Preamble

DIFFERENTIAL EQUATIONS

APRIL 1967, VOLUME III, NO. 4

ON REPRESENTATIONS OF SOLUTIONS FOR SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS IN THE NEIGHBORHOOD OF AN IRREGULAR SINGULAR POINT

Consider the system of linear differential equations
$$ \frac{dy}{dx} = A(x)y \tag{0} $$
where $x$ is a complex variable, $y$ is an $n$-dimensional vector, and $A(x)$ is an $n \times n$ matrix whose elements are analytic in the neighborhood of the point $x = \infty$ and can be represented by convergent series
$$ A(x) = \sum_{k=0}^{\infty} A_k x^{-k} \tag{1} $$
The point $x = \infty$ is, in general, an irregular singular point for system $(0)$.

As is well known, if the eigenvalues of the matrix $A_0$ are distinct, then system $(0)$ has a formal fundamental solution of the form
$$ Y(x) = P(x) x^W \exp(Q(x)) \tag{2} $$
where $Q(x)$ is a diagonal matrix whose elements are polynomials in $x$, $W$ is a constant diagonal matrix, and $P(x)$ is a formal power series in $x^{-1}$.

In the general case, when the eigenvalues of $A_0$ are not necessarily distinct, the structure of the formal solution becomes more complex. Specifically, it may involve fractional powers of $x$. The classical theory developed by Hukuhara, Turrittin, and others establishes that there exists a formal transformation $y = T(x)z$ that reduces the system to a simpler form, often referred to as the "canonical form."

The primary objective of this paper is to investigate the analytical representation of solutions to system $(0)$ in sectors of the complex plane centered at the irregular singular point. We aim to provide a rigorous derivation of the asymptotic behavior of these solutions and to establish the conditions under which the formal series $P(x)$ represents an actual solution in the sense of asymptotic expansions.

[FIGURE:1]

We shall assume that the matrix $A(x)$ satisfies certain regularity conditions in a sector $S$. Under these assumptions, we demonstrate that for any formal solution $(2)$, there exists a true solution $y(x)$ of the system $(0)$ such that $y(x) \approx \Phi(x)$.

§ 1. Integral Representations

$$\phi(x, f) = \int \sin(at) \psi(t) dt \tag{1.1}$$

The integral exists if $\phi(t) \to 0$ monotonically as $t \to \infty$. Let $\phi'(t) \to 0$ as $t \to \infty$ also monotonically, and $|\phi'(t)/\phi(t)| \to 0$ as $t \to \infty$. Then, from the expression for $\cos(at)\phi(t)$, we can identify the principal term of this function, namely:

$$\cos(at) + o(\phi'/\phi) \Phi(t), \tag{1.2}$$

where $o(\phi'/\phi)$ is an infinitesimal as $t \to \infty$ of at least the order of $\phi'/\phi$ or higher. Let us now consider $\sin(a + \rho \ln t)$. Let $a + \rho \ln t = \tau$. Since $d\tau/dt = \rho/t > 0$ as $t \to \infty$, the function $\tau = \tau(t)$ is increasing. Here, the functions $\phi(\tau)$ and $\psi(\tau)$ decrease as $\tau \to \infty$; therefore, the integral exists and, according to (1.2), we have:
$$\sin(a + \rho \ln t) \tag{1.4}$$
It is necessary to use the estimates provided by Fikhtengolts \cite{2}. Here, $o(\gamma(t))$ is a scalar function of the order of smallness of $\gamma(t)$ as $t \to \infty$, and $O(\Gamma(t))$ is a matrix with elements of the form $O(\gamma(t))$.

Let us consider the case where the integral exists in $H$. The expression can be represented as $p \ln(f)(t-i)$. To obtain $s^*(\alpha + \beta \ln t)$, we have:
$$\begin{aligned} s^*(\alpha + \beta \ln t) \end{aligned} \tag{1.6}$$
We will obtain analogous results if the term $\cos(\alpha + \beta \ln t)$ is used under the integral sign instead of $e^{i(\alpha + \beta \ln t)}$:

$$\Phi_{\alpha, \beta}(t) = \int \cos(\alpha + \beta \ln t) \frac{dt}{t}$$

$$\sin (at + \rho \ln t) + o(t^{-1}) t^{-1}, \quad a > 0 \tag{1.4'}$$

$$\Phi_3(t) = \cos(at + \rho \ln t)$$
$$\Phi_3(t) = \sin(at + \rho \ln t) + o(1)$$
$$\Phi_4(t) = \cos(at + \rho \ln t), \quad t > 0$$

V. V. Khoroshilov investigates a system of two equations of the form
$$ \dot{x} = (P_0 + P_1 t^{-1} + P_2 t^{-2} + \dots)x \tag{2.1} $$
where $P_i$ $(i = 0, 1, \dots)$ are constant second-order matrices. We shall first focus on the case where
$$ P_0 = \begin{pmatrix} ia & 0 \\ 0 & -ia \end{pmatrix} $$
where $a > 0$ is a real number. V. V. Khoroshilov seeks the solution to system (2.1) in the following form:
$$ X = \exp \left( \int P_0(t) dt \right) Z(t) \tag{2.3} $$
$$ Z(t) = I + \sum_{i=1}^{\infty} Z_i(t) \tag{2.5} $$
The terms are determined by:
$$ \exp \left( - \int \frac{1}{f} \, df \right) \, dt \tag{2.7} $$
$$ Z_2(t) = \int P_1(t) \exp \left( -\int P_0 dt \right) dt \tag{2.8} $$

Khoroshilov proved that $4P \to 0$ as $l \to \infty$. Regarding the small elements of the matrix, we shall show that one can indeed obtain $Z_{ui} = 0$ by appropriately choosing the limits of integration. In accordance with formula (2.8), Khoroshilov assumes that if $(p_{jj}) < 1$, then the integral in (2.6) can be replaced. In the case where $p = 1$, Khoroshilov’s approach would yield:

$$ P_{ii}^* - P_{22} = P + P_{r-} \tag{2.9} $$
$$ \exp(i(2at + \rho \ln t)) (1 + O(t^{-1})) $$
$$ \exp(-(2at + \rho \ln t)i) t^{-\rho} (1 + o(t^{-1})) $$
$$ P_{ij}^*(t) = p_{ij} t^{-2} + o(t^{-3}) $$

We shall now write formulas (2.6) and (2.7):
$$ Z_1(t) = \int \dots, \quad Z_2(t) = \int \dots $$
$$ \hat{Q} = \exp(-r_{12}) \int \exp(r_{12}) \rho_{12}(t) dt \tag{2.10} $$
Based on the preceding formulas, we establish (2.10) for the case where $-2 < 0$.

§ 2. Asymptotic Estimates

We have:
$$ z_{t}(t) = \exp\left( -(2\alpha t + \rho \ln t)i \right) \left( 1 + o\left( \frac{1}{t} \right) \right) $$
$$ \int_{0}^{t} (t - \tau) \exp(2\alpha \tau) \rho_{12} d\tau $$
These are the equations found here:
$$ \left( E \frac{\partial}{\partial t} - A \right) y(t) = \rho_{12}(t) $$

We demonstrate that the first term is a small quantity. Let us consider the quantity $\phi(t) = \int [\cos (2at + \rho \ln t) + i \sin (2at + \rho \ln t)] dt$. Based on formulas (1.4') and (1.4), we have:
$$ Z_l(t) = \dots + o(t^{-1}) \tag{2.13} $$

Case 2: $\rho - 2 = 0$. In this case, we can write:
$$ I^{(0)}(t) = \exp(-(2at + \rho \ln t)i) \int g (1 + o(t^{-1})) dt \tag{2.14} $$
According to the previous formulas, the second and third terms are of the order $o(t^{-1})$. We obtain the first term according to formulas (1.5) and (1.5'). Finally, we obtain:
$$ z^{(2)}(t) = \exp(-(2at + \rho \ln t)i) \int t^n [\cos(2at + \rho \ln t) + \sin(2at + \rho \ln t)] dt + o(t^{-1}) $$
$$ = \exp(-(2at + \rho \ln t)i) t^{-2\rho} [ \sin (2 \alpha t + \rho \ln t) - t \cos (2 \alpha t + \rho \ln t) ] + O(t^{-k}) \tag{2.15} $$

Case 3: $\rho - 2 > 0$. Then we have:
$$ \Gamma(t) = \exp \left( -(2\alpha t + \rho \ln t) \Gamma_p (1 + \alpha t^{-1}) \right) \tag{2.16} $$

This equation describes a functional form where the exponent is governed by a combination of linear and logarithmic time components. The $\rho \ln t$ term suggests a power-law correction to the primary exponential decay $2\alpha t$. The term $(1 + \alpha t^{-1})$ indicates sensitivity to short-term fluctuations that diminishes as $t \to \infty$.

$$ \Gamma(t) = \Gamma_0 \exp \left( - (2\alpha t + \beta \ln t) \right) $$
$$ \left[ \cos \left( \frac{2a}{l} + \rho \ln t \right) + i \sin \left( \frac{2a}{l} + \rho + o(t^{-n}) \right) \right] $$
$$ = \exp\left(-(2\alpha t + \rho \ln t)i\right) \Gamma \tag{2.17} $$

From this, we obtain:
$$ z_{ij}(t) = o(t^{-2}) \tag{2.19} $$

When determining $z^i(t)$, we follow an analogous line of reasoning, substituting $a = -a$ and $\bar{\rho} = -\rho$. Consequently, we obtain:
$$ z^i(t) = i \cdot t^{-k} \tag{2.20} $$
This allows us to express the solution to (2.3) in the form:
$$ X = \exp \left( \int P_0(t) dt \right) + O(t^{-k}) \tag{2.21} $$

§ 3. Series Convergence and Matrix Estimates

In our method of estimation, let $\rho = 3 < 3$, where $l > 1$ is an integer. Then, from (2.7), we have:
$$ \int_{0}^{t} P_{21}(t) \Phi_{12}(t) \, dt $$
Since $z_{\nu}(t) = \exp((2at + \eta)i) + o(t^{-k})$, we obtain:
$$ |z(\zeta, l; t)| < N \tag{3.2} $$
$$ z(t) \exp(-r) \int \exp(r(\zeta)) \, d\zeta < C t^{3/2} + O(t^{1/2}) \tag{3.3} $$

The elements are defined by the formula $\langle b + \langle n \rangle \rangle (N_0 - U)$. Evaluating the quantity (3.6), we obtain:
$$ \frac{|4-\rho|(6-\rho) \dots (2\nu-\rho)}{5 \cdot 7 \dots (2\nu+1)} \tag{3.10} $$

Using this approach, it is possible to easily estimate the magnitude of the parameters. The method provides a streamlined framework for calculating values without requiring intensive computational overhead.

The elements are defined by the formula $J_f(t) = \int_0^t \exp\left( -\int_{t'}^t \chi p(\tau) z_i^2(\tau) d\tau \right) dt'$ (3.12). By rounding the estimates, we obtain:
$$ (3k+2m) (b^k-x-2) \dots (3k+2m) \tag{3.15} $$
Alternative estimates can also be obtained:
$$ (aqyi - (3.16)) \frac{\rho + \dots}{5 \cdot 7 \dots (2l+1)} \tag{3.16} $$

The series converges absolutely and uniformly. It is straightforward to estimate the remainder of the series if we retain the first $N$ terms. Using the estimates (3.13) and (3.14), we obtain:
$$ |r_n(t)| < M \frac{(At)^n}{n!} \exp(At) $$
$$ | Z_{21} | \leq \frac{(\rho+4)(\rho+6) \dots (\rho+20)}{5 \cdot 7 \dots (2l+1)} (s+2l+2) \tag{3.26} $$

When all elements of the series (2.5) are determined according to our method, all series converge at the rate of an exponential function. Our estimates are significantly simpler than those in Khoroshilov's work, where some series converge only as a geometric progression.

§ 4. System Analysis

Consider the equation:
$$ Q(0) = [a + bi, a - bi] \tag{4.1} $$
Let us denote $X(t) = V(t)$. For $V(t)$, we obtain:
$$ \rho_{22} = \pi(2)\Lambda(2) + I - 2b $$
$$ \rho_{21} = \dots, \quad \rho_{12} = \dots \tag{4.4} $$

Let us introduce a new unknown matrix $V(t)$. This equation takes the form of equation (2.1). Based on the established theoretical framework, we can analyze the properties of this expression.

[FIGURE:1]

The time evolution of the operator $V(t)$ can be expressed as:
$$ V(t) = \exp \left( J \left[ \kappa + \left( \langle H \rangle - \frac{i}{2} \gamma \right) t - \frac{1}{2} \int_0^t \alpha(t - \tau) d\tau \right] \right) $$
The term $\langle H \rangle$ represents the expectation value of the Hamiltonian. The term involving $\gamma$ accounts for decay or decoherence rates. The integral component suggests memory effects or non-Markovian interaction.

The relationship between the variables $n$, $D$, and $Y$ suggests a constraint:
$$ n(D Y n(2) < 2K $$
The expression $(2) - (1)$ indicates a step-by-step derivation where the difference between two states is calculated to isolate a specific rate of change.

§ 5. Transformation and Independent Variables

Following Khoroshilov, we introduce a new independent variable $t$ and a new unknown matrix $X(t)$ using the transformation formula:
$$ X(t) = \dots \tag{5.2} $$
Then we obtain:
$$ A = \dots -2y \dots \tag{5.3} $$
Let $a > 0$. We obtain formula (4.6) with $b = -2a, a = 0$.

The solution to equation (5.1) can be obtained as:
$$ \Phi(l) = \exp \left[ \int_{l_0}^{l} \frac{1}{n} \, dl \right] \tag{5.6} $$
The elements are found using formulas (5.4), (4.2), and (5.5).

§ 6. Application to the Bessel Equation

Consider the Bessel equation (6.1), which, after the substitution $y - t$, transforms into equation (6.2). We write this in matrix form:
$$ \dot{y} = A y \tag{6.3} $$
Next, we perform the substitution:
$$ V = X S \tag{6.4} $$
Thus, $a = 1$, $\rho = 0$, and $\beta = 0$. According to formula (2.21), we have $X(t) = \exp [ (b - b_0) t ]$. By applying formula (6.4), we obtain:
$$ \exp \left( \frac{6.7}{t} \right) \left[ \cos \left( t + \frac{\dots}{t} \right) + i \sin \left( t + \frac{\dots}{t} \right) \right] + O\left( \frac{1}{t^2} \right) $$
We obtain two linearly independent real-valued solutions to equation (6.1):
$$ y_1 = \cos[t + \dots], \quad y_2 = \sin[t + \dots] \tag{6.8} $$

The series in positive powers of $x = t$ converge uniformly. By applying the estimation methods detailed in § 1, we obtain:
$$ z_{kj}(t) = -i \dots o(t^{-1}) \tag{6.9} $$
$$ \frac{7n^4 + 9n^3 \sqrt{b}}{24} \tag{6.10} $$

The primary governing equation for this transition is given by:
$$ \frac{2I - I}{9 \cdot (-K_{27\pi}) + 1} \gg K \left( -f(6.11) + 1 \right)! $$
This expression characterizes the boundary conditions necessary for the stability of the model.

$$ \begin{aligned} & \frac{\Gamma(k + 9/2)}{\Gamma(2k)} \left( k + \frac{9}{2n} \right) \tag{6.14} \\ & \Gamma(n = k + 1; j + 1) \tag{6.15} \\ & I < k \left( 3 + \frac{\sqrt{2}}{k} \right) \Gamma(k, m=1, \dots) \tag{6.16} \end{aligned} $$

Substituting (6.19) into (6.8), we obtain the final estimates (6.20).

References

1. Khoroshilov, V. V. PMM (Applied Mathematics and Mechanics), 15, 37–54, 1951.
2. Fikhtengolts, G. M. Course of Differential and Integral Calculus, Vol. II, Fizmatgiz, 1959, p. 568.