamma$; one obtains the monodromy group of the differential equation. I.A. Lappo-Danilevskii [5] has studied the problem of Riemann: Let $A(t)$ be a rational function of $t$ and let the singularities of the fundamental matrix $X(t)$ be known, find $A(t)$.

5) Let the function $z=\phi(t)$ be a conformal mapping of the upper half-plane $\operatorname{Im} t > 0$ onto the interior of a polygon, the boundary of which consists of a finite number of segments of straight lines and circular arcs. The function $\phi(t)$ will then satisfy the Schwarz equation:

$$ \{ z , t \} \equiv \frac{z'''}{z'} - \frac{3}{2} \left( \frac{z''}{z'} \right)^2 = R(t) . \tag{5} $$

where $R(t)$ is a rational function, and the equation

$$ w'' + \frac{1}{2} R(t) w = 0 \tag{6} $$

is Fuchsian. Any solution of equation \eqref{5} may be represented in the form $z = w_1 / w_2$, where $w_1$ and $w_2$ are linearly independent solutions of equation \eqref{6}. Let $G$ be an infinite discrete group and let $\phi(t)$ be an automorphic function of $G$, then $\phi(t)$ can be represented as $\phi = w_1 / w_2$, where $w_1$, $w_2$ are linearly independent solutions of equation \eqref{6} and $R(t)$ is some algebraic function.

Non-linear theory.

1) Consider the Cauchy problem:

$$ \tag{7} \dot{x} \ = \ f ( t ,\ x ) , \ \ x ( t _{0} ) \ = \ x ^{0} , $$

where $ t \in \mathbf C (t),\ x = ( x _{1} \dots x _{n} ) \in \mathbf C ^{n} (x) $, $ f = ( f _{1} \dots f _{n} ) $.

Cauchy's theorem: Let the function $ f(t,\ x) $ be holomorphic in $ (t,\ x) $ in a region $ G \subset \mathbf C (t) \times \mathbf C ^{n} (x) $ and let the point $ (t _{0} ,\ x ^{0} ) \in G $. Then there exists a $ \delta > 0 $ such that in the domain $ | t - t _{0} | < \delta $ there exists a solution $ x (t; \ t _{0} ,\ x ^{0} ) $ of the Cauchy problem (7), which is unique and holomorphic.

An analytic continuation of the solution $ x(t; \ t _{0} ,\ x ^{0} ) $ will also be a solution of the system (7), but the function obtained as a result of the continuation may have singularities and, in the general case, is a many-valued function of $ t $. The problems which arise are: What singularities may this function have and how can one construct the general solution? In the linear case these questions have been conclusively answered. In the non-linear case the situation is much more complicated and has not been fully clarified even when the $ f _{j} (t,\ x) $ are rational functions of $ t,\ x $.

2) Consider the differential equation:

$$ \tag{8} \frac{dx}{dt} \ = \ \frac{P ( t ,\ x )}{Q ( t ,\ x )} , $$

where $ t \in \mathbf C ,\ x \in \mathbf C , $ and $ P $ and $ Q $ are holomorphic functions of $ (t,\ x) $ in a certain region $ G $. A point $ (t _{0} ,\ x _{0} ) $ is called an (essentially) singular point of equation (8) if $ P (t _{0} ,\ x _{0} ) =0 $, $ Q (t _{0} ,\ x _{0} ) = 0 $. Below the structure of the solutions in a neighbourhood of a singular point of the equation is clarified. Develop $ P $ and $ Q $ into Taylor series:

$$ P ( t ,\ x ) \ = \ a _{11} ( x - x _{0} ) + a _{12} ( t - t _{0} ) + \dots , $$

$$ Q ( t ,\ x ) \ = \ a _{21} ( x - x _{0} ) + a _{22} ( t - t _{0} ) + \dots , $$

and let $ \lambda _{1} ,\ \lambda _{2} $ be the eigen values of the matrix $ \| a _{ij} \| $. The following theorem holds. Let $ \lambda _{j} \neq 0 $ and let none of the numbers $ \lambda _{1} / \lambda _{2} ,\ \lambda _{2} / \lambda _{1} $ be either a non-negative integer or a real negative number. Then there exists a neighbourhood $ U $ of the point $ (t _{0} ,\ x _{0} ) $, a neighbourhood $ V $ of the point $ \widetilde{t} = 0 ,\ \widetilde{x} = 0 $, and functions $ \widetilde{t} = \widetilde{t} (t,\ x) $ and $ \widetilde{x} = \widetilde{x} (t,\ x) $ such that the mapping $ U \rightarrow V $ defined by these functions is biholomorphic, and the differential equation (8) in the new variables assumes the form [6]:

$$ \frac{d \widetilde{x} }{d \widetilde{t} } \ = \ \frac{\lambda _{1} \widetilde{x} }{\lambda _{2} \widetilde{t} } . $$

All solutions of equation (8) in the new variables are written in the form $ \widetilde{x} = C \widetilde{t} {} ^ {\lambda _{1} / \lambda _ 2} $ and $ \widetilde{x} \equiv 0 $. Thus, a singular point of the equation is a branching point of infinite order for all solutions of equation (8) (except for the trivial solutions). The singular points of the solution which coincide with the singular points of the equation are called stationary. As distinct from the linear case, the solution of a non-linear equation may have singular points not only at the singular points of the equation; such singular points of the solution are called movable. Painlevé's theorem is valid: The solutions of the equation

$$ P ( t ,\ x ,\ \dot{x} ) \ = \ 0 , $$

where $ P $ is a polynomial in $ x $ and $ \dot{x} $ with holomorphic coefficients in $ t $, has no movable transcendental singular points [7].

If, in equation (8), $ P $ and $ Q $ are polynomials in $ t,\ x $, then, in view of Painlevé's theorem, all movable singular points are algebraic. On substituting $ t = 1/ t ^ \prime $, $ x = x ^ \prime / t ^ \prime $, equation (8) assumes the form

$$ \frac{d x ^ \prime}{d t ^ \prime} \ = \ \frac{P _{1} ( t ^ \prime ,\ x ^ \prime )}{Q _{1} ( t ^ \prime ,\ x ^ \prime )} , $$

where $ P _{1} $ and $ Q _{1} $ are polynomials. Let $ x _{j} $ be the roots of the equation $ P _{1} (0,\ x ^ \prime ) = 0 $. The points $ (0,\ x _{j} ^ \prime ) $ are called infinitely-remote singular points of equation (8); the structure of the solutions in a neighbourhood of these points is described by the theorem quoted above [6].

Let $ P $ and $ Q $ be polynomials of degree $ n $. Since $ P,\ Q $ are defined by their coefficients and the pair $ ( \lambda P ,\ \lambda Q ) $ defines the same equation, one obtains a one-to-one correspondence between equations (8) and the points of the complex projective space $ \mathbf C P ^{N} $, $ N = (n + 1 ) (n - 2) - 1 $. The following theorem is valid: If some set of measure zero is removed from $ \mathbf C P ^{N} $, the remaining equations (8) will have the following property: All solutions $ x = x(t) $ are everywhere dense in $ \mathbf C ^{2} = \mathbf C (t) \times \mathbf C (x) $[8].

3) Consider the autonomous system

$$ \tag{9} \dot{x} \ = \ f ( x ) , $$

$ t \in \mathbf C (t) ,\ x \in \mathbf C ^{n} (x),\ f = (f _{1} \dots f _{n} ) $. A point $ x ^{0} $ will then be a singular point of the system (9) if $ f ( x ^{0} ) = 0 $. Poincaré's theorem is valid: Let $ x ^{0} $ be a singular point of the autonomous system (9). Also let a) the elementary divisors of the Jacobi matrix $ f ^ {\ \prime} ( x ^{0} ) $ be prime divisors; and b) the eigen values $ \lambda _{j} $ of this matrix lie on one side of some straight line in $ \mathbf C ( \lambda ) $ passing through the coordinate origin. Then there exists neighbourhoods $ U,\ V $ of the points $ x = x ^{0} ,\ \widetilde{x} = 0 $ and a biholomorphic mapping $ x = x ( \widetilde{x} ) : \ V \rightarrow U $ such that the system (9) expressed in the variable $ \widetilde{x} $ assumes the form [9]:

$$ \frac{d \widetilde{x} _ j}{dt} \ = \ \lambda _{j} \widetilde{x} _{j} , \ \ 1 \leq j \leq n . $$

If only condition a) is satisfied, it is possible, by using a transformation $ x = \phi ( \widetilde{x} ) $, where $ \phi ( \widetilde{x} ) $ is a formal power series, to convert system (9) in a neighbourhood of a singular point into a system which can be integrated in quadratures [9], [10]. However, the convergence of these series has been proved on assumptions close to a) and b). If the function $ f (x) $ and the transformation $ x = \phi ( \widetilde{x} ) $ are real for real $ x,\ \widetilde{x} $, a theorem similar to the theorem of Poincaré has been proved [11]. The structure of the solutions of the autonomous system (9) in general, where $ f _{j} (x) $ are polynomials and $ n \geq 3 $, has not yet (1970s) been studied.

References

[1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) MR0069338 Zbl 0064.33002
[2]	W. Wazov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[3]	G.D. Birkhoff, "Singular points of ordinary linear differential equations" Trans. Amer. Math. Soc. , 10 (1909) pp. 436–470 MR1500848 Zbl 46.0695.02 Zbl 45.0484.01 Zbl 44.0373.01
[4]	W.J. Trjitzinsky, "Analytic theory of linear differential equations" Acta Math. , 62 (1934) pp. 167–226 MR1555383 Zbl 60.1109.01
[5]	J.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953)
[6]	L. Bieberbach, "Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt" , Springer (1965) MR0176133 Zbl 0124.04603
[7]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119
[8]	M.G. Khudai-Verenov, "On a property of solutions of a differential equation" Mat. Sb. , 56 (98) : 3 (1962) pp. 301–308 (In Russian)
[9]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[10]	A.D. Bryuno, "Local methods in nonlinear differential equations" , Springer (1989) (Translated from Russian) MR0993771
[11]	C.L. Siegel, "Ueber die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung" Nachrichten Akad. Wissenschaft. Göttingen (1952) pp. 21–30 Zbl 0047.32901
[12]	H. Poincaré, , Oeuvres de H. Poincaré , 3 , Gauthier-Villars (1916–1965)
[13]	L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) MR1522111 Zbl 55.0810.04 Zbl 46.0621.01 Zbl 45.0693.07

Comments

The Riemann monodromy problem mentioned above is of great importance in the modern theory of completely integrable or soliton equations. Cf. Soliton.

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07
[a2]	E. Hille, "Ordinary differential equations in the complex domain" , Wiley (Interscience) (1976)