# Mokhonko A. A.

### On meromorphic solutions of the systems of linear differential equations with meromorphic coefficients

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1227-1240

UDC 517.925.7

For systems of linear differential equations whose dimension can be decreased, we establish estimates for the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of coefficients of the system.

### On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 139-144

Assume that the coefficients and solutions of the equation $f^{(n)}+p_{n−1}(z)f^{(n−1)} +...+ p_{s+1}(z)f^{(s+1)} +...+ p_0(z)f = 0$ have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients $p_j , j = s+1, . . . ,n−1$, increase slower (in terms of the Nevanlinna characteristics) than $p_s(z)$. It is proved that this equation has at most $s$ linearly independent solutions of finite order.

### Malmquist Theorem for the Solutions of Differential Equations in the Vicinity of a Branching Point

Mokhonko A. A., Mokhonko A. Z.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1286–1290

An analog of the Malmquist theorem on the growth of solutions of the differential equation $f' = P(z, f)/Q(z, f)$, where $P(z, f)$ and $Q(z, f)$ are polynomials in all variables, is proved for the case where the coefficients and solutions of this equation have a branching point in infinity (e.g., a logarithmic singularity).

### Deficiency Values for the Solutions of Differential Equations with Branching Point

Mokhonko A. A., Mokhonko A. Z.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 939–957

We study the distribution of values of the solutions of an algebraic differential equation *P*(*z, f, f′, . . . , f* ^{(s)}) = 0 with the property that its coefficients and solutions have a branching point at infinity (e.g., a logarithmic singularity). It is proved that if *a* ∈ ℂ is a deficiency value of *f* and *f* grows faster than the coefficients, then the following identity takes place: *P*(*z, a,* 0*, . . . ,* 0) ≡ 0*, z* ∈ {*z* : *r* _{0} ≤ *|z| <* ∞}*.* If *P*(*z, a,* 0*, . . . ,* 0) is not identically equal to zero in the collection of variables *z* and *a,* then only finitely many values of *a* can be deficiency values for the solutions *f* ∈ *M* _{ b } with finite order of growth.

### Nonisospectral flows on semiinfinite unitary block Jacobi matrices

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 521–544

It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately,
then the corresponding operator $\textbf{J}(t)$ satisfies the generalized
Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$,
where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix.

The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$.
It is mapped into the unitary operator of multiplication $L(t)$ in the isomorphic space $L^2({\mathbb T}, d\rho)$, where ${\mathbb T} = {z: |z| = 1}$.
This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation.
A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented.

The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices
(from self-adjoint case on ${\mathbb R}$)
and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory).

### On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 505–513

The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where *P*(*z, f*) and *Q*(*z, f*) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.

### On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 356–365

We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated.

### Malmquist theorem for solutions of differential equations in a neighborhood of a logarithmic singular point

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 476–483

The Malmquist theorem (1913) on the growth of meromorphic solutions of the differential equation *f ′ = P(z,f) / Q(z,f)*, where *P(z,f)* and *Q(z,f)* are polynomials in all variables, is proved for the case of meromorphic solutions with logarithmic singularity at infinity.