# Nizhnik L. P.

### Differential Equations with Bistable Nonlinearity

Nizhnik L. P., Samoilenko A. M.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 517-554

We study bounded solutions of differential equations with bistable nonlinearity by numerical and analytic methods. A simple mechanical model of circular pendulum with magnetic suspension in the upper equilibrium position is regarded as a bistable dynamical system simulating a supersensitive seismograph. We consider autonomous differential equations of the second and fourth orders with discontinuous piecewise linear and cubic nonlinearities. Bounded solutions with finitely many zeros, including solitonlike solutions with two zeros and kinklike solutions with several zeros are studied in detail. It is shown that, to within the sign and translation, the bounded solutions of the analyzed equations are uniquely determined by the integer numbers \( n=\left[\frac{d}{l}\right] \) where *d* is the distance between the roots of these solutions and *l* is a constant characterizing the intensity of nonlinearity. The existence of bounded chaotic solutions is established and the exact value of space entropy is found for periodic solutions.

### Spectral Analysis of Some Graphs with Infinite Rays

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1193–1204

We perform a detailed spectral analysis of countable graphs formed by joining semibounded infinite chains to vertices of a finite graph. The spectrum of a self-adjoint operator generated by the adjacency matrix of the graph is characterized, the spectral measure is constructed, the eigenvectors are presented in the explicit form, and the spectral expansion in eigenvectors is obtained.

### Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Myroslav L’vovych Horbachuk (on his 75 th birthday)

Berezansky Yu. M., Gerasimenko V. I., Khruslov E. Ya., Kochubei A. N., Mikhailets V. A., Nizhnik L. P., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 451-454

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Adamyan V. M., Berezansky Yu. M., Khruslov E. Ya., Kochubei A. N., Kuzhel' S. A., Marchenko V. O., Mikhailets V. A., Nizhnik L. P., Ptashnik B. I., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

### Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 723–743

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied in the scale of Hilbert spaces associated with $A$. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of $A$ by the same formula. As an application the one-dimensional Schrodinger operator with generalized zero-range potential is considered in the Sobolev space $W_2^p(R),\quad p \in N$.

### Evgen Yakovich Khruslov (on his 75 th birthday)

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Pastur L. A., Samoilenko A. M., Sharko V. V.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 549-550

### Yurij Makarovich Berezansky (the 80th anniversary of his birth)

Gorbachuk M. L., Gorbachuk V. I., Kondratiev Yu. G., Kostyuchenko A. G., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11

### Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 659–668

We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators.

### Approximation of general zero-range potentials

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 582-589

A norm resolvent convergence result is proved for approximations of general Schrodinger operators with zero-range potentials. An approximation of the δ’-interaction by nonlocal non-Hermitian potentials (without a renormalization of the coupling constant) is also constructed.

### Scattering problem for a multidimensional system of first-order partial differential equations

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1065–1076

We construct transformation operators, which enables us to study a scattering problem and investigate the properties of a scattering operator for a multidimensional system of first-order partial differential equations.

### On point interaction in quantum mechanics

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1557–1560

For the Schrödinger operator corresponding to the point interaction, a direct definition is given in terms of a singular perturbation.

### Stable difference scheme for a nonlinear Klein-Gordon equation

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 857–859

For a nonlinear Klein-Gordon equation, we obtain a stable difference scheme for large time intervals. We prove that this scheme has the sixth order of accuracy.

### Nonstationary inverse scattering problem for a system of second-order equations

Ukr. Mat. Zh. - 1982. - 34, № 6. - pp. 718—724

### Boundary-value problems for the heat equation with a time derivative in the matching conditions

Nizhnik L. P., Taraborkin L. A.

Ukr. Mat. Zh. - 1982. - 34, № 1. - pp. 120-126

### Conditional stability of a nonstationary inverse scattering problem

Nizhnik L. P., Romanenko R. V.

Ukr. Mat. Zh. - 1981. - 33, № 5. - pp. 694—696

### Reverse scattering problem for a transport equation discrete with respect to directions

Ukr. Mat. Zh. - 1980. - 32, № 5. - pp. 678–683

### The inverse scattering problem on a half-line with a non-seleconjugate potential matrix

Ukr. Mat. Zh. - 1974. - 26, № 4. - pp. 469–486

### Inverse nonstationary scattering problem for the Dirac equation

Ukr. Mat. Zh. - 1972. - 24, № 1. - pp. 110–114

### Analytic continuation of the resolvent of a self-adjoint operator across its continuous spectrum

Dyuzhenkova L. I., Nizhnik L. P.

Ukr. Mat. Zh. - 1968. - 20, № 6. - pp. 759–765

### Correct problem for the wave equation without initial data

Ukr. Mat. Zh. - 1968. - 20, № 6. - pp. 802–813

### Spectral structure and self - conjugation of perturbations of differential operators with constant coefficients

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 385-399

If $P(D) + \sum^k_{j=1}c_j(x)Q_j(D)$ is a formally self-conjugate differential expression with sufficiently smooth and rapidly decreasing toward infinity coefficients $c_j(x)$ there exist absolutely summable throughout the space of derivatives $c_j(x)$ to the order $\max \left\{n + 1, q_j\right\}$, where $n$ is the dimension of the space, and $q_j$ is the degree of the polynomial $Q_j(\xi)$, and $$\lim_{[\eta]\rightarrow\infty}\int_{|\xi - \eta|\leq1}\frac{\sum^k_{j=1}|Q_j(\xi)|^2}{1 + |P(\xi)|^2}$$ the closure in $L_2(E^n)$ of the differential operator, determined on a class of infinitely differentiable finite functions $c^{\infty}_0$ by means of the differential exit pression $P(D) + \sum^k_{j=1}c_j(x) Q_j(D)$ is a self-conjugate operator, the limiting spectrum of which coincides with the set of values of the polynomial $P(\xi)$.

### Infinitesimal bends of piecewise regular surfaces of rotation of negative curvature

Ukr. Mat. Zh. - 1962. - 14, № 4. - pp. 426-432

### Problem of Dispersion for a Schrodinger Equation

Ukr. Mat. Zh. - 1960. - 12, № 2. - pp. 209 - 212